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Statistics By Jim

Making statistics intuitive

Interpreting Correlation Coefficients

By Jim Frost 145 Comments

What are Correlation Coefficients?

Correlation coefficients measure the strength of the relationship between two variables. A correlation between variables indicates that as one variable changes in value, the other variable tends to change in a specific direction.  Understanding that relationship is useful because we can use the value of one variable to predict the value of the other variable. For example, height and weight are correlated—as height increases, weight also tends to increase. Consequently, if we observe an individual who is unusually tall, we can predict that his weight is also above the average.

In statistics , correlation coefficients are a quantitative assessment that measures both the direction and the strength of this tendency to vary together. There are different types of correlation coefficients that you can use for different kinds of data . In this post, I cover the most common type of correlation—Pearson’s correlation coefficient.

Before we get into the numbers, let’s graph some data first so we can understand the concept behind what we are measuring.

Graph Your Data to Find Correlations

Scatterplots are a great way to check quickly for correlation between pairs of continuous data. The scatterplot below displays the height and weight of pre-teenage girls. Each dot on the graph represents an individual girl and her combination of height and weight. These data are actual data that I collected during an experiment.

At a glance, you can see that there is a correlation between height and weight. As height increases, weight also tends to increase. However, it’s not a perfect relationship. If you look at a specific height, say 1.5 meters, you can see that there is a range of weights associated with it. You can also find short people who weigh more than taller people. However, the general tendency that height and weight increase together is unquestionably present—a correlation exists.

Pearson’s correlation coefficient takes all of the data points on this graph and represents them as a single number. In this case, the statistical output below indicates that the Pearson’s correlation coefficient is 0.694.

What do the Pearson correlation coefficient and p-value mean? We’ll interpret the output soon. First, let’s look at a range of possible correlation coefficients so we can understand how our height and weight example fits in.

Related posts : Using Excel to Calculate Correlation and Guide to Scatterplots

How to Interpret Pearson Correlation Coefficients

Pearson’s correlation coefficient is represented by the Greek letter rho ( ρ ) for the population parameter and r for a sample statistic. This correlation coefficient is a single number that measures both the strength and direction of the linear relationship between two continuous variables. Values can range from -1 to +1.

The greater the absolute value of the Pearson correlation coefficient, the stronger the relationship.

• The extreme values of -1 and 1 indicate a perfectly linear relationship where a change in one variable is accompanied by a perfectly consistent change in the other. For these relationships, all of the data points fall on a line. In practice, you won’t see either type of perfect relationship.
• A coefficient of zero represents no linear relationship. As one variable increases, there is no tendency in the other variable to either increase or decrease.
• When the value is in-between 0 and +1/-1, there is a relationship, but the points don’t all fall on a line. As r approaches -1 or 1, the strength of the relationship increases and the data points tend to fall closer to a line.

The sign of the Pearson correlation coefficient represents the direction of the relationship.

• Positive coefficients indicate that when the value of one variable increases, the value of the other variable also tends to increase. Positive relationships produce an upward slope on a scatterplot.
• Negative coefficients represent cases when the value of one variable increases, the value of the other variable tends to decrease. Negative relationships produce a downward slope.

Statisticians consider Pearson’s correlation coefficients to be a standardized effect size because they indicate the strength of the relationship between variables using unitless values that fall within a standardized range of -1 to +1. Effect sizes help you understand how important the findings are in a practical sense. To learn more about unstandardized and standardized effect sizes, read my post about Effect Sizes in Statistics .

Learn how to calculate correlation in my post, Correlation Coefficient Formula Walkthrough .

Covariance is an unstandardized form of correlation. Learn about it in my posts:

• Covariance: Definition, Formula & Example
• Covariances vs Correlation: Understanding the Differences

Examples of Positive and Negative Correlation Coefficients

A positive correlation example is the relationship between the speed of a wind turbine and the amount of energy it produces. As the turbine speed increases, electricity production also increases.

A negative correlation example is the relationship between outdoor temperature and heating costs. As the temperature increases, heating costs decrease.

Graphs for Different Correlation Coefficients

Graphs always help bring concepts to life. The scatterplots below represent a spectrum of different Pearson correlation coefficients. I’ve held the horizontal and vertical scales of the scatterplots constant to allow for valid comparisons between them.

Discussion about the Scatterplots

For the scatterplots above, I created one positive correlation between the variables and one negative relationship between the variables. Then, I varied only the amount of dispersion between the data points and the line that defines the relationship. That process illustrates how correlation measures the strength of the relationship. The stronger the relationship, the closer the data points fall to the line. I didn’t include plots for weaker correlation coefficients that are closer to zero than 0.6 and -0.6 because they start to look like blobs of dots and it’s hard to see the relationship.

A common misinterpretation is assuming that negative Pearson correlation coefficients indicate that there is no relationship. After all, a negative correlation sounds suspiciously like no relationship. However, the scatterplots for the negative correlations display real relationships. For negative correlation coefficients, high values of one variable are associated with low values of another variable. For example, there is a negative correlation coefficient for school absences and grades. As the number of absences increases, the grades decrease.

Earlier I mentioned how crucial it is to graph your data to understand them better. However, a quantitative measurement of the relationship does have an advantage. Graphs are a great way to visualize the data, but the scaling can exaggerate or weaken the appearance of a correlation. Additionally, the automatic scaling in most statistical software tends to make all data look similar .

Fortunately, Pearson’s correlation coefficients are unaffected by scaling issues. Consequently, a statistical assessment is better for determining the precise strength of the relationship.

Graphs and the relevant statistical measures often work better in tandem.

Pearson’s Correlation Coefficients Measure Linear Relationship

Pearson’s correlation coefficients measure only linear relationships. Consequently, if your data contain a curvilinear relationship, the Pearson correlation coefficient will not detect it. For example, the correlation for the data in the scatterplot below is zero. However, there is a relationship between the two variables—it’s just not linear.

This example illustrates another reason to graph your data! Just because the coefficient is near zero, it doesn’t necessarily indicate that there is no relationship.

Spearman’s correlation is a nonparametric alternative to Pearson’s correlation coefficient. Use Spearman’s correlation for nonlinear, monotonic relationships and for ordinal data. For more information, read my post Spearman’s Correlation Explained !

Hypothesis Test for Correlation Coefficients

Correlation coefficients have a hypothesis test. As with any hypothesis test, this test takes sample data and evaluates two mutually exclusive statements about the population from which the sample was drawn. For Pearson correlations, the two hypotheses are the following:

• Null hypothesis: There is no linear relationship between the two variables. ρ = 0.
• Alternative hypothesis: There is a linear relationship between the two variables. ρ ≠ 0.

Correlation coefficients that equal zero indicate no linear relationship exists. If your p-value is less than your significance level , the sample contains sufficient evidence to reject the null hypothesis and conclude that the Pearson correlation coefficient does not equal zero. In other words, the sample data support the notion that the relationship exists in the population.

Related post : Overview of Hypothesis Tests

Interpreting our Height and Weight Correlation Example

Now that we have seen a range of positive and negative relationships, let’s see how our Pearson correlation coefficient of 0.694 fits in. We know that it’s a positive relationship. As height increases, weight tends to increase. Regarding the strength of the relationship, the graph shows that it’s not a very strong relationship where the data points tightly hug a line. However, it’s not an entirely amorphous blob with a very low correlation. It’s somewhere in between. That description matches our moderate correlation coefficient of 0.694.

For the hypothesis test, our p-value equals 0.000. This p-value is less than any reasonable significance level. Consequently, we can reject the null hypothesis and conclude that the relationship is statistically significant. The sample data support the notion that the relationship between height and weight exists in the population of preteen girls.

Correlation Does Not Imply Causation

I’m sure you’ve heard this expression before, and it is a crucial warning. Correlation between two variables indicates that changes in one variable are associated with changes in the other variable. However, correlation does not mean that the changes in one variable actually cause the changes in the other variable.

Sometimes it is clear that there is a causal relationship. For the height and weight data, it makes sense that adding more vertical structure to a body causes the total mass to increase. Or, increasing the wattage of lightbulbs causes the light output to increase.

However, in other cases, a causal relationship is not possible. For example, ice cream sales and shark attacks have a positive correlation coefficient. Clearly, selling more ice cream does not cause shark attacks (or vice versa). Instead, a third variable, outdoor temperatures, causes changes in the other two variables. Higher temperatures increase both sales of ice cream and the number of swimmers in the ocean, which creates the apparent relationship between ice cream sales and shark attacks.

Beware of spurious correlations!

In statistics, you typically need to perform a randomized, controlled experiment to determine that a relationship is causal rather than merely correlation. Conversely, Correlational Studies will find relationships quickly and easily but they are not suitable for establishing causality.

Learn more about Correlation vs. Causation: Understanding the Differences .

Related posts : Using Random Assignment in Experiments and Observational Studies

How Strong of a Correlation is Considered Good?

What is a good correlation? How high should correlation coefficients be? These are commonly asked questions. I have seen several schemes that attempt to classify correlations as strong, medium, and weak.

However, there is only one correct answer. A Pearson correlation coefficient should accurately reflect the strength of the relationship. Take a look at the correlation between the height and weight data, 0.694. It’s not a very strong relationship, but it accurately represents our data. An accurate representation is the best-case scenario for using a statistic to describe an entire dataset.

The strength of any relationship naturally depends on the specific pair of variables. Some research questions involve weaker relationships than other subject areas. Case in point, humans are hard to predict. Studies that assess relationships involving human behavior tend to have correlation coefficients weaker than +/- 0.6.

However, if you analyze two variables in a physical process, and have very precise measurements, you might expect correlations near +1 or -1. There is no one-size fits all best answer for how strong a relationship should be. The correct values for correlation coefficients depend on your study area.

Taking Correlation to the Next Level with Regression Analysis

Wouldn’t it be nice if instead of just describing the strength of the relationship between height and weight, we could define the relationship itself using an equation? Regression analysis does just that. That analysis finds the line and corresponding equation that provides the best fit to our dataset. We can use that equation to understand how much weight increases with each additional unit of height and to make predictions for specific heights. Read my post where I talk about the regression model for the height and weight data .

Regression analysis allows us to expand on correlation in other ways. If we have more variables that explain changes in weight, we can include them in the model and potentially improve our predictions. And, if the relationship is curved, we can still fit a regression model to the data.

Additionally, a form of the Pearson correlation coefficient shows up in regression analysis. R-squared is a primary measure of how well a regression model fits the data. This statistic represents the percentage of variation in one variable that other variables explain. For a pair of variables, R-squared is simply the square of the Pearson’s correlation coefficient. For example, squaring the height-weight correlation coefficient of 0.694 produces an R-squared of 0.482, or 48.2%. In other words, height explains about half the variability of weight in preteen girls.

If you’re learning about statistics and like the approach I use in my blog, check out my Introduction to Statistics book! It’s available at Amazon and other retailers.

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Reader Interactions

August 17, 2024 at 2:43 pm

Great, thank you!

August 15, 2024 at 9:33 am

Hi Jim. I had a query. Like if we say there is a correlation of 0.68 between x and y variable, then what exactly does this “0.68” as a “number” indicate apart from the fact that we can say there is a moderate association between x and y.

May 7, 2024 at 9:18 am

Is there any benefit to doing both a correlation and a regression test? I don’t think there is – I believe that a regression output will give you the same information a correlation output would plus more. Please could you let me know if that is correct or am I missing something?

May 7, 2024 at 2:08 pm

Hi Charlotte,

In general, you are correct for simple regression, where you have one independent variable and the dependent variable. The R-square for that model is literally the square of the Pearson’s correlation (r) for those two variables. As you mention, regression gives you additional output along with the strength of the relationship.

But there are a few caveats.

Regression is much more flexible than correlation because it allows you to add other variables, fit curvature and include interaction effects. For example, regression allows you to fit curvature between the two variables using polynomials. So, there are cases where using Pearson’s correlation is inappropriate because the data violate some of the assumptions but regression analysis can handle those data acceptably.

But what you say is correct when you’re looking at a straight line relationship between a pair of variables. In that specific case, simple regression and Pearson’s correlation provide consistent information with regression providing more details.

March 12, 2024 at 4:11 am

Hi If you are finding the trend between one type of quantitative discrete data and one type of qualitative ordinal data, what correlation test do you use?

September 9, 2023 at 4:46 am

It could be that the sharks are using ice cream as bait. Maybe the sharks are smarter than we think… Seriously, the ice cream as a cause is not likely, but sometimes a perfectly sensible hypothesis with lots of data behind it can be just plain wrong.

September 9, 2023 at 11:43 pm

It can be wrong in causal sense but if ice cream cones has a non-causal correlation with the number of shark attacks, it can still help you make predictions. Now, if you thought limiting ice cream sales will reduce shark attacks, that’s not going to work!

June 9, 2023 at 1:56 am

What is to be done when two positive items show a negative correlation within one variable.. e.g increase in house help decreases no interruptions in work?? It’s confusing as both r positive questions

June 10, 2023 at 1:09 am

It’s possibly the result of other variables, known as confounding variables (or confounders) that you might not even have recorded. For example, there might be some other variable that correlates with both “house help” and “interruptions at work” that explain the unexpected negative correlation. Perhaps individuals with house help have more activities occurring throughout the day at home. Those activities would then cause more interruptions. So, you might have chain of correlations where the “home activities” and “house help” have positive correlations. Additionally, “home activities” and “interruptions” might have a negative correlation. Given this arrangement, it wouldn’t be surprising to see a negative correlation between “home activities” and “interruptions.”

It goes to show that you need to understand the larger context when analyzing data. Technically, this phenomenon is known as omitted variable bias . Your model (pairwise correlation) omits an important variable (a confounder) which is biasing the results. Click the link to learn more.

The answer is to identify and record the confounding variables and include them in your model, likely a regression model or partial correlation.

May 8, 2023 at 12:58 pm

What if my pearson’s r is 0.187 and p-value is 0.001 do i reject the null hypothesis?

May 8, 2023 at 2:56 pm

Yes! That p-value is below any reasonable significance level. Hence, you can reject the null hypothesis. However, be aware that while the correlation is statistically significant, it is so weak that it probably isn’t practically significant in the real world. In other words, it probably exists in the population you’re assessing but it is too weak to be noticeable/meaningful.

November 30, 2022 at 4:53 am

Thank you, Jim. I really appreciate your help. I will read your post about statistical v practical significance – that sounds really useful. I love how you explain things in such an accessible way.

I have one more question that I was hoping you would be able to help me with, please?

If I have done a correlation test and I have found an extremely weak negative relationship (e.g., -.02) but the relationship is not statistically significant, would this mean that although I have found that there is a very weak negative correlation between the variables in the sample data, this would unlikely to be found in the population. Therefore, I would fail to reject the null hypothesis that the correlation in the population equals zero.

Thank you again for your help and for this wonderful blog.

December 1, 2022 at 1:57 am

You’re very welcome!

In the case where the correlation is not significant, it indicates that you have insufficient evidence to conclude that it does not equal zero. That’s a mouthful but there’s a reason for the convoluted wording. Insignificant results don’t prove that there is no effect, it just indicates that your test didn’t detect an effect in the population. It could be that the effect doesn’t exist in the population OR it could be that your sample size was too small or there’s too much variability in the data.

In short, we say that you failed to reject the null hypothesis.

Basically, you can’t prove a negative (no effect). All you can say is that your study didn’t detect an effect. In this case, it didn’t detect a non-zero correlation.

You can read more about the reason behind the wording failing to reject the null hypothesis and what it means precisely.

November 29, 2022 at 12:39 pm

Thank you for this webpage. It is great. I have a question, which I was hoping you’d be able to help me with please.

I have carried out a correlation test, and from my understanding a null hypothesis would be that there is no relationship between the two variables (the variables are independent – there is no correlation).

The p value is statistically significant (.000), and the Pearson correlation result is -.036.

My understanding is that if there is a statically significant relationship then I would reject the null hypothesis (which suggests there is no relationship between the two variables). My issue is then whether -.036 suggests a very weak relationship or no relationship at all given how close to 0 it is. If it is the latter, would I then say I have failed to reject the null hypothesis even though there is a statisicially significant relationship? Or would I say that I have rejected the null hypothesis because there is a statically significant relationship, but the correlation is very weak.

Any help would be appreciated. Kind regards.

November 29, 2022 at 4:10 pm

What you’re seeing is the difference between statistical significance and practically significance. Yes, your results are statistically significant. You can reject the null hypothesis that rho (the correlation in the population) does not equal zero. Your data provide enough evidence to conclude that the negative correlation exists in the population (not just your sample).

However, as you say, it’s an extremely weak relationship. Even though it’s not zero it is essentially zero in a practical sense. Statistically significant results don’t automatically mean that the effect size (correlation is this case) is meaningful in the real-world. When a test has very high statistical power (e.g., sometimes due to a very large sample size), it can detect trivial effects. Those effects are real but they’re small in size.

I write more about this in my post about statistical vs. practical significance . But, in a nutshell, your correlation coefficient is statistically significant, but it is not a meaningful effect in the real world.

September 28, 2022 at 10:44 am

I have a simple question, only to frame how to use correlation. Imagine a trial with plants, testing different phosphate (Pi) concentrations (like 8) and its effect on plant growth (assessed as mean plant size per Pi concentration, from enough replicates and data validity to perform classical parametric statistics).

In case A, I have a strong (positive) and significant Pearson correlation between these two parameters, and in particular, the 8 average size values show statistical significant differences (ANOVA) between all the Pi concentrations tested.

In case B, I have the same strong (positive) significant Pearson correlation, but there is no any statistical significant difference in term of size between any Pi concentration tested.

My guess is that it may be possible to interpret the case A as Pi is correlated with plant growth; but in case B, no interpretation can be provided given that no significant difference is seen between Pi concentrations on plant size, even if a correlation is obtained. Is this right ? But in this case, if I have 3 out the 8 Pi concentrations which I obtained significant difference on plant size, should I perform correlation only between significant Pi groups or could I still take all the 8 Pi groups to make interpretations ? Thanks in advance !

September 29, 2022 at 7:02 pm

I don’t fully understand your trial. You say that you have a continuous measure of Pi concentration and then average plant sizes. Pearson correlations work with two continuous measures–not a group average. So, you’d need to correlate the Pi concentration with plant size, not average plant size. Or perhaps I’m misunderstanding your description. Please clarify your process. Thanks!

In a more general sense, you have to remember that statistical significance doesn’t necessarily indicate there is a real-world, practical significance to your results. That’s possibly what you’re finding in case B. Although again it’s hard to say if you’re applying correlation to averages.

Statistical significance just indicates that you have reason to believe that a relationship/effect exists in the population. It doesn’t necessarily mean that the effect is large enough to be practically meaningful. For more information, read my post about Practical vs. Statistical Significance .

August 16, 2022 at 11:16 am

This was very educative and easy to follow through for a statistics noob such as me. Thanks! I like your books. Which one is most suited for a beginner level of knowledge?

August 17, 2022 at 12:20 am

My Introduction to Statistics book is the best to get started with for beginners. Click the link to see a post where I discuss it and included a full table of contents.

After reading that, you’d be ready to read both of my two other books: Hypothesis Testing Regression Analysis

May 16, 2022 at 2:45 pm

Jim, Nassim Taleb makes the point on YouTube (search for Taleb and correlation) that an r = 0.10 is much closer to zero than to r = 0.20) implying that the distribution function for r is very dependent on the r in the population, and the sample size and that the scale of -1.0 to +1.0 is not a scale separated by equal units. He then warns of significance tests because r is a random variable and subject to sampling fluctuations and r = .25 could easily be zero due to sampling error (especially for small sample sizes). Can you please discuss if the scale of r = -1.0 to 1.0 is set in equidistant units, or units that only superficially look like they are equidistant?

May 16, 2022 at 6:41 pm

I did a quick search and found a video where he’s talking about using correlation in the financial and investment areas. He seems to be saying that correlation is not the correct tool for that context. I can’t talk to that point because I’m not familiar with the context.

However, yes, I can help you out with most of the other points!

I’ll start with the fact that the scale of -1 to +1 is, in some ways, not consistent. To start, correlation coefficients are a standardized effect. As such, they are unitless. You can’t link them to anything real, but they help you compare between disparate types of studies. In other words, they excel at providing a standard basis of comparison between studies. However, they’re not as good for knowing what the statistic actually means, except for a few specific values, -1, +1, and 0. And perhaps that’s why Taleb isn’t fond of them. (At 20 minutes, I didn’t watch the entire video.)

However, we can convert r to R-squared and it becomes more meaningful. R-squared tells us how much of the variance the relationship accounts for. And, as the name implies, you simply square r to get R-squared. It’s in R-squared where you see that the difference between r of 0.1 and 0.2 is different from say 0.8 and 0.9. When you go from 0.1 to 0.2, R-squared increases from 0.01 to 0.04, an increase of 3%. And note that at those correlations, we’re only explaining between 1 – 4% of the variance. Virtually nothing! Now, if we look at going from an r of 0.8 to 0.9, R-squared increases from 0.64 to 0.81, or 17%. So, we have the same size increase in r (0.1) in both cases, but R-squared increases by 3% in one case and 17% in the other. Also, notice how at a r of 0.5, you’re only accounting for 25% of the variance. That’s not very much. You need an r of 0.707 to explain half the variance (50%). Another way to think of it is that the range of r [0, 0.7] accounts for half the variance while r [0.7, 1] accounts for the other half.

I agree with the point that r = 0.1 is virtually nothing. In fact, you need an r of 0.316 to explain even a tenth (10%) of the variability. I also agree that fixed differences in r (e.g., 0.1) indicates different changes in the strength of the relationship, as I illustrate above. I think those points are valid.

Below, I include a graph showing r vs. R-squared and the curved line indicates that the relationship between the two statistics changes (the inconsistency you mention). If the relationship was consistent, it would be a straight line. For me, R-squared is the better statistic, particularly in conjunction with regression analysis, which provides more information about the nature of the relationships. Of course, the negative range of r produces the mirror graph but the same ideas apply.

I think correlation coefficients (r) have some other shortcomings. They describe the strength of the relationship but not the actual relationship. And they don’t account for other variables. Regression analysis handles those aspects and I generally prefer that methodology. For me, simple correlation just doesn’t provide enough information by itself in most cases. You also typically don’t get residual plots so you can be sure that you’re satisfying the assumptions (Pearson’s correlation (r) is essentially a linear model).

The sample r does depend on the relationship in the population. But that’s true for all sample statistics–as I write in my post, Sample Statistics Are Always Wrong to Some Extent! I don’t think it’s any worse for correlation than other types of sample statistics. As you increase your sample size, the estimate’s precision will increase (i.e., the error bars become smaller).

I think significance tests are valid for correlation. Yes, it’s subject to sampling fluctuations ( sampling error ) but so are all sample based statistics. Hypothesis testing is designed to factor that in. In fact, significance testing specifically helps you distinguish between cases where the sample r = 0.25 might represent 0 in the population vs. cases where that is unlikely. That’s the very intention of significance testing, so I strongly disagree with that point!

April 9, 2022 at 2:20 am

Thank you for the fast response!! I have alaso read the Spearman’s Rho article (very insightful). In my scatterplot it is suggesting that there is no correlation (completely random distribution). However, I would still like to test the correlation but in the Spearmans’s Rho article you mentioned that if it is there is no correlation, both the spearman’s Rho value and Pearson’s correlation value would be close to zero. Is it also possible that one value is positive and one is negative? My results right now are R2 Linear= 0.003, Pearson correlation= .058, and Spearman’s correlation coefficient= -0.19. Should I base the rejection of either of my hypothesises on Spaerman’s value or Pearson’s value

Thank you so much!!!

April 9, 2022 at 10:42 pm

I’m glad that it was helpful! It’s definitely possible for correlations to switch directions like that. That’s especially true because both correlations are barely different from zero. So, it wouldn’t take much to cause them to be on opposite sides of zero. The R-squared is telling you that the Pearson’s correlation explains hardly any of the variability.

April 8, 2022 at 7:05 pm

Thank you for this post!! I was wondering, I did a scatterplot which gave me a R2 value of 0.003. The fitline showed a really weak positive correlation which I wanted to test with the Spearmans rho. However, this value is showing a negative value (negative relationship). Do you maybe know why it is showing different correlations since I am using the exact same values?

April 8, 2022 at 7:51 pm

The R-squared value and slope you’re seeing are related to Pearson’s correlation, which differs from Spearmans rho. They’re different statistical measures using different methods, so it’s not surprising that their values can be different. For more information, read my post about Spearman’s Rho .

April 6, 2022 at 3:37 am

Hi Jim, I had a question. It’s kinda complicated but I try my best to explain it well.

I run a correlation test between objective social isolation and subjective social isolation. To measure OSI, I used an instrument called LSNS-6, while I used R-UCLA Loneliness Scale to measure the SSI. Here is the scoring guide for the instruments: * higher score obtained in LSNS-6 = low objective social isolation * higher score obtained in R-UCLA Loneliness scale = high subjective social isolation

After I run the correlation test, I found the value was r= -.437.

My question is, did the value represents correlation between variables (meaning when someone is objectively isolated, they are less likely to be subjectively isolated and vice versa) OR the value represents correlation between scores of instruments used (meaning when someone score higher in LSNS-6, they will had a lower scores for R-UCLA Loneliness Scale and vice versa)? I had confusions due to the scoring guide. I hope you can help me.

Thank you Jim!

April 8, 2022 at 8:17 pm

This specific correlation is a bit tricky because, based on what you wrote, the LSNS-6 is inverted. High LSNS-6 scores correspond to low objective social isolation. Let’s work through this example.

The negative correlation (-0.437) indicates that high LSNS-6 scores tend to correlate with low R-UCLA scores. Now, if we “translate” the instrument measures into what the scores mean as constructs, low objective social isolation tends to correspond low subjective social isolation.

In other words, there is a negative correlation between the instrument scores. However, there is a positive correlation between the concepts of objective social isolation and subjective isolation, which makes theoretical sense.

The reason why the instrument scores have a negative correlation and the constructs having a positive correlation goes back to the fact that high LSNs-6 scores relate to low objective isolation.

I hope that helps!

April 2, 2022 at 7:16 am

Thanks so much for the highly helpful statistical resources on this website. I am a bit confused about an analysis I carried out. My scatter plot show a kind of negative relationship between two variables but my Pearson’s correlation coefficient results tend to say something different. r= -0.198 and p-value of 0.082. I would appreciate clarification on this.

April 4, 2022 at 3:56 pm

I’m not sure what is surprising you? Can you be more specific?

It sounds like your scatterplot displays a negative correlation and your negative correlation is also negative, which sounds consistent. It’s a fairly weak correlation. The p-value indicates that your data don’t provide quite enough evidence to conclude that the correlation you see in the sample via the scatterplot and correlation coefficient also exists in the population. It might just be sampling error.

January 14, 2022 at 8:31 am

Hi Jim, Andrew here.

I am using a Pearson test for two variables: LifeSatisfaction and JobSatisfaction. I have gotten a P-Value 0.000 whilst my R-Value is 0.338. Can you explain to me what relation this is? Am I right in thinking that is strong significance with a weak correlation? And that there is no significant correlation between the two.

January 14, 2022 at 4:59 pm

What you’re running in to is the difference between statistical significance and practical significance in the real world. A statistically significant results, such as your correlation, suggests that the relationship you observe in your sample also exists in the population as a whole. However, statistical significance says nothing about how important that relationship is in a practical sense.

Your correlation results suggest that a positive correlation exists between life satisfaction and job satisfaction amongst the population from which you drew your sample. However, the fairly weak correlation of 0.338 might not be of practical significant. People with satisfying jobs might be a little happier but perhaps not to a noticeable degree.

So, for your correlation, statistical significance–yes! Practical significant–maybe not.

For more information, read my post about statistical significance vs. practical significance where I go into it in more detail.

January 7, 2022 at 7:07 pm

Thank you, Jim, will do.

January 7, 2022 at 5:07 pm

Hello Jim, I just came across this website. I have a query.

I wrote the following for a report: Table 5 shows the associations between all the domains. The correlation coefficients between the environment and the economy, social, and culture domains are rs=0.335 (weak), rs=0.427 (low) and rs=0.374 (weak), respectively. The correlation coefficient between the economy and the social and culture domains are rs=0.224 and rs=0.157, respectively and are negligible. The correlation coefficient (rs =0.451) between the social and the culture domains is low, positive, and significant. These weak to low correlation coefficient values imply that changes in one domain are not correlated strongly with changes in the related domain.

The comment I received was: Correlation studies are meant to see relationships- not influence- even if there is a positive correlation between x and y, one can never conclude if x or y is the reason for such correlation. It can never determine which variables have the most influence. Thus the caution and need to re-word for some of the lines above. A correlation study also does not take into account any extraneous variables that might influence the correlation outcome.

I am not sure how I should reword? I have checked several sources and their interpretations are similar to mine, Please advise. Thank you

January 7, 2022 at 9:25 pm

Personally, I think your wording is fine. Appropriately, you don’t suggest that correlation implies causation. You state that there is correlation. So, I’m not sure why the reviewer has an issue with it.

Perhaps the reviewer wants an explicit statement to that effect? “As with all correlation studies, these correlations do not necessarily represent causal relationships.”

The second portion of the review comment about extraneous variables is, in my opinion, more relevant. Pairwise correlations don’t control for the effects of other variables. Omitted variable bias can affect these pairs. I write about this in a post about omitted variable bias . These biases can exaggerate or minimize the apparent strength of pairwise correlations.

You can avoid that problem by using partial correlations or multiple regression analysis. Although, it’s not necessarily a problem. It’s just a possibility.

January 5, 2022 at 8:52 pm

Is it possible to compare two correlation coefficients? For example, let’s say that I have three data points (A, B, and C) for each of 75 subjects. If I run a Pearson’s on the A&B survey points and receive a result of .006, while the Pearson’s on the A&C survey points is .215…although both are not significant, can I say that there is a stronger correlation between A&C than between A&B? thank you!

January 6, 2022 at 8:31 pm

I am not aware of test that will assess whether the difference between two correlation coefficients is statistically significant. I know you can do that with regression coefficients , so you might want to determine whether you can use that approach. Click the link to learn more.

However, I can guess that your two coefficients probably are not significantly different and thus you can’t say one is higher. Each of your hypothesis tests are assessing whether one of the coefficients is significantly different from zero. In both cases (0.006 and 0.215), neither are significantly different from zero. Because both of your coefficients are on the same side of zero (positive) the distance between them is even smaller than your larger coefficients (0.215) distance from zero. Hence, that difference probably is also not statistically significant. However, one muddling issue is that with the two datasets combined you have a larger total sample size than either alone, which might allow a supposed combined test to determine that the smaller difference is significant. But that’s uncertain and probably unlikely.

There’s a more fundamental issue to consider beyond statistical significance . . . practical significance. The correlation of 0.006 is so small it might as well be zero. The other is 0.215 (which according to the hypothesis test, also might as well be zero). However, in practical terms, a correlation of 0.215 is also a very weak correlation. So, even if its hypothesis test said it was statistically significant from zero, it’s a puny correlation that doesn’t provide much predictive power at all. So, you’re looking at the difference between two practically insignificant correlations. Even if the larger sample size for a combined test did indicate the difference is statistically significant, that difference (0.215 – 0.006 = 0.209) almost certainly is not practically significant in a real-world sense.

But, if you really want to know the statistical answer, look into the regression method.

May 16, 2022 at 2:57 pm

JIm – here is a YT purporting to demonstrate how to compare correlation coefficients for statistical significance. I’m not a statistician and cannot vouch for the contents. https://www.youtube.com/watch?v=ipqUoAN2m4g

May 16, 2022 at 7:22 pm

That seems like a very non-standard approach in the YT video. And, with a sample size of 200 (100 males, 100 females), even very small effect sizes should be significant. So, I have some doubts about that process, but I haven’t dug into it. It might be totally valid, but it seems inefficient in terms of statistical power for the sample size.

Here’s how I would’ve done that analysis. Instead of correlation, I’d use regression with an interaction effect. I’d want to model the relationship between the amount time studying for a test and the scores. Additionally, I also gather 100 males and females and want to see if the relationship between time studying and test scores differs between genders. In regression, that’s an interaction effect. It’s the same question the YT video assesses, but using a different approach that provides a whole lot more answers.

To see that approach in action, read my post about Comparing Regression Lines Using Hypothesis Tests . In that post, I refer to comparing the relationships between two conditions, A and B. You can equate those two conditions to gender (male and female). And I look at the relationship between Input and Output, which you can equate to Time Studying and Test Score, respectively. While reading that post, notice how much more information you obtain using that approach than just the two correlation coefficients and whether they’re significantly different.

That’s what I mean by generally preferring regression analysis over simple correlation.

December 9, 2021 at 7:33 pm

salut Jim merci beaucoup pour cette explication je travaille sur un article et je veux calculer la taille d’echantillon pour critiquer la taille d’echantillon utulisé est ce que c posiible de deduire le P par le graphqiue et puis appliquer la regle pour d”duire N ?

December 12, 2021 at 11:57 pm

Unfortunately, I don’t speak French. However, I used Google Translate and I think I understand your question.

No, you can’t calculate the p-value by looking at a graph. You need the actual data values to do that. However, there is another approach you can use to determine whether they have a reasonable sample size.

You can use power and sample size software (such as the free G*Power ) to determine a good sample size. Keep in mind that the sample size you need depends on the strength of the correlation in the population. If the population has a correlation of 0.3, then you’ll need 67 data points to obtain a statistical power of 0.8. However, if the population correlation is higher, the required sample size declines while maintaining the statistical power of 0.8. For instance, for population correlations of 0.5 and 0.8, you’ll only need sample sizes of 23 and 8, respectively.

Using this approach, you’ll at least be able to determine whether they’re using a reasonable sample size given the size of correlation that they report even though you won’t know the p-value.

Hopefully, the reported the sample size, but, if not, you can just count the number of dots on the scatterplot.

November 19, 2021 at 4:47 pm

Hi Jim. How do I interpret r(12) = -.792, p < .001 for Pearson Coefficiient Correlation?

October 26, 2021 at 4:53 am

Hi If the correlation between the two independent constructs/variables and the dependent variable/constructs is medium or large, what must the manager to improve the two independent constructs/variables

October 7, 2021 at 1:12 am

Hi Jim, First of all thank you, this is an excellent resource and has really helped clarify some queries I had. I have run a Pearson’s r test on some stats software to analyse relationship between increasing age and need for friendship. The return is r = 0.052 and p = 0.381. Am I right in assuming there is a very slight positive correlation between the variables but one that is not statistically significant so the null hypothesis cannot be rejected? Kind regards

October 7, 2021 at 11:26 pm

Hi Victoria,

That correlation is so close to 0 that it essentially means that there is no relationship between your two variables. In fact, it’s so close to zero that calling it a very slight positive correlation might be exaggerating by a bit.

As for the p-value, you’re correct. It’s testing the null hypothesis that the correlation equals zero. Because your p-value is greater than any reasonable significance level, you fail to reject the null. Your data provide insufficient evidence to conclude that the correlation doesn’t equal zero (no effect).

If you haven’t, you should graph your data in a scatterplot. Perhaps there’s a U shaped relationship that Pearson’s won’t detect?

July 21, 2021 at 11:23 pm

No Jim, I mean to ask, let’s assume correlation between variable x and y is 0.91, how do we interpret the remaining 0.09 assuming correlation at 1 is strong positive linear correlation. ?

Is this because of diversification, correlation residual or any error term?

July 21, 2021 at 11:29 pm

Oh, ok. Basically, you’re asking why it’s not a perfect correlation of 1? What explains that difference of 0.09 between the observed correlation and 1? There are several reasons. The typical reason is that most relationships aren’t perfect. There’s usually a certain amount of inherent uncertainty between two variables. It’s the nature of the relationship. Occasionally, you might find very near perfect correlations for relationships governed by physical laws.

If you were to have pair of variables that should have a perfect correlation for theoretical reasons, you might still observe an imperfect correlation thanks to measurement error.

July 20, 2021 at 12:49 pm

If two variable has a correlation of 0.91 what is 0.09, in the equation?

July 21, 2021 at 10:59 pm

I’d need more information/context to be able to answer that question. Is it a regression coefficient?

June 30, 2021 at 4:21 pm

You are a great resource. Thank you for being so responsive. I’m sure I’ll be bugging you some more in the future.

June 30, 2021 at 12:48 pm

Jim, using Excel, I just calculated that the correlation between two variables (A and B) is .57, which I believe you would consider to be “moderate.” My question is, how can I translate that correlation into a statement that predicts what would happen to B if A goes up by 1 point. Thanks in advance for your help and most especially for your clarity.

June 30, 2021 at 2:59 pm

Hi Gerry, to get that type of information, you’ll need use regression analysis. Read my post about using Excel to perform regression for details . For your example, be sure to use A as the independent variable and B as the dependent variable. Then look at the regression coefficient for A to get your answer!

May 24, 2021 at 11:51 pm

Hey Man, I’m taking my stats final this week and I’m so glad I found you! Thank you for saving random college kids like me!

May 19, 2021 at 8:38 am

Hi, I am Nasib Zaman The Spearman correlation between high temperature and COVID-19 cases was significant ( r = 0.393). Correlation between UV index and COVID-19 cases was also significant ( r = 0.386). Is it true?

May 20, 2021 at 1:31 am

Both suggests that as temperature and UV increase that the number of COVID cases increases. Although it is a weak correlation. I don’t know whether that’s true or not. You’d have to assess the validity of the data to make that determination. Additionally, their might be confounding variables at play, which could bias the correlations. I have no way of knowing.

April 12, 2021 at 1:49 pm

I am using Pearson’s correlation co-efficient to to express the strength of relationship between my two variables on happiness, would this be an appropriate use?

Happiness Diet RelationshipSatisfaction

Pearson Correlation

Happiness 1.000 .310 . 416 Diet .310 1.000 .193 RelationshipSatisfaction .416 .193 1.000

Sig. (1-tailed) 0.00 0.00 Happiness Diet 0.00 0.00 RelationshipSatisfaction 0.00 0.00

N Happiness 1297 1297 1297 Diet 1297 1297 1297 RelationshipSatisfaction 1297 1297 1297

If so, would I be right to say that because the coefficient was r= (.193), it suggests that there is not too strong a relationship between the two independent variables. Can I use anything else to indicate significance levels?

March 29, 2021 at 3:12 am

I just want to say that your posts are great, but the QA section in the comments is even greater!

Congrats, Jim.

March 29, 2021 at 2:57 pm

Thanks so much!! 🙂

And, I’m really glad you enjoy the QA in the comments. I always request readers to post their questions in the comments section of the relevant post so the answers benefit everyone!

March 24, 2021 at 1:16 am

Thank you very much. This question was troubling me since last some days , thanks for helping.

Have a nice day…

March 24, 2021 at 1:34 am

You’re very welcome, Ronak! I’m glad to help!

March 22, 2021 at 12:56 pm

Nalin here. I found your article to be very clarifying conceptually. I had a doubt.

So there is this dataset I have been working on and I calculated the Pearson correlation coefficient between the target variable and the predictor variables. I found out that none of the predictor variables had a correlation >0.1 and <-0.1 with the target variable, hence indicating that no linear relationship exists between them.

How can I verify whether or not any non-linear relationships exist between these pairs of variables or not? Will a scatterplot confirm my claims?

March 23, 2021 at 3:09 pm

Yes, graphing the data in a scatterplot is always a good idea. While you might not have a linear relationship, you could have a curvilinear relationship. A scatterplot would reveal that.

One other thing to watch out for is omitted variable bias. When you perform correlation on a pair of variables, you’re not factoring in other relevant variables that can be confounding the results. To see what I mean, read my post about omitted variable bias . In it, I start with a correlation that appear to be zero even though there actually is a relationship. After I accounted for another variable, there was a significant relationship between the original pair of variables! Just another thing to watch out for that isn’t obvious!

March 20, 2021 at 3:23 am

Yes, I am also doing well…

I am having some subsequent queries…

By overall trend you mean that correlation coefficient will capture how y is changing with respect to x (means y is increasing or decreasing with increase or decrease in x), am i interpreting correctly ?

March 22, 2021 at 12:25 am

This is something should be clear by examining the scatterplot. Will a straight line fit the dots? Do the dots fall randomly about a straight line or are there patterns? If a straight line fits the data, Pearson’s correlation is valid. However, if it does not, then Pearson’s is not valid. Graphing is the best way to make the determination.

Thanks for the image.

March 23, 2021 at 3:41 pm

Hi again Ronak!

On your graph, the data points are the red line (actually lots and lots of data points and not really a line!). And, the green line is the linear fit. You don’t usually think of Pearson’s correlation as modeling the data but it uses a linear fit. So, the green line is how Pearson’s correlation models your data. You can see that the model doesn’t fit the data adequately. There are systematic (i.e., non-random departures) from the data points. Right there you know that Pearson’s correlation is invalid for these data.

Your data has an upward trend. That is, as X increases, Y also increases. And Pearson’s partially captures that trend. Hence, the positive slope for the green line and the positive correlation you calculated. But, it’s not perfect. You need a better model! In terms of correlation, the graph displays a monotonic relationship and Spearman’s correlation would be a good candidate. Or, you could use regression analysis and include a polynomial to model the curvature . Either of these methods will produce a better fit and more accurate results!

March 18, 2021 at 11:01 am

i am ronak from india. how are you?…hoping corona has not troubled you much. you have simplified concept very well. you are doing amazing job ,great work. i have one doubt and want to clarify it.

Question : whenever we talk correlation coefficient we talk in terms of linear relationship. but i have calculated correlation coefficient for relationship Y vs X^3.

X variable : 1 to 10000 Y = X^3

and correlation coefficient is coming around 0.9165. it is strange even relationship is not linear still it is giving me very high correlation coefficient.

March 19, 2021 at 3:53 pm

I’m doing well here. Just hunkering down like everyone else! I hope you’re doing well too! 🙂

For your data, I’d recommend graphing them in a scatterplot and fit a linear trend line. You can do that in Excel. If your data follow an S-shaped cubic relationship, it is still possible to get a relatively strong correlation. You’ll be able to see how that happens in the scatterplot with trend line. There’s an overall trend to the data that your line follows, but it does hug the curves. However, if you fit a model with a cubic term to fit the curves, you’ll get a better model.

So, let’s switch from a correlation to R-squared. Your correlation of 0.9165 corresponds to an R-squared of 0.84. I’m literally squaring your correlation coefficient to get the R-squared value. Now, fit a regression model with the quadratic and cubic terms to fit your data. You’ll find that your R-squared for this model is higher than for the linear model.

In short, the linear correlation is capturing the overall trend in the data but doesn’t fit the data points as well as the model designed for curvilinear data. Your correlation seems good but it doesn’t fully fit the data.

March 11, 2021 at 10:56 am

Hi Jim Do the partial correlation include the continuous (scale) variables all times? Is it possible to include other types of variables (as nominal or ordinal)? Regards Jagar

March 16, 2021 at 12:30 am

Pearson correlations are for continuous data that follow a linear relationship. If you have ordinal data or continuous data that follow a monotonic relationship, you can use Spearman’s correlation.

There are correlations specifically for nominal data. I need to write a blog post about those!

March 10, 2021 at 11:45 am

if the correlation coefficient is 0.153 what type of correlation is it?

February 14, 2021 at 1:49 pm

February 12, 2021 at 8:09 pm

If my r value when finding correlation between two things is -0.0258 what would that be negative weak correlation or something else?

February 14, 2021 at 12:08 am

Hi Dez, your correlation coefficient is essentially zero, which indicates no relationship between the variables. As one variable increases, there is no tendency for the variable to either increase or decrease. There’s just no relationship between them according to your data.

January 9, 2021 at 12:10 pm

my coefficient correlation between my independent variables (anger, anxiety, happiness, satisfaction) and a dependent variable(entrepreneurial decision making behavior) is 0.401, 0.303, 0.369, 0.384.

what does this mean? how do i interpret explain this? what’s the relationship?

January 10, 2021 at 1:33 am

It means that separately each independent variable (IV) has a positive correlation with the dependent variable (DV). As each IV increases, the DV tends to increase. However, it is a fairly weak correlation. Additionally, these correlations don’t control for confounding variables. You should perform a regression analysis because you have your IVs and DV. Your model will tell how much variability the IVs account for in the DV collectively. And, it will control for the other variables in the model, which can help reduce omitted variable bias.

The information in this post should help you interpret your correlation coefficients. Just read through it carefully.

January 4, 2021 at 6:20 am

Hello there, If one were to find out the correlation between the average grade and a variable, could this coefficient be used? Thanks!

January 4, 2021 at 4:03 pm

If you mean something like an average grade per student and the other variable is something like the number of hours each student studies, yes, that’s fine. You just need to be sure that the average grade applies to one person and that the other variable applies to the same person. You can’t use a class average and then the other variable is for individuals.

December 27, 2020 at 8:27 am

I’m helping a friend working on a paper and don’t have the variables. The question centers around the nature of Criterion Referenced Tests, in general, i.e. correlations of CRT vs. Norm Referenced Tests. As you know, Norm Referenced compares students to each other across a wide population. In this paper, the student is creating a teacher made CRT. It is measuring proficiency of students of more similar abilities and smaller population to criteria and not to each other. I suspect, in general, the CRT doesn’t distinguish as well between students with similar abilities and knowledge. Therefore, the reliability coefficients, in general, are less reliable. How does this effect high or low correlations?

December 26, 2020 at 9:40 pm

high or lower correlation on a CRT proficiency test good or bad?

December 27, 2020 at 1:30 am

Hi Raymond, I’d have to know more about the variables to have an idea about what the correlation means.

December 8, 2020 at 11:02 pm

I have zero statistics experience but I want to spice up a paper that I’m writing with some quants. And so learned the basics about Pearson correlation on SPSS and I plugged in my data. Now, here’s where it gets “interesting.” Two sets of numbers show up: One on the Pearson Correlation row and below that is the Sig. (2-tailed) row.

I’m too embarrassed to ask folks around me (because I should already know this!). So, let me ask you: which of the row of numbers should I use in my analysis about the correlations between two variables? For example, my independent variable correlates with the dependent variable at -.002 on the first (Pearson Correlation) row. But below that is the Sig. (2-tailed) .995. What does that mean? And is it necessary to have both numbers?

I would really appreciate your response … and will acknowledge you (if the paper gets published).

Many thanks from an old-school qualitative researcher struggling in the times of quants! 🙂

December 9, 2020 at 12:32 am

The one you want to use for a measure of association is the Pearson Correlation. The other value is the p-value. The p-value is for a hypothesis test that determines whether your correlation value is significantly different from zero (no correlation).

If we take your -0.002 correlation and it’s p-value (0.995), we’d interpret that as meaning that your sample contains insufficient evidence to conclude that the population correlation is not zero. Given how close the correlation is to zero, that’s not surprising! Zero correlation indicates there is no tendency for one variable to either increase or decrease as the other variable increases. In other words, there is no relationship between them.

November 24, 2020 at 7:55 am

Thank you for the good explanation. I am looking for the source or an article that states that most correlations regarding human behaviour are around .6. What source did you use?

Kind regards, Amy

November 13, 2020 at 5:27 am

This is an informative article and I agree with most of what is said, but this particular sentence might be misleading to readers: “R-squared is a primary measure of how well a regression model fits the data.”. R-squared is in fact based on the assumption that the regression model fits the data to a reasonable extent therefore it cannot also simultaneously be a measure of the goodness of said fit.

The rest of the claims regarding R-squared I completely agree with.

Cheers, Georgi

November 13, 2020 at 2:48 pm

Yes, I make that exact point repeatedly throughout multiple blog posts, particularly my post about R-squared .

Additionally, R-squared is a goodness-of-fit measure, so it is not misleading to say that it measures how well the model fits the data. Yes, it is not a 100% informative measure by itself. You’d also need to assess residual plots in conjunction with the R-squared. Again, that’s a point that I make repeatedly.

I don’t mind disagreements, but I do ask that before disagreeing, you read what I write about a topic to understand what I’m saying. In this case, you would’ve found in my various topics about R-squared and residual plots that we’re saying the same thing.

November 7, 2020 at 12:31 pm

Thank you very much!

November 6, 2020 at 7:34 pm

Hi Jim, I have a question for you – and thank you in advance for responding to it 🙂

Set A has the correlation coefficient of .25 and Set B has the correlation of .9, Which set has the steeper trend line? A or B?

November 6, 2020 at 8:41 pm

Set B has a stronger relationship. However, that’s not quite equivalent to saying it has a steeper trend line. It means the data points fall closer to the line.

If you look at the examples in this post, you’ll notice that all the positive correlations have roughly equal slopes despite having different correlations. Instead, you see the points moving closer to the line as the strength of the relationship increases. The only exception is that a correlation of zero has a slope of zero.

The point being that you can’t tell from the correlation alone which trend line is steeper. However, the relationship in Set B is much stronger than the relationship in Set A.

October 19, 2020 at 6:33 am

Thank you 😊. Now I understand.

October 11, 2020 at 4:49 am

hi, I’m a little confused.

What does it indicating, If there is positive correlation, but negative coefficient from multiple regression outcome? in this situation, how to interpret? the relationship is negative or positive?

October 13, 2020 at 1:32 pm

This is likely a case of omitted variable bias. A pairwise correlation involves just two variables. Multiple regression analysis involves three variables at a minimum (2 IVs and a DV). Correlation doesn’t control for other variables while regression analysis controls for the other variables in the model. That can explain the different relationships. Omitted variable bias occurs under specific conditions. Click the link to read about when it occurs. I include an example where I first look at a pair of variables and then three variables and shows how that changes the results, similar to your example.

September 30, 2020 at 4:26 pm

Hi Jim, I have 4 objective in my research and when I did the correlation between first one and others the result is: ob1 with ob2 is (0.87) – ob1 with ob3 is (0.84) – ob1 with ob4 is ( 0.83). My question is what is that meaning and can I do Correlation Coefficient with all of them in one time.

September 28, 2020 at 4:06 pm

Which best describes the correlation coefficient for r=.08?

September 30, 2020 at 4:29 pm

Hi Jolette,

I’d say that is an extremely weak correlation. I’d want to see its p-value. If it’s not significant, then you can’t conclude that the correlation is different from zero (no correlation). Is there something else particular you want to know about it?

September 15, 2020 at 11:50 am

Correlation result between Vul and FCV

t = 3.4535, df = 306, p-value = 0.0006314 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: 0.08373962 0.29897226 sample estimates: cor 0.1936854

What does this mean?

September 17, 2020 at 2:53 am

Hi Lakshmi,

It means that your correlation coefficient is ~0.19. That’s the sample estimate. However, because you’re working with a sample, there’s always sample error and so the population correlation is probably not exactly equal to the sample value. The confidence interval indications that you can be 95% confident that the true population correlation falls between ~0.08 and 0.30. The p-value is less than any common significance level. Consequently, you can reject the null hypothesis that the population correlation equals zero and conclude that it does not equal zero. In other words, the correlation you see in the sample is likely to exist in the population.

A correlation of 0.19 is a fairly weak relationship. However, even though it is weak, you have enough evidence to conclude that it exists in the population.

September 1, 2020 at 8:16 am

Hi Jim Thank you for your support. I have a question that is. Testing criteria for Validity by Pearson correlation, r table determine by formula DF=N-2 – If it is Valid the correlation value less that Pearson correlation value. (Pearson correlation > r table ) – if it is Invalid the correlation value greater that Pearson correlation value. (Pearson correlation < r table ) I got the above information on SPSS tutorial Video about Pearson correlation.

but I didn't get on other literature please can you recommend me some literature that refers about this? or can you clarify more about how to check Validity by Pearson correlation?

August 31, 2020 at 3:21 am

HI JIM i am zia from pakistan i wanna finding correlation of two factoer i have find 144.6 of 66.93 thats is postive relation?

August 31, 2020 at 12:39 pm

Hi Zia, I’m sorry but I’m not clear about what you’re asking. Correlation coefficients range between -1 and +1, so those two values are not correlation coefficients. Are they regression coefficients?

August 16, 2020 at 6:47 am

Warmest greetings.

My name is Norshidah Nordin and I am very grateful if you could provide me some answers to the following questions.

1) Can I used two different set of samples (for e.g. students academic performance (CGPA) as dependent variable and teacher’s self efficacy as dependent variable) to run on a Pearson correlation analysis. If yes, could you elaborate on this aspect.

2) what is the minimum sample size to use in multiple regression analysis.

August 17, 2020 at 9:06 pm

Hi Norshidah,

For correlations, you need to have multiple measurements on the same item or person. In your scenario, it sounds like you’re taking different measurements on different people. Pearson’s correlation would not be appropriate.

The minimum sample size for multiple regression depends on the number of terms you need to include in your model. Read my post about overfitting regression models , which occurs when you have too few observations for the number of model terms.

I hope this helps!

July 29, 2020 at 5:27 pm

Greetings sir, question…. Can you do an accurate regression with a Pearson’s correlation coefficient of 0.10? Why or Why not?

July 31, 2020 at 5:33 pm

Hi Monique,

It is possible. First, you should determine whether that correlation is statistically significant. You’re seeing a correlation in your sample, but you want to be confident that is also exists in the large population you’re studying. There’s a possibility that the correlation only exists in your sample by random chance and does not exist in the population–particularly with such a low coefficient. So, check the p-value for the coefficient. If it’s significant, you have reason to proceed with the regression analysis. Additionally, graph your data. Pearson’s only is for linear relationships. Perhaps your coefficient is low because the relationship is curved?

You can fit the regression model to your data. A correlation of 0.10 equates to an R-squared of only 0.01, which is very low. Perhaps adding more independent variables will increase the R-squared. Even if the r-squared stays very low, if your independent variable is significant, you’re still learning something from your regression model. To understand what you can learn in this situation, read my post about regression models with significant variables and a low R-squared values .

So, it is possible to do a valid regression and learn useful information even when the correlation is so low. But, you need to check for significance along the way.

July 8, 2020 at 4:55 am

Hello Jim, first and foremost thank you for giving us a comprehensive information regarding this! This totally help me. But I have a question; my pearson results showing that there’s a moderate positive relationship between my variables which is Parasocial Interaction and the fans’ purchase intention.

But the thing is, if I look at the answer majority of my participants are mostly answering Neutral regarding purchase intention.

What does this means? could you help me to figure out this T.T thanks you in advance! I’m a student currently doing thesis from Malaysia.

July 8, 2020 at 4:00 pm

Hi Titania,

Have you graphed your data using a scatterplot? I’d highly recommend that because I think it will probably clarify what your data are telling you. Also, are both of your variables continuous variables? I’m wonder if purchase intention is ordinal if one of the values is Neutral. If that’s the case, you’d need to use Spearman’s Rank Correlation rather than Pearson’s.

June 18, 2020 at 8:57 am

Hello Jim ! I have a question . I calculated a correlation coefficient between the scale variables and got 0.36, which is relatively weak since it gives a 0.12 if quared. What does the interpretation of correlation concern ? The sample taken or the type of data measurement ? or anything else?

I hope you got my question. Thank you for your help!!

June 18, 2020 at 5:06 pm

I’m not clear what you’re asking exactly. Please clarify. The correlation measures the strength of the relationship between the two continuous variables, as I explain in this article.

Yes, that it is a weak relationship. If you’re going to include this is a regression analysis, you might want to read my article about interpreting low R-squared values .

I’m not sure what you mean by scale variables. However, if these are Likert scale items, you’ll need to use Spearman’s correlation instead of Pearson’s correlation.

May 26, 2020 at 12:08 am

Hi Jim I am very new to statistics and data analysis. I am doing a quantitative study and my sample size is 200 participants. So far I have only obtained 50 complete responses. . Using G*Power a simple linear regression with a medium effect size, an alpha of .05, and a power level of .80 can I do a data analysis with this small sample.

May 26, 2020 at 3:52 am

Please repost your question in the comments section of the appropriate article. It has nothing to do with correlation coefficients. Use the search bar part way down in the right column and search for power. I have a post about power analysis that is a good fit.

May 24, 2020 at 9:02 pm

Thank you Mr.Jim, it was a great answer for me!😉 Take good care~

May 24, 2020 at 9:46 am

I am a student from Malaysia.

I have a question to ask Mr.Jim about how to determine the validity (the accurate figure) of the data for analysis purpose base on the table of Pearson’s Correlation Coefficient? Do it has any method?

For example, since the coefficient between one independent variable with the other variable is below 0.7, thus the data valid for analysis purpose.

However, I have read the table there is a figure which more than 0.7. I am not sure about that.

Hope to hearing from Mr.Jim soon. Thank you.

May 24, 2020 at 4:20 pm

Hi, I hope you’re doing well!

There is no single correlation coefficient value that determines whether it is valid to study. It partly depends on your subject area. I low noise physical process might often have a correlation in the very high 0.9s and 0.8 would be considered unacceptable. However, in a study of human behavior, it’s normal and acceptable to have much lower correlations. For example a correlation of 0.5 might be considered very good. Of course, I’m writing the positive values, but the same applies to negative correlations too.

It also depends on what the purpose of your study. If you’re doing something practical, such as describing the relationship between material composition and strength, there might be very specific requirements about how strong that relationship must be for it to be useful. It’s based on real-world practicalities. On the other hand, if you’re just studying something for the sake of science and expanding knowledge, lower correlations might still be interesting.

So, there’s not single answer. It depends on the subject-area you are studying and the purpose of your study.

February 17, 2020 at 3:49 pm

HI Jim, what could be the implication of my result if I obtained a weak relationship between industry experience and instructional effectiveness? thanks in advance

February 20, 2020 at 11:29 am

The best way to think of it is to look at the graphs in this article and compare the higher correlation graphs to the lower correlation graphs. In the higher correlation graphs, if you know the value of one variable, you have a more precise prediction of the value of the other variable. Look along the x-axis and pick a value. In the higher correlation graphs, the range of y-values that correspond to your x-value is narrower. That range is relatively wide for lower correlations.

For your example, I’ll assume there is a positive correlation. As industry experience increases, instructional effectiveness also increases. However, because that relationship is weak, the range of instructional effectiveness for any given value of industry experience is relatively wide.

November 25, 2019 at 9:05 pm

if correlation between X and Y is 0.8 .what is the correlation of -X and -Y

November 26, 2019 at 4:59 pm

If you take all the values of X and multiply them by -1 and do the same for Y, your correlation would still be 0.8.

November 7, 2019 at 3:51 am

This is very helpful, thank you Jim!

November 6, 2019 at 3:16 am

Hi, My data is continuous – the variables are individual shares volatility and oil prices and they were non-normal. I used Kendall’s Tau and did not rank the data or alter it in any way. Can my results be trusted?

November 6, 2019 at 3:32 pm

Hi Lorraine,

Kendall’s Tau is a correlation coefficient for ranked data. Even though you might not have ranked your data, your statistical software must have created the ranks behind the scenes.

Typically, you’ll use Pearson’s correlation when you have continuous data that have a straight line relationship. If your data are ordinal, ranked, or do not have a straight line relationship, using something other than Pearson’s correlation is necessary.

You mention that your data are nonnormal. Technically, you want to graph your data and look at the shape of the relationship rather than assessing the distribution for each variable. Although, nonnormality can make a linear relationship less likely. So, graph your data on a scatterplot and see what it looks like. If it is close to a straight line, you should probably use Pearson’s correlation. If it’s not a straight line relationship, you might need to use something like Kendall’s Tau or Spearman’s rho coefficient, both of which are based on ranked data. While Spearman’s rho is more commonly used, Kendall’s Tau has preferable statistical properties.

October 24, 2019 at 11:56 pm

Hi, Jim. If correlations between continuous variables can be measured using Pearson’s, how is correlation between categorical variables measured? Thank you.

October 25, 2019 at 2:38 pm

There are several possible methods, although unlike with continuous data, there doesn’t seem to be a consensus best approach.

But, first off, if you want to determine whether the relationship between categorical variables is statistically significant, use the chi-square test of independence . This test determines whether the relationship between categorical variables is significant, but it does not tell you the degree of correlation.

For the correlation values themselves, there are different methods, such as Goodman and Kruskal’s lambda, Cramér’s V (or phi) for categorical variables with more than 2 levels, and the Phi coefficient for binary data. There are several others that are available as well. Offhand I don’t know the relative pros and cons of each methodology. Perhaps that would be a good post for the future!

August 29, 2019 at 7:31 pm

Thanks, great explanations.

April 25, 2019 at 11:58 am

In a multi-variable regression model, is there a method for determining where two predictor variables are correlated in their impact on the outcome variable?

If so, then how is this type of scenario determined, and handled?

Thanks, Curt

April 25, 2019 at 1:27 pm

When predictors are correlated, it’s known as multicollinearity. This condition reduces the precision of the coefficient estimates. I’ve written a post about it: Multicollinearity: Detection, Problems, and Solutions . That post should answer all your questions!

February 3, 2019 at 6:45 am

Hi Jim: Great explanations. One quick thing, because the probability distribution is asymptotic, there is no p=.000. The probability can never be zero. I see students reporting that or p<.000 all of the time. The actual number may be p <.00000001, so setting a level of p < .001 is usually the best thing to do and seems like journal editors want that when reporting data. Your thoughts?

February 4, 2019 at 12:25 am

Hi Susan, yes, you’re correct about that. You can’t have a p-value that equals zero. Sometimes software will round down when it’s a very small value. The underlying issue is that no matter how large the difference between your sample value and the null hypothesis value, there is a non-zero probability that you’d obtain the observed results when the null is true.

January 9, 2019 at 6:41 pm

Sir you are love. Such a nice share

November 21, 2018 at 11:17 am

Awesome stuff, really helpful

November 9, 2018 at 11:48 am

What do you do when you can’t perform randomized controlled experiments, like in the cases of social science or societal wide health issues? Apropos to gun violence in America, there appears to be correlation between the availability of guns in a society and the number of gun deaths in a society, where as the number of guns in the society goes up the number of gun deaths go up. This is true of individual states in the US where gun availability differs, and also in countries where gun availability differs. But, when/how can you come to a determination that lowering the number of guns available in a society could reasonably be said to lower the number of gun deaths in that society.

November 9, 2018 at 12:20 pm

Hi Patrick,

It is difficult proving causality using observational studies rather than randomized experiments.

In my mind, the following approach can help when you’re trying to use observational studies to show that A causes B.

In observational study, you need to worry about confounding variables because the study is not randomized. These confounding variables can provide alternative explanations for the effect/correlations. If you can include all confounding variables in the analysis, it makes the case stronger because it helps rule out other causes. You must also show that A precedes B. Further, it helps if you can demonstrate the mechanism by which A causes B. That mechanism requires subject-area knowledge beyond just a statistical test.

Those are some ideas that come to my mind after brief reflection. There might well be more and, of course, there will be variations based on the study-area.

September 19, 2018 at 4:55 am

Thank you so much, I am learning a lot of thing from you!

Please, keep doing this great job!

Best regards

September 19, 2018 at 11:45 pm

You bet, Patrik!

September 18, 2018 at 6:04 am

Another question is: should I consider transform my variable before using person correlation, if they do not follow normal distribution or if the two variable do not have a clear liner relationship? What is the implication of that transformation? How to interpret the relationship if used transformed variable (let“s say log)?

September 18, 2018 at 4:44 pm

Because the data need to follow the bivariate normal distribution to use the hypothesis test, I’d assume the transformation process would be more complex than transforming each variable individually. However, I’m not sure about this.

However, if you just want to make a straight line for the correlation to assess, I’d be careful about that too. The correlation of the transformed data would not apply to the untransformed data. One solution would be to use Spearman’s rank order correlation. Another would be to use regression analysis. In regression analysis, you can fit curves, use transformations, etc., and the assumption is that the residual follow a normal distribution (along with some other assumptions) is easy to check.

If you’re not sure that your data fit the assumptions for Pearson’s correlation, consider using regression instead. There are more tools there for you to use.

September 18, 2018 at 5:36 am

Hi Jim, I am always here following your posts.

I would like if you could clarify something to me, please! What is the assumptions for person correlation that must hold true, in order to apply correlation coefficient?

I have read something on the internet, but there is many confusion. Some people are saying that the dependent variable (if have) must be normally distributed, other saying both (dependent and independent) must be following normal distribution. Therefore, I dont know which one I should follow. I would appreciate a lot your kind contribution. This is something that I am using for my paper.

Thank you in advance!

September 18, 2018 at 4:34 pm

I’m so glad to see that you’re hear reading and learning!

This issue turns out to be a bit complicated!

The assumption is actually that the two variables follow a bivariate normal distribution. I won’t go into that here in much detail, but a bivariate normal distribution is more complex than just each variable following a normal distribution. In a nutshell, if you plot data that follow a bivariate normal distribution on a scatterplot, it’ll appear as an elliptical shape.

In terms of the the correlation coefficient, that simply describes the relationship between the data. It is what it is and the data don’t need to follow a bivariate normal distribution as long as you are assessing a linear relationship.

On the other hand, the hypothesis test of Pearson’s correlation coefficient does assume that the data follow a bivariate normal distribution. If you want to test whether the coefficient equals zero, then you need to satisfy this assumption. However, one thing I’m not sure about is whether the test is robust to departures from normality. For example, a 1-sample t-test assumes normality, but with a large enough sample size you don’t need to satisfy this assumption. I’m not sure if a similar sample size requirement applies to this particular test.

I hope this clarifies this issue a bit!

August 29, 2018 at 8:04 am

Hello, thanks for the good explanation. Do variables have to be normally distributed to be analyzed in a Pearson’s correlation? Thanks, Moritz

August 30, 2018 at 1:41 pm

No, the variables do not need to follow a normal distribution to use Pearson’s correlation. However, you do need to graph the data on a scatterplot to be sure that the relationship between the variables is linear rather than curved. For curved relationships, consider using Spearman’s rank correlation.

June 1, 2018 at 9:08 am

Pearson’s correlation measures only linear relationships. But regression can be performed with nonlinear functions, and the software will calculate a value of R^2. What is the meaning of an R^2 value when it accompanies a nonlinear regression?

June 1, 2018 at 9:49 am

Hi Jerry, you raise an important point. R^2 is actually not a valid measure in nonlinear models. To read about why, read my post about R-squared in nonlinear models . In that post, I write about why it’s problematic that many statistical software packages do calculate R-squared values for nonlinear regression. Instead, you should use a different goodness-of-fit measure, such as the standard error of the regression .

May 30, 2018 at 11:59 pm

Hi, fantastic blog, very helpful. I was hoping I could ask a question? You talk about correlation coefficients but I was wondering if you have a section that talks about the slope of an association? For example, am I right in thinking that the slope is equal to the standardized coefficient from a regression?

I refer to the paper of Cameron et al., (The Aging of Elastic and Muscular Arteries. Diabetes Care 26:2133–2138, 2003) where in table 3 they report a correlation and a slope. Is the correlation the r value and the slope the beta value?

Many thanks, Matt

May 31, 2018 at 12:13 pm

Thanks and I’m glad you found the blog to be helpful!

Typically, you’d use regression analysis to obtain the slope and correlation to obtain the correlation coefficient. These statistics represent fairly different types of information. The correlation coefficient (r) is more closely related to R^2 in simple regression analysis because both statistics measure how close the data points fall to a line. Not surprisingly if you square r, you obtain R^2.

However, you can use r to calculate the slope coefficient. To do that, you’ll need some other information–the standard deviation of the X variable and the standard deviation of the Y variable.

The formula for the slope in simple regression = r(standard deviation of Y/standard deviation of X).

For more information, read my post about slope coefficients and their p-values in regression analysis . I think that will answer a lot of your questions.

April 12, 2018 at 5:19 am

Nice post ! About pitfalls regarding correlation’s interpretation, here’s a funny database:

http://www.tylervigen.com/spurious-correlations

And a nice and poetic illustration of the concept of correlation:

https://www.youtube.com/watch?v=VFjaBh12C6s&t=0s&index=4&list=PLCkLQOAPOtT1xqDNK8m6IC1bgYCxGZJb_

Have a nice day

April 12, 2018 at 1:57 pm

Thanks for sharing those links! It always fun finding strange correlations like that.

The link for spurious correlations illustrates an important point. Many of those funny correlations are for time series data where both variables have a long-term trend. If you have two variables that you measure over time and they both have long term trends, those two variables will have a strong correlation even if there is no real connection between them!

April 3, 2018 at 7:05 pm

“In statistics, you typically need to perform a randomized, controlled experiment to determine that a relationship is causal rather than merely correlation.”

Would you please provide an example where you can reasonably conclude that x causes y? And how do you know there isn’t a z that you didn’t control for?

April 3, 2018 at 11:00 pm

That’s a great question. The trick is that when you perform an experiment, you should randomly assign subjects to treatment and control groups. This process randomly distributes any other characteristics that are related to the outcome variable (y). Suppose there is a z that is correlated to the outcome. That z gets randomly distributed between the treatment and control groups. The end result is that z should exist in all groups in roughly equal amounts. This equal distribution should occur even if you don’t know what z is. And, that’s the beautiful thing about random assignment. You don’t need to know everything that can affect the outcome, but random assignment still takes care of it all.

Consequently, if there is a relationship between a treatment and the outcome, you can be pretty certain that the treatment causes the changes in the outcome because all other correlation-only relationships should’ve been randomized away.

I’ll be writing about random assignment in the near future. And, I’ve written about the effectiveness of flu shots , which is based on randomized controlled trials.

Comments and Questions Cancel reply

12.4 Testing the Significance of the Correlation Coefficient

The correlation coefficient, r , tells us about the strength and direction of the linear relationship between x and y . However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient r and the sample size n , together.

We perform a hypothesis test of the "significance of the correlation coefficient" to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute r , the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r , is our estimate of the unknown population correlation coefficient.

• The symbol for the population correlation coefficient is ρ , the Greek letter "rho."
• ρ = population correlation coefficient (unknown)
• r = sample correlation coefficient (known; calculated from sample data)

The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is "close to zero" or "significantly different from zero". We decide this based on the sample correlation coefficient r and the sample size n .

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is "significant."

• Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.
• What the conclusion means: There is a significant linear relationship between x and y . We can use the regression line to model the linear relationship between x and y in the population.

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is "not significant".

• Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is not significantly different from zero."
• What the conclusion means: There is not a significant linear relationship between x and y . Therefore, we CANNOT use the regression line to model a linear relationship between x and y in the population.
• If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of x that are within the domain of observed x values.
• If r is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
• If r is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed x values in the data.

PERFORMING THE HYPOTHESIS TEST

• Null Hypothesis: H 0 : ρ = 0
• Alternate Hypothesis: H a : ρ ≠ 0

WHAT THE HYPOTHESES MEAN IN WORDS:

• Null Hypothesis H 0 : The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship (correlation) between x and y in the population.
• Alternate Hypothesis H a : The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

DRAWING A CONCLUSION: There are two methods of making the decision. The two methods are equivalent and give the same result.

• Method 1: Using the p -value
• Method 2: Using a table of critical values

In this chapter of this textbook, we will always use a significance level of 5%, α = 0.05

Using the p -value method, you could choose any appropriate significance level you want; you are not limited to using α = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, α = 0.05. (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

METHOD 1: Using a p -value to make a decision

Using the ti-83, 83+, 84, 84+ calculator.

To calculate the p -value using LinRegTTEST : On the LinRegTTEST input screen, on the line prompt for β or ρ , highlight " ≠ 0 " The output screen shows the p-value on the line that reads "p =". (Most computer statistical software can calculate the p -value.)

• Decision: Reject the null hypothesis.
• Conclusion: "There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero."
• Decision: DO NOT REJECT the null hypothesis.
• Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is NOT significantly different from zero."
• You will use technology to calculate the p -value. The following describes the calculations to compute the test statistics and the p -value:
• The p -value is calculated using a t -distribution with n - 2 degrees of freedom.
• The formula for the test statistic is t = r n − 2 1 − r 2 t = r n − 2 1 − r 2 . The value of the test statistic, t , is shown in the computer or calculator output along with the p -value. The test statistic t has the same sign as the correlation coefficient r .
• The p -value is the combined area in both tails.

An alternative way to calculate the p -value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

• Consider the third exam/final exam example .
• The line of best fit is: ŷ = -173.51 + 4.83 x with r = 0.6631 and there are n = 11 data points.
• Can the regression line be used for prediction? Given a third exam score ( x value), can we use the line to predict the final exam score (predicted y value)?
• H 0 : ρ = 0
• H a : ρ ≠ 0
• The p -value is 0.026 (from LinRegTTest on your calculator or from computer software).
• The p -value, 0.026, is less than the significance level of α = 0.05.
• Decision: Reject the Null Hypothesis H 0
• Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

METHOD 2: Using a table of Critical Values to make a decision

The 95% Critical Values of the Sample Correlation Coefficient Table can be used to give you a good idea of whether the computed value of r r is significant or not . Compare r to the appropriate critical value in the table. If r is not between the positive and negative critical values, then the correlation coefficient is significant. If r is significant, then you may want to use the line for prediction.

Example 12.7

Suppose you computed r = 0.801 using n = 10 data points. df = n - 2 = 10 - 2 = 8. The critical values associated with df = 8 are -0.632 and + 0.632. If r < negative critical value or r > positive critical value, then r is significant. Since r = 0.801 and 0.801 > 0.632, r is significant and the line may be used for prediction. If you view this example on a number line, it will help you.

Try It 12.7

For a given line of best fit, you computed that r = 0.6501 using n = 12 data points and the critical value is 0.576. Can the line be used for prediction? Why or why not?

Example 12.8

Suppose you computed r = –0.624 with 14 data points. df = 14 – 2 = 12. The critical values are –0.532 and 0.532. Since –0.624 < –0.532, r is significant and the line can be used for prediction

Try It 12.8

For a given line of best fit, you compute that r = 0.5204 using n = 9 data points, and the critical value is 0.666. Can the line be used for prediction? Why or why not?

Example 12.9

Suppose you computed r = 0.776 and n = 6. df = 6 – 2 = 4. The critical values are –0.811 and 0.811. Since –0.811 < 0.776 < 0.811, r is not significant, and the line should not be used for prediction.

Try It 12.9

For a given line of best fit, you compute that r = –0.7204 using n = 8 data points, and the critical value is = 0.707. Can the line be used for prediction? Why or why not?

THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Consider the third exam/final exam example . The line of best fit is: ŷ = –173.51+4.83 x with r = 0.6631 and there are n = 11 data points. Can the regression line be used for prediction? Given a third-exam score ( x value), can we use the line to predict the final exam score (predicted y value)?

• Use the "95% Critical Value" table for r with df = n – 2 = 11 – 2 = 9.
• The critical values are –0.602 and +0.602
• Since 0.6631 > 0.602, r is significant.
• Conclusion:There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

Example 12.10

Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if r is significant and the line of best fit associated with each r can be used to predict a y value. If it helps, draw a number line.

• r = –0.567 and the sample size, n , is 19. The df = n – 2 = 17. The critical value is –0.456. –0.567 < –0.456 so r is significant.
• r = 0.708 and the sample size, n , is nine. The df = n – 2 = 7. The critical value is 0.666. 0.708 > 0.666 so r is significant.
• r = 0.134 and the sample size, n , is 14. The df = 14 – 2 = 12. The critical value is 0.532. 0.134 is between –0.532 and 0.532 so r is not significant.
• r = 0 and the sample size, n , is five. No matter what the dfs are, r = 0 is between the two critical values so r is not significant.

Try It 12.10

For a given line of best fit, you compute that r = 0 using n = 100 data points. Can the line be used for prediction? Why or why not?

Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between x and y in the population.

The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatterplot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.

• The relationship between the variables being correlated should be linear. The data points should fall along an approximate straight-line pattern when plotted as ( x , y ) data points on a scatter plot.
• The y values for any particular x value are normally distributed about the line. This implies that there are more y values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of y values lie on the line.
• The standard deviations of the population y values about the line are equal for each value of x . In other words, each of these normal distributions of y values has the same shape and spread about the line.
• The residual errors are mutually independent (no pattern).
• The data are produced from a well-designed, random sample or randomized experiment.

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• Authors: Barbara Illowsky, Susan Dean
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• Book title: Introductory Statistics 2e
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Conducting a hypothesis test for the population correlation coefficient p.

There is one more point we haven't stressed yet in our discussion about the correlation coefficient r and the coefficient of determination r 2 — namely, the two measures summarize the strength of a linear relationship in samples only . If we obtained a different sample, we would obtain different correlations, different r 2 values, and therefore potentially different conclusions. As always, we want to draw conclusions about populations , not just samples. To do so, we either have to conduct a hypothesis test or calculate a confidence interval. In this section, we learn how to conduct a hypothesis test for the population correlation coefficient ρ (the Greek letter "rho").

Incidentally, where does this topic fit in among the four regression analysis steps?

• Model formulation
• Model estimation
• Model evaluation

It's a situation in which we use the model to answer a specific research question, namely whether or not a linear relationship exists between two quantitative variables

In general, a researcher should use the hypothesis test for the population correlation ρ to learn of a linear association between two variables, when it isn't obvious which variable should be regarded as the response. Let's clarify this point with examples of two different research questions.

We previously learned that to evaluate whether or not a linear relationship exists between skin cancer mortality and latitude, we can perform either of the following tests:

• t -test for testing H 0 : β 1 = 0
• ANOVA F -test for testing H 0 : β 1 = 0

That's because it is fairly obvious that latitude should be treated as the predictor variable and skin cancer mortality as the response. Suppose we want to evaluate whether or not a linear relationship exists between a husband's age and his wife's age? In this case, one could treat the husband's age as the response:

or one could treat wife's age as the response:

Pearson correlation of HAge and WAge = 0.939

In cases such as these, we answer our research question concerning the existence of a linear relationship by using the t -test for testing the population correlation coefficient H 0 : ρ = 0.

Let's jump right to it! We follow standard hypothesis test procedures in conducting a hypothesis test for the population correlation coefficient ρ . First, we specify the null and alternative hypotheses:

Null hypothesis H 0 : ρ = 0 Alternative hypothesis H A : ρ ≠ 0 or H A : ρ < 0 or H A : ρ > 0

Second, we calculate the value of the test statistic using the following formula:

Test statistic :  \(t^*=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\) 

Third, we use the resulting test statistic to calculate the P -value. As always, the P -value is the answer to the question "how likely is it that we’d get a test statistic t* as extreme as we did if the null hypothesis were true?" The P -value is determined by referring to a t- distribution with n -2 degrees of freedom.

Finally, we make a decision:

• If the P -value is smaller than the significance level α, we reject the null hypothesis in favor of the alternative. We conclude "there is sufficient evidence at the α level to conclude that there is a linear relationship in the population between the predictor x and response y ."
• If the P -value is larger than the significance level α, we fail to reject the null hypothesis. We conclude "there is not enough evidence at the α level to conclude that there is a linear relationship in the population between the predictor x and response y ."

Let's perform the hypothesis test on the husband's age and wife's age data in which the sample correlation based on n = 170 couples is r = 0.939. To test H 0 : ρ = 0 against the alternative H A : ρ ≠ 0, we obtain the following test statistic:

\[t^*=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}=\frac{0.939\sqrt{170-2}}{\sqrt{1-0.939^2}}=35.39\]

To obtain the P -value, we need to compare the test statistic to a t -distribution with 168 degrees of freedom (since 170 - 2 = 168). In particular, we need to find the probability that we'd observe a test statistic more extreme than 35.39, and then, since we're conducting a two-sided test, multiply the probability by 2. Minitab helps us out here:

Incidentally, we can let statistical software like Minitab do all of the dirty work for us. In doing so, Minitab reports:

It should be noted that the three hypothesis tests we learned for testing the existence of a linear relationship — the t -test for H 0 : β 1 = 0, the ANOVA F -test for H 0 : β 1 = 0, and the t -test for H 0 : ρ = 0 — will always yield the same results. For example, if we treat the husband's age ("HAge") as the response and the wife's age ("WAge") as the predictor, each test yields a P -value of 0.000... < 0.001:

And similarly, if we treat the wife's age ("WAge") as the response and the husband's age ("HAge") as the predictor, each test yields of P -value of 0.000... < 0.001:

Technically, then, it doesn't matter what test you use to obtain the P -value. You will always get the same P -value. But, you should report the results of the test that make sense for your particular situation:

• If one of the variables can be clearly identified as the response, report that you conducted a t -test or F -test results for testing H 0 : β 1 = 0. (Does it make sense to use x to predict y ?)
• If it is not obvious which variable is the response, report that you conducted a t -test for testing H 0 : ρ = 0. (Does it only make sense to look for an association between x and y ?)

One final note ... as always, we should clarify when it is okay to use the t -test for testing H 0 : ρ = 0? The guidelines are a straightforward extension of the "LINE" assumptions made for the simple linear regression model. It's okay:

• When it is not obvious which variable is the response.
• For each x , the y 's are normal with equal variances.
• For each y , the x 's are normal with equal variances.
• Either, y can be considered a linear function of x .
• Or, x can be considered a linear function of y .
• The ( x , y ) pairs are independent

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• Knowledge Base
• Correlation Coefficient | Types, Formulas & Examples

Correlation Coefficient | Types, Formulas & Examples

Published on August 2, 2021 by Pritha Bhandari . Revised on June 22, 2023.

A correlation coefficient is a number between -1 and 1 that tells you the strength and direction of a relationship between variables .

In other words, it reflects how similar the measurements of two or more variables are across a dataset.

Table of contents

What does a correlation coefficient tell you, using a correlation coefficient, interpreting a correlation coefficient, visualizing linear correlations, types of correlation coefficients, pearson’s r, spearman’s rho, other coefficients, other interesting articles, frequently asked questions about correlation coefficients.

Correlation coefficients summarize data and help you compare results between studies.

Summarizing data

A correlation coefficient is a descriptive statistic . That means that it summarizes sample data without letting you infer anything about the population. A correlation coefficient is a bivariate statistic when it summarizes the relationship between two variables, and it’s a multivariate statistic when you have more than two variables.

If your correlation coefficient is based on sample data, you’ll need an inferential statistic if you want to generalize your results to the population. You can use an F test or a t test to calculate a test statistic that tells you the statistical significance of your finding.

Comparing studies

A correlation coefficient is also an effect size measure, which tells you the practical significance of a result.

Correlation coefficients are unit-free, which makes it possible to directly compare coefficients between studies.

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In correlational research , you investigate whether changes in one variable are associated with changes in other variables.

After data collection , you can visualize your data with a scatterplot by plotting one variable on the x-axis and the other on the y-axis. It doesn’t matter which variable you place on either axis.

Visually inspect your plot for a pattern and decide whether there is a linear or non-linear pattern between variables. A linear pattern means you can fit a straight line of best fit between the data points, while a non-linear or curvilinear pattern can take all sorts of different shapes, such as a U-shape or a line with a curve.

There are many different correlation coefficients that you can calculate. After removing any outliers , select a correlation coefficient that’s appropriate based on the general shape of the scatter plot pattern. Then you can perform a correlation analysis to find the correlation coefficient for your data.

You calculate a correlation coefficient to summarize the relationship between variables without drawing any conclusions about causation .

Both variables are quantitative and normally distributed with no outliers, so you calculate a Pearson’s r correlation coefficient .

The value of the correlation coefficient always ranges between 1 and -1, and you treat it as a general indicator of the strength of the relationship between variables.

The sign of the coefficient reflects whether the variables change in the same or opposite directions: a positive value means the variables change together in the same direction, while a negative value means they change together in opposite directions.

The absolute value of a number is equal to the number without its sign. The absolute value of a correlation coefficient tells you the magnitude of the correlation: the greater the absolute value, the stronger the correlation.

There are many different guidelines for interpreting the correlation coefficient because findings can vary a lot between study fields. You can use the table below as a general guideline for interpreting correlation strength from the value of the correlation coefficient.

While this guideline is helpful in a pinch, it’s much more important to take your research context and purpose into account when forming conclusions. For example, if most studies in your field have correlation coefficients nearing .9, a correlation coefficient of .58 may be low in that context.

The correlation coefficient tells you how closely your data fit on a line. If you have a linear relationship, you’ll draw a straight line of best fit that takes all of your data points into account on a scatter plot.

The closer your points are to this line, the higher the absolute value of the correlation coefficient and the stronger your linear correlation.

If all points are perfectly on this line, you have a perfect correlation.

If all points are close to this line, the absolute value of your correlation coefficient is high .

If these points are spread far from this line, the absolute value of your correlation coefficient is low .

Note that the steepness or slope of the line isn’t related to the correlation coefficient value. The correlation coefficient doesn’t help you predict how much one variable will change based on a given change in the other, because two datasets with the same correlation coefficient value can have lines with very different slopes.

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You can choose from many different correlation coefficients based on the linearity of the relationship, the level of measurement of your variables, and the distribution of your data.

For high statistical power and accuracy, it’s best to use the correlation coefficient that’s most appropriate for your data.

The most commonly used correlation coefficient is Pearson’s r because it allows for strong inferences. It’s parametric and measures linear relationships. But if your data do not meet all assumptions for this test, you’ll need to use a non-parametric test instead.

Non-parametric tests of rank correlation coefficients summarize non-linear relationships between variables. The Spearman’s rho and Kendall’s tau have the same conditions for use, but Kendall’s tau is generally preferred for smaller samples whereas Spearman’s rho is more widely used.

The table below is a selection of commonly used correlation coefficients, and we’ll cover the two most widely used coefficients in detail in this article.

The Pearson’s product-moment correlation coefficient, also known as Pearson’s r, describes the linear relationship between two quantitative variables.

These are the assumptions your data must meet if you want to use Pearson’s r:

• Both variables are on an interval or ratio level of measurement
• Data from both variables follow normal distributions
• Your data have no outliers
• Your data is from a random or representative sample
• You expect a linear relationship between the two variables

The Pearson’s r is a parametric test, so it has high power. But it’s not a good measure of correlation if your variables have a nonlinear relationship, or if your data have outliers, skewed distributions, or come from categorical variables. If any of these assumptions are violated, you should consider a rank correlation measure.

The formula for the Pearson’s r is complicated, but most computer programs can quickly churn out the correlation coefficient from your data. In a simpler form, the formula divides the covariance between the variables by the product of their standard deviations .

Pearson sample vs population correlation coefficient formula

When using the Pearson correlation coefficient formula, you’ll need to consider whether you’re dealing with data from a sample or the whole population.

The sample and population formulas differ in their symbols and inputs. A sample correlation coefficient is called r , while a population correlation coefficient is called rho, the Greek letter ρ.

The sample correlation coefficient uses the sample covariance between variables and their sample standard deviations.

The population correlation coefficient uses the population covariance between variables and their population standard deviations.

Spearman’s rho, or Spearman’s rank correlation coefficient, is the most common alternative to Pearson’s r . It’s a rank correlation coefficient because it uses the rankings of data from each variable (e.g., from lowest to highest) rather than the raw data itself.

You should use Spearman’s rho when your data fail to meet the assumptions of Pearson’s r . This happens when at least one of your variables is on an ordinal level of measurement or when the data from one or both variables do not follow normal distributions.

While the Pearson correlation coefficient measures the linearity of relationships, the Spearman correlation coefficient measures the monotonicity of relationships.

In a linear relationship, each variable changes in one direction at the same rate throughout the data range. In a monotonic relationship, each variable also always changes in only one direction but not necessarily at the same rate.

• Positive monotonic: when one variable increases, the other also increases.
• Negative monotonic: when one variable increases, the other decreases.

Monotonic relationships are less restrictive than linear relationships.

Spearman’s rank correlation coefficient formula

The symbols for Spearman’s rho are ρ for the population coefficient and r s for the sample coefficient. The formula calculates the Pearson’s r correlation coefficient between the rankings of the variable data.

To use this formula, you’ll first rank the data from each variable separately from low to high: every datapoint gets a rank from first, second, or third, etc.

Then, you’ll find the differences (d i ) between the ranks of your variables for each data pair and take that as the main input for the formula.

If you have a correlation coefficient of 1, all of the rankings for each variable match up for every data pair. If you have a correlation coefficient of -1, the rankings for one variable are the exact opposite of the ranking of the other variable. A correlation coefficient near zero means that there’s no monotonic relationship between the variable rankings.

The correlation coefficient is related to two other coefficients, and these give you more information about the relationship between variables.

Coefficient of determination

When you square the correlation coefficient, you end up with the correlation of determination ( r 2 ). This is the proportion of common variance between the variables. The coefficient of determination is always between 0 and 1, and it’s often expressed as a percentage.

The coefficient of determination is used in regression models to measure how much of the variance of one variable is explained by the variance of the other variable.

A regression analysis helps you find the equation for the line of best fit, and you can use it to predict the value of one variable given the value for the other variable.

A high r 2 means that a large amount of variability in one variable is determined by its relationship to the other variable. A low r 2 means that only a small portion of the variability of one variable is explained by its relationship to the other variable; relationships with other variables are more likely to account for the variance in the variable.

The correlation coefficient can often overestimate the relationship between variables, especially in small samples, so the coefficient of determination is often a better indicator of the relationship.

Coefficient of alienation

When you take away the coefficient of determination from unity (one), you’ll get the coefficient of alienation. This is the proportion of common variance not shared between the variables, the unexplained variance between the variables.

A high coefficient of alienation indicates that the two variables share very little variance in common. A low coefficient of alienation means that a large amount of variance is accounted for by the relationship between the variables.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Chi square test of independence
• Statistical power
• Descriptive statistics
• Degrees of freedom
• Pearson correlation
• Null hypothesis

Methodology

• Double-blind study
• Case-control study
• Research ethics
• Data collection
• Hypothesis testing
• Structured interviews

Research bias

• Hawthorne effect
• Unconscious bias
• Recall bias
• Halo effect
• Self-serving bias
• Information bias

A correlation reflects the strength and/or direction of the association between two or more variables.

• A positive correlation means that both variables change in the same direction.
• A negative correlation means that the variables change in opposite directions.
• A zero correlation means there’s no relationship between the variables.

A correlation is usually tested for two variables at a time, but you can test correlations between three or more variables.

A correlation coefficient is a single number that describes the strength and direction of the relationship between your variables.

Different types of correlation coefficients might be appropriate for your data based on their levels of measurement and distributions . The Pearson product-moment correlation coefficient (Pearson’s r ) is commonly used to assess a linear relationship between two quantitative variables.

These are the assumptions your data must meet if you want to use Pearson’s r :

Correlation coefficients always range between -1 and 1.

The sign of the coefficient tells you the direction of the relationship: a positive value means the variables change together in the same direction, while a negative value means they change together in opposite directions.

No, the steepness or slope of the line isn’t related to the correlation coefficient value. The correlation coefficient only tells you how closely your data fit on a line, so two datasets with the same correlation coefficient can have very different slopes.

To find the slope of the line, you’ll need to perform a regression analysis .

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How to Write a Hypothesis for Correlation

How to Calculate a P-Value

A hypothesis is a testable statement about how something works in the natural world. While some hypotheses predict a causal relationship between two variables, other hypotheses predict a correlation between them. According to the Research Methods Knowledge Base, a correlation is a single number that describes the relationship between two variables. If you do not predict a causal relationship or cannot measure one objectively, state clearly in your hypothesis that you are merely predicting a correlation.

Research the topic in depth before forming a hypothesis. Without adequate knowledge about the subject matter, you will not be able to decide whether to write a hypothesis for correlation or causation. Read the findings of similar experiments before writing your own hypothesis.

Identify the independent variable and dependent variable. Your hypothesis will be concerned with what happens to the dependent variable when a change is made in the independent variable. In a correlation, the two variables undergo changes at the same time in a significant number of cases. However, this does not mean that the change in the independent variable causes the change in the dependent variable.

Construct an experiment to test your hypothesis. In a correlative experiment, you must be able to measure the exact relationship between two variables. This means you will need to find out how often a change occurs in both variables in terms of a specific percentage.

Establish the requirements of the experiment with regard to statistical significance. Instruct readers exactly how often the variables must correlate to reach a high enough level of statistical significance. This number will vary considerably depending on the field. In a highly technical scientific study, for instance, the variables may need to correlate 98 percent of the time; but in a sociological study, 90 percent correlation may suffice. Look at other studies in your particular field to determine the requirements for statistical significance.

State the null hypothesis. The null hypothesis gives an exact value that implies there is no correlation between the two variables. If the results show a percentage equal to or lower than the value of the null hypothesis, then the variables are not proven to correlate.

Record and summarize the results of your experiment. State whether or not the experiment met the minimum requirements of your hypothesis in terms of both percentage and significance.

Related Articles

How to determine the sample size in a quantitative..., how to calculate a two-tailed test, how to interpret a student's t-test results, how to know if something is significant using spss, quantitative vs. qualitative data and laboratory testing, similarities of univariate & multivariate statistical..., what is the meaning of sample size, distinguishing between descriptive & causal studies, how to calculate cv values, how to determine your practice clep score, what are the different types of correlations, how to calculate p-hat, how to calculate percentage error, how to calculate percent relative range, how to calculate a sample size population, how to calculate bias, how to calculate the percentage of another number, how to find y value for the slope of a line, advantages & disadvantages of finding variance.

• University of New England; Steps in Hypothesis Testing for Correlation; 2000
• Research Methods Knowledge Base; Correlation; William M.K. Trochim; 2006
• Science Buddies; Hypothesis

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Hypothesis Testing for Correlation ( AQA A Level Maths: Statistics )

Revision note.

Hypothesis Testing for Correlation

You should be familiar with using a hypothesis test to determine bias within probability problems. It is also possible to use a hypothesis test to determine whether a given product moment correlation coefficient calculated from a sample could be representative of the same relationship existing within the whole population.  For full information on hypothesis testing, see the revision notes from section 5.1.1 Hypothesis Testing

Why use a hypothesis test?

• This would involve having data on each individual within the whole population
• It is very rare that a statistician would have the time or resources to collect all of that data
• The PMCC for a sample taken from the population is denoted r
• A hypothesis test would be conducted using the value of to r determine whether the population can be said to have positive, negative or zero correlation

How is a hypothesis test for correlation carried out?

• Most of the time the hypothesis test will be carried out by using a critical value
• You won't be expected to calculate p-values but you might be given a p-value
• The hypothesis test could either be a one-tailed test or a two-tailed test
• You will be given the critical value in the question  
• If  r  is not in the critical region the null hypothesis should be accepted and the alternative hypothesis should be rejected

Or: Compare the p - value with the significance level

• If the p - value is less than the significance level the test is significant and the null hypothesis should be rejected
• If the p - value is greater than the significance level the null hypothesis should be accepted and the alternative hypothesis should be rejected
• Use the wording in the question to help you write your conclusion
• If rejecting the null hypothesis your conclusion should state that there is evidence to accept the context of the alternative hypothesis at the level of significance of the test only
• If accepting the null hypothesis your conclusion should state that there is not enough evidence to accept the context of the alternative hypothesis at the level of significance of the test only

Worked example

A student believes that there is a positive correlation between the number of hours spent studying for a test and the percentage scored on it.

The student takes a random sample of 10 of his friends and records the amount of revision they did and percentage they score in the test.

Given that the critical value for this test is 0.5494, carry out a hypothesis test at the 5% level of significance to test whether the student’s claim is justified.

• Make sure you read the question carefully to determine whether the test you are carrying out is for a one-tailed or a two-tailed test and use the level of significance accordingly. Be careful when comparing negative values of r with a negative critical value, it is easy to make an error with negative numbers when in an exam situation.

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Module 12: Linear Regression and Correlation

Testing the significance of the correlation coefficient, learning outcomes.

• Calculate and interpret the correlation coefficient

The correlation coefficient,  r , tells us about the strength and direction of the linear relationship between x and y . However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient r and the sample size n , together.

We perform a hypothesis test of the “ significance of the correlation coefficient ” to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute  r , the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only have sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r , is our estimate of the unknown population correlation coefficient.

• The symbol for the population correlation coefficient is ρ , the Greek letter “rho.”
• ρ = population correlation coefficient (unknown)
• r = sample correlation coefficient (known; calculated from sample data)

The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is “close to zero” or “significantly different from zero”. We decide this based on the sample correlation coefficient r and the sample size n .

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is “significant.” Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero. What the conclusion means: There is a significant linear relationship between x and y . We can use the regression line to model the linear relationship between x and y in the population.

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is “not significant.”

Conclusion: “There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is not significantly different from zero.” What the conclusion means: There is not a significant linear relationship between x and y . Therefore, we CANNOT use the regression line to model a linear relationship between x and y in the population.

• If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of x that are within the domain of observed x values.
• If r is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
• If r is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed x values in the data.

Performing the Hypothesis Test

• Null Hypothesis: H 0 : ρ = 0
• Alternate Hypothesis: H a : ρ ≠ 0

What the Hypotheses Mean in Words

• Null Hypothesis H 0 : The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship(correlation) between x and y in the population.
• Alternate Hypothesis H a : The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

Drawing a Conclusion

There are two methods of making the decision. The two methods are equivalent and give the same result.

• Method 1: Using the p -value
• Method 2: Using a table of critical values

In this chapter of this textbook, we will always use a significance level of 5%,  α = 0.05

Using the  p -value method, you could choose any appropriate significance level you want; you are not limited to using α = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, α = 0.05. (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

Method 1: Using a p -value to make a decision

To calculate the  p -value using LinRegTTEST:

• On the LinRegTTEST input screen, on the line prompt for β or ρ , highlight “≠ 0”
• The output screen shows the p-value on the line that reads “p =”.
• (Most computer statistical software can calculate the p -value.)

If the p -value is less than the significance level ( α = 0.05)

• Decision: Reject the null hypothesis.
• Conclusion: “There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.”

If the p -value is NOT less than the significance level ( α = 0.05)

• Decision: DO NOT REJECT the null hypothesis.
• Conclusion: “There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is NOT significantly different from zero.”

Calculation Notes:

• You will use technology to calculate the p -value. The following describes the calculations to compute the test statistics and the p -value:
• The p -value is calculated using a t -distribution with n – 2 degrees of freedom.
• The formula for the test statistic is [latex]\displaystyle{t}=\frac{{{r}\sqrt{{{n}-{2}}}}}{\sqrt{{{1}-{r}^{{2}}}}}[/latex]. The value of the test statistic, t , is shown in the computer or calculator output along with the p -value. The test statistic t has the same sign as the correlation coefficient r .
• The p -value is the combined area in both tails.

An alternative way to calculate the  p -value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

Method 2: Using a table of Critical Values to make a decision

The 95% Critical Values of the Sample Correlation Coefficient Table can be used to give you a good idea of whether the computed value of is significant or not. Compare  r to the appropriate critical value in the table. If r is not between the positive and negative critical values, then the correlation coefficient is significant. If r is significant, then you may want to use the line for prediction.

Suppose you computed  r = 0.801 using n = 10 data points. df = n – 2 = 10 – 2 = 8. The critical values associated with df = 8 are -0.632 and + 0.632. If r < negative critical value or r > positive critical value, then r is  significant . Since r = 0.801 and 0.801 > 0.632, r is significant and the line may be used for prediction. If you view this example on a number line, it will help you.

For a given line of best fit, you computed that  r = 0.6501 using n = 12 data points and the critical value is 0.576. Can the line be used for prediction? Why or why not?

If the scatter plot looks linear then, yes, the line can be used for prediction, because  r > the positive critical value.

Suppose you computed  r = –0.624 with 14 data points. df = 14 – 2 = 12. The critical values are –0.532 and 0.532. Since –0.624 < –0.532, r is significant and the line can be used for prediction

For a given line of best fit, you compute that  r = 0.5204 using n = 9 data points, and the critical value is 0.666. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction, because  r < the positive critical value.

Suppose you computed  r = 0.776 and n = 6. df = 6 – 2 = 4. The critical values are –0.811 and 0.811. Since –0.811 < 0.776 < 0.811, r is not significant, and the line should not be used for prediction.

–0.811 <  r = 0.776 < 0.811. Therefore, r is not significant.

For a given line of best fit, you compute that  r = –0.7204 using n = 8 data points, and the critical value is = 0.707. Can the line be used for prediction? Why or why not?

Yes, the line can be used for prediction, because  r < the negative critical value.

Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if  r is significant and the line of best fit associated with each r can be used to predict a y value. If it helps, draw a number line.

• r = –0.567 and the sample size, n , is 19. The df = n – 2 = 17. The critical value is –0.456. –0.567 < –0.456 so r is significant.
• r = 0.708 and the sample size, n , is nine. The df = n – 2 = 7. The critical value is 0.666. 0.708 > 0.666 so r is significant.
• r = 0.134 and the sample size, n , is 14. The df = 14 – 2 = 12. The critical value is 0.532. 0.134 is between –0.532 and 0.532 so r is not significant.
• r = 0 and the sample size, n , is five. No matter what the dfs are, r = 0 is between the two critical values so r is not significant.

For a given line of best fit, you compute that  r = 0 using n = 100 data points. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction no matter what the sample size is.

Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between x and y in the population.

The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatterplot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.

The assumptions underlying the test of significance are:

• There is a linear relationship in the population that models the average value of y for varying values of x . In other words, the expected value of y for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
• The y values for any particular x value are normally distributed about the line. This implies that there are more y values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of y values lie on the line.
• The standard deviations of the population y values about the line are equal for each value of x . In other words, each of these normal distributions of y values has the same shape and spread about the line.
• The residual errors are mutually independent (no pattern).
• The data are produced from a well-designed, random sample or randomized experiment.

The  y values for each x value are normally distributed about the line with the same standard deviation. For each x value, the mean of the y values lies on the regression line. More y values lie near the line than are scattered further away from the line.

Concept Review

Linear regression is a procedure for fitting a straight line of the form [latex]\displaystyle\hat{{y}}={a}+{b}{x}[/latex] to data. The conditions for regression are:

• Linear: In the population, there is a linear relationship that models the average value of y for different values of x .
• Independent: The residuals are assumed to be independent.
• Normal: The y values are distributed normally for any value of x .
• Equal variance: The standard deviation of the y values is equal for each x value.
• Random: The data are produced from a well-designed random sample or randomized experiment.

The slope  b and intercept a of the least-squares line estimate the slope β and intercept α of the population (true) regression line. To estimate the population standard deviation of y , σ , use the standard deviation of the residuals, s .

[latex]\displaystyle{s}=\sqrt{{\frac{{{S}{S}{E}}}{{{n}-{2}}}}}[/latex] The variable ρ (rho) is the population correlation coefficient.

To test the null hypothesis  H 0 : ρ = hypothesized value , use a linear regression t-test. The most common null hypothesis is H 0 : ρ = 0 which indicates there is no linear relationship between x and y in the population.

The TI-83, 83+, 84, 84+ calculator function LinRegTTest can perform this test (STATS TESTS LinRegTTest).

Formula Review

Least Squares Line or Line of Best Fit: [latex]\displaystyle\hat{{y}}={a}+{b}{x}[/latex]

where  a = y -intercept,  b = slope

Standard deviation of the residuals:

[latex]\displaystyle{s}=\sqrt{{\frac{{{S}{S}{E}}}{{{n}-{2}}}}}[/latex]

SSE = sum of squared errors

n = the number of data points

• OpenStax, Statistics, Testing the Significance of the Correlation Coefficient. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:83/Introductory_Statistics . License : CC BY: Attribution
• Introductory Statistics . Authored by : Barbara Illowski, Susan Dean. Provided by : Open Stax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

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Pearson Correlation Coefficient: Formula, Examples

In the world of data science , understanding the relationship between variables is crucial for making informed decisions or building accurate machine learning models. Correlation is a fundamental statistical concept that measures the strength and direction of the relationship between two variables. However, without the right tools and knowledge, calculating correlation coefficients and p-values can be a daunting task for data scientists. This can lead to suboptimal decision-making, inaccurate predictions, and wasted time and resources.

In this post, we will discuss what Pearson’s r represents, how it works mathematically ( formula ), its interpretation, statistical significance , and importance for making decisions in real-world applications  such as business forecasting or medical diagnosis. We will also explore some examples of using Pearson’s r (correlation coefficient) and p-value (used for statistical significance) with real data sets so you can see how this powerful statistic works in action. We will learn to use Python’s scipy.stats pearsonr method which is a simple and effective way to calculate the correlation coefficient and p-value between two variables. As a data scientist , it is very important to understand Pearson’s r and its implications for making decisions based on data.

Table of Contents

What is Pearson Correlation Coefficient?

Pearson correlation coefficient is a statistical measure that describes the linear relationship between two variables. It is typically represented by the symbol ‘r’. Pearson correlation coefficient can take on values from -1 to +1 and it is used to determine how closely two variables are related. It measures the strength of their linear relationship, which means that it indicates whether one variable increases or decreases as the other variable increases or decreases. A Pearson correlation coefficient of 1 indicates a perfect positive (direct) linear relationship, while a Pearson correlation coefficient of -1 indicates a perfect negative (inverse) linear relationship. Furthermore, when Pearson’s r is 0 there is no linear relationship between the two variables. 

It’s important to note that correlation does not imply causation . A significant Pearson’s r value indicates a linear association, but it doesn’t mean that one variable causes the other. Other factors, known as confounding variables, may influence this relationship. Additionally, Pearson’s r only measures linear relationships. If the relationship is non-linear, other statistical methods may be more appropriate to describe the association. A study finds a significant positive Pearson correlation coefficient ( r ) between monthly ice cream sales and the number of drowning incidents. The data show that as ice cream sales increase, the number of drowning incidents also increases. If we mistakenly infer causation from this correlation , we might conclude that eating ice cream leads to an increased risk of drowning. 

The increase in both ice cream sales and drowning incidents might both be caused by a third variable ( confounding variable ): the temperature or season (i.e., summer). During summer months, temperatures are higher, which leads to more people buying ice cream. Simultaneously, more people are likely to engage in swimming activities, which increases the risk of drowning incidents. Temperature acts as a confounding variable that is associated with both ice cream sales and drowning incidents.

Pearson Correlation Coefficient vs Plots

The following plots represent linear relationship vis-a-vis different values of Pearson correlation coefficient.

The following is the explanation for the above plots:

Direct Linear Relationship ( r close to +1) : The first plot shows a clear upward trend, indicating that as x increases, y also increases. The points are closely aligned around a straight line, suggesting a strong positive linear relationship. The Pearson Correlation Coefficient for such a dataset would be close to +1, implying that the variables move together in the same direction.

No Linear Relationship ( r close to 0) : The second plot shows a scatter of points with no apparent pattern. There is no discernible slope, and the points do not align around any line. This randomness suggests that there is no linear relationship between x and y . In such a case, the Pearson Correlation Coefficient would be close to 0, indicating no linear correlation between the variables.

Inverse Linear Relationship ( r close to -1) : The third plot shows a clear downward trend, indicating that as x increases, y decreases. The points are closely aligned around a straight line, but this time the line slopes downwards, suggesting a strong negative linear relationship. The Pearson Correlation Coefficient for such a dataset would be close to -1, implying that the variables move in opposite directions.

Pearson Correlation Coefficient – Different Values vs Strength of Relationship

When assessing the linear relationship between two variables using correlation analysis, the magnitude of the correlation coefficient (ignoring the sign) provides insight into the strength of the relationship . Here’s a more detailed guide to interpreting the absolute value of the correlation coefficient:

± 1.00 : This represents a perfect correlation , indicating that for every change in one variable, there is a predictable and exact corresponding change in the other variable. In a graph, the data points would lie exactly on a straight line, either upwards or downwards, depending on the sign.

± 0.80 : When the correlation coefficient approaches this value, it is considered a strong correlation . This suggests a high degree of predictability in the relationship, where changes in one variable are closely followed by changes in the other, though not perfectly.

± 0.50 : This value signifies a moderate correlation . The relationship between the variables is evident and can be described as substantial, but there are other factors and variability influencing the relationship.

± 0.20 : This is indicative of a weak correlation , where there is a slight, possibly inconsistent association between the variables. The predictability is low, and while there may be a relationship, it is not strong and could be easily influenced by other variables.

0 : A zero or close to zero correlation coefficient means there is no linear correlation between the variables. There’s no predictable association that can be discerned from the data; any relationship is likely due to chance or randomness.

Pearson Correlation Coefficient – Real-world Examples

Pearson correlation coefficient can be used to examine relationships between variables in a variety of real-world applications such as some of the following:

• In medicine, Pearson’s r can be used to measure the strength of the relationship between patient age and cholesterol levels.
• In finance, Pearson’s r can be used to measure the strength of the relationship between stock prices and earnings per share.
• In business forecasting, Pearson’s r can be used to measure the strength of the relationship between sales and marketing efforts.
• In lifestyle research, Pearson’s r can be used to measure the strength of the relationship between exercise habits and obesity rates.
• Another example is measuring correlation between customer loyalty against customer satisfaction levels and ascertain whether customers who report higher levels of satisfaction also demonstrate higher levels of loyalty or vice versa. Another example could include studying height against weight wherein one might use Pearson’s correlation coefficient to measure if taller individuals tend to weigh more than their shorter counterparts on average or if there is no obvious connection present at all between height and weight when considering real-world data sets.

Pearson’s correlation coefficient has implications for hypothesis testing as well as other decision-making processes. By measuring the strength of a linear relationship between two variables, researchers can make informed decisions based on their findings which can help guide future research studies or inform corporate policies and practices. Pearson’s correlation coefficient also provides a basis for making predictions about future outcomes when given certain inputs or conditions–which is incredibly valuable in various business settings where predicting customer behavior or market trends is critical for success.

Pearson Correlation Coefficient – Formula

The Pearson Correlation Coefficient formula is given as the following:

Pearson Correlation Coefficients should not be taken as definitive proof that there is a relationship between two variables; rather they should only serve as indicators for further investigation which can then lead to more conclusive results regarding such relationships. In addition, Pearson Correlation Coefficients are considered reliable only when sample sizes are large enough and data points are normally distributed; if these conditions are not met then other statistical tests may be necessary in order to determine the significance of any indicated correlations.

Scatterplots & Pearson Correlation Coefficient

Scatterplots are a powerful way of visualizing data and relationships between two variables.

They are graphs that display data points in which the values for two variables are plotted against each other. The x-axis usually displays one variable, and the y-axis displays the other variable. Each point on a scatter plot represents one data set composed of the independent and dependent variables being studied; when plotted in relation to each other, these points form clusters or patterns which allow us to analyze the strength and direction of the relationship between these variables.

When plotting scatter plots, Pearson’s correlation coefficient can be used to determine how closely related two variables are to each other by measuring the degree of association between them.

Pearson’s correlation coefficient is calculated using the formula:

r = ∑(x – x̅)(y – y̅) / √∑(x – x̅)²∑ (y – y̅)²

where x̅ and y̅ represent mean values for the respective x and y values.

By examining how closely points cluster together on a scatter plot, one can measure both linearity and strength in order to determine Pearson’s correlation coefficient value. The picture below represents correlation coefficient in three different scatter plots.

The picture below might represent a very high correlation coefficient closer to 1.

The picture below might represent decently high correlation coefficient closer to 0.5.

The picture below might represent a very low correlation coefficient closer to 0.

Pearson Correlation Coefficient Examples

This example will illustrate how to use Pearson correlation coefficient (PCC) to determine the correlation between two continuous variables. In the example below, the marks of mathematics and science for a class of students in a school are considered for evaluating correlation. Based on the value of the PCC, data scientists can identify linear relationships between these two variables, providing invaluable insights about the data. Note the usage of PCC formula defined in earlier section.

Statistical Significance of Pearson Correlation Coefficient

In order to determine whether any given Pearson correlation coefficient has a statistically significant result or not, we will need to go through the following steps:

• Determine null & alternate hypothesis : The null hypothesis can be stated that there is no relationship between the two variables (r = 0) while the alternate hypothesis is that there is a relationship (r != 0).
• Determine statistics for hypothesis testing : We will calculate t-statistics and perform t-test with (n-2) degree of freedom
• Determine level of significance : The level of significance chosen is 0.05
• Calculate & compare t-statistics with critical value : We will test the significance by evaluating t-statistics and comparing it with critical value read from t-distribution table at 0.05 significance level. If the t-statistics value is greater than the critical value at 0.05, the null hypothesis can be rejected. This would mean that there is enough evidence to support the alternate hypothesis that there is some relationship between two variables.

The following is the formula for calculating the value of t-statistics for determining statistical significance of Pearson correlation coefficient:

In the above formula, r is correlation coefficient value and n is sample size. In the example given in earlier section, the t-value will come out to be  based on the following calculation. The value of n = 7 and value of r = 0.724.

$$ t = \frac{0.724\sqrt(7-2)}{\sqrt(1 – 0.724^2)} $$

$$t = 2.347 $$

One can also calculate p-value and compare it with 0.05 significance level. If p-value is less than 0.05, the Pearson correlation coefficient can be considered to be statistically significant and the null hypothesis rejected in favor of the alternate hypothesis. The degree of freedom = (N-2). 

In above example, degree of freedom = (7-2) = 5.

Thus, p-value by looking at the t-distribution table for a two-tailed test with t-value as 2.347, df = 5 comes out to be 0.0658. Thus, at a 0.05 level of significance, we don’t have enough evidence to reject the null hypothesis (no relationship between the marks). Thus, based on given evidence, one can conclude that there is a statistical evidence that the linear variables such as marks of maths and science are not strongly correlated.

Recall that a P-value is a statistic that tells us how likely an observed result is due to chance alone. If a Pearson Correlation Coefficient has an associated P-value below 0.05, then it can be considered statistically significant. This means that the Pearson Correlation Coefficient is unlikely to have occurred by chance and thus supports the hypothesis that there is indeed some kind of relationship between the two variables being studied.

Calculating Correlation Coefficient & P-value using PearsonR

Here is the Python code for calculating the correlation coefficient and p-value. The data used in the code below is NPX & PeptideAbundance which can be accessed from Kaggle competition ( Parkinson disease prediction ) . NPX (Normalized protein expression) is the frequency of the protein’s occurrence in the sample. PeptideAbundance is the frequency of the amino acid in the sample

If the p-value is less than 0.05, we reject the null hypothesis and conclude that there is a significant correlation between the two variables. Otherwise, we fail to reject the null hypothesis, and we conclude that there is no significant correlation.

In the above code, the correlation coefficient and the p-value is printed. If the p-value is less than 0.05, it is printed that the correlation is significant; otherwise, it gets printed that the correlation is not significant.

In conclusion, Pearson correlation coefficient is a powerful tool that allows for measuring the strength of linear relationships between two variables. It has implications for decision-making processes and research studies as well as real-world applications such as medicine, finance, business forecasting and lifestyle research. Pearson’s correlation coefficient also provides statistical significance testing which helps researchers make informed decisions based on their findings. Thus, Pearson correlation coefficient is an invaluable resource to have when conducting any form of quantitative analysis or data exploration.

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What is a hypothesis?

A hypothesis is an assumption that is neither proven nor disproven. In the research process, a hypothesis is made at the very beginning and the goal is to either reject or not reject the hypothesis. In order to reject or or not reject a hypothesis, data, e.g. from an experiment or a survey, are needed, which are then evaluated using a hypothesis test .

Usually, hypotheses are formulated starting from a literature review. Based on the literature review, you can then justify why you formulated the hypothesis in this way.

An example of a hypothesis could be: "Men earn more than women in the same job in Austira."

To test this hypothesis, you need data, e.g. from a survey, and a suitable hypothesis test such as the t-test or correlation analysis . Don't worry, DATAtab will help you choose the right hypothesis test.

How do I formulate a hypothesis?

In order to formulate a hypothesis, a research question must first be defined. A precisely formulated hypothesis about the population can then be derived from the research question, e.g. men earn more than women in the same job in Austria.

Hypotheses are not simple statements; they are formulated in such a way that they can be tested with collected data in the course of the research process.

To test a hypothesis, it is necessary to define exactly which variables are involved and how the variables are related. Hypotheses, then, are assumptions about the cause-and-effect relationships or the associations between variables.

What is a variable?

A variable is a property of an object or event that can take on different values. For example, the eye color is a variable, it is the property of the object eye and can take different values (blue, brown,...).

If you are researching in the social sciences, your variables may be:

• Attitude towards environmental protection

If you are researching in the medical field, your variables may be:

• Body weight
• Smoking status

What is the null and alternative hypothesis?

There are always two hypotheses that are exactly opposite to each other, or that claim the opposite. These opposite hypotheses are called null and alternative hypothesis and are abbreviated with H0 and H1 .

Null hypothesis H0:

The null hypothesis assumes that there is no difference between two or more groups with respect to a characteristic.

The salary of men and women does not differ in Austria.

Alternative hypothesis H1:

Alternative hypotheses, on the other hand, assume that there is a difference between two or more groups.

The salary of men and women differs in Austria.

The hypothesis that you want to test or that you have derived from the theory usually states that there is an effect e.g. gender has an effect on salary . This hypothesis is called an alternative hypothesis.

The null hypothesis usually states that there is no effect e.g. gender has no effect on salary . In a hypothesis test, only the null hypothesis can be tested; the goal is to find out whether the null hypothesis is rejected or not.

Types of hypotheses

What types of hypotheses are available? The most common distinction is between difference and correlation hypotheses , as well as directional and non-directional hypotheses .

Differential and correlation hypotheses

Difference hypotheses are used when different groups are to be distinguished, e.g., the group of men and the group of women. Correlation hypotheses are used when the relationship or correlation between variables is to be tested, e.g., the relationship between age and height.

Difference hypotheses

Difference hypotheses test whether there is a difference between two or more groups.

Examples of difference hypotheses are:

• The "group" of men earn more than the "group" of women.
• Smokers have a higher risk of heart attack than non-smokers
• There is a difference between Germany, Austria and France in terms of hours worked per week.

Thus, one variable is always a categorical variable, e.g., gender (male, female), smoking status (smoker, nonsmoker), or country (Germany, Austria, and France); the other variable is at least ordinally scaled, e.g., salary, percent risk of heart attack, or hours worked per week.

Correlation hypotheses

Correlation hypotheses test correlations between two variables, for example height and body weight

Correlation hypotheses are, for example:

• The taller a person is, the heavier he is.
• The more horsepower a car has, the higher its fuel consumption.
• The better the math grade, the higher the future salary.

As can be seen from the examples, correlation hypotheses often take the form "The more..., the higher/lower...". Thus, at least two ordinally scaled variables are being examined.

Directional and non-directional hypotheses

Hypotheses are divided into directional and non-directional or one-sided and two-sided hypotheses. If the hypothesis contains words like "better than" or "worse than", the hypothesis is usually directional.

In the case of a non-directional hypothesis, one often finds building blocks such as "there is a difference between" in the formulation, but it is not stated in which direction the difference lies.

• With a non-directional hypothesis , the only thing of interest is whether there is a difference in a value between the groups under consideration.
• In a directional hypothesis , what is of interest is whether one group has a higher or lower value than the other.

Non-directional hypotheses

Non-directional hypotheses test whether there is a relationship or a difference, and it does not matter in which direction the relationship or difference goes. In the case of a difference hypothesis, this means there is a difference between two groups, but it does not say whether one of the groups has a higher value.

• There is a difference between the salary of men and women (but it is not said who earns more!).
• There is a difference in the risk of heart attack between smokers and non-smokers (but it is not said who has the higher risk!).

In regard to a correlation hypothesis, this means there is a relationship or correlation between two variables, but it is not said whether this relationship is positive or negative.

• There is a correlation, between height and weight.
• There is a correlation between horsepower and fuel consumption in cars.

In both cases it is not said whether this correlation is positive or negative!

Directional hypotheses

Directional hypotheses additionally indicate the direction of the relationship or the difference. In the case of the difference hypothesis a statement is made which group has a higher or lower value.

• Men earn more than women

In the case of a correlation hypothesis, a statement is made as to whether the correlation is positive or negative.

• The taller a person is the heavier he is
• The more horsepower a car has, the higher its fuel economy

The p-value for directional hypotheses

Usually, statistical software always calculates the non-directional test and then also outputs the p-value for this.

To obtain the p-value for the directional hypothesis, it must first be checked whether the effect is in the right direction. Then the p-value must be divided by two. This is because the significance level is not split on two sides, but only on one side. More about this in the tutorial about the p-value .

If you select a directed alternative hypothesis in DATAtab for the calculated hypothesis test, the conversion is done automatically and you only need to read the result.

Step-by-step instructions for testing hypotheses

• Literature research
• Formulation of the hypothesis
• Define scale level
• Determine significance level
• Determination of hypothesis type
• Which hypothesis test is suitable for the scale level and hypothesis type?

Next tutorial about hypothesis testing

The next tutorial is about hypothesis testing. You will learn what hypothesis tests are, how to find the right one and how to interpret it.

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Cite DATAtab: DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net

Testing the Significance of Correlations

• Comparison of correlations from independent samples
• Comparison of correlations from dependent samples
• Testing linear independence (Testing against 0)
• Testing correlations against a fixed value
• Calculation of confidence intervals of correlations
• Fisher-Z-Transformation
• Calculation of the Phi correlation coefficient r Phi for categorial data
• Calculation of the weighted mean of a list of correlations
• Transformation of the effect sizes r , d , f , Odds Ratio and eta square
• Calculation of Linear Correlations

1. Comparison of correlations from independent samples

Correlations, which have been retrieved from different samples can be tested against each other. Example: Imagine, you want to test, if men increase their income considerably faster than women. You could f. e. collect the data on age and income from 1 200 men and 980 women. The correlation could amount to r = .38 in the male cohort and r = .31 in women. Is there a significant difference in the correlation of both cohorts?

(Calculation according to Eid, Gollwitzer & Schmidt, 2011 , pp. 547; single sided test)

2. Comparison of correlations from dependent samples

• 85 children from grade 3 have been tested with tests on intelligence (1), arithmetic abilities (2) and reading comprehension (3). The correlation between intelligence and arithmetic abilities amounts to r 12 = .53, intelligence and reading correlates with r 13 = .41 and arithmetic and reading with r 23 = .59. Is the correlation between intelligence an arithmetic abilities higher than the correlation between intelligence and reading comprehension?

(Calculation according to Eid et al., 2011 , S. 548 f.; single sided testing)

3. Testing linear independence (Testing against 0)

With the following calculator, you can test if correlations are different from zero. The test is based on the Student's t distribution with n - 2 degrees of freedom. An example: The length of the left foot and the nose of 18 men is quantified. The length correlates with r = .69. Is the correlation significantly different from 0?

(Calculation according to Eid et al., 2011 , S. 542; two sided test)

4. Testing correlations against a fixed value

With the following calculator, you can test if correlations are different from a fixed value. The test uses the Fisher-Z-transformation.

(Calculation according to Eid et al., 2011 , S. 543f.; two sided test)

5. Calculation of confidence intervals of correlations

The confidence interval specifies the range of values that includes a correlation with a given probability (confidence coefficient). The higher the confidence coefficient, the larger the confidence interval. Commonly, values around .9 are used.

based on Bonett & Wright (2000); cf. simulation of Gnambs (2022)

6. Fisher-Z-Transformation

The Fisher-Z-Transformation converts correlations into an almost normally distributed measure. It is necessary for many operations with correlations, f. e. when averaging a list of correlations. The following converter transforms the correlations and it computes the inverse operations as well. Please note, that the Fisher-Z is typed uppercase.

r Phi is a measure for binary data such as counts in different categories, e. g. pass/fail in an exam of males and females. It is also called contingency coefficent or Yule's Phi. Transformation to d Cohen is done via the effect size calculator .

8. Calculation of the weighted mean of a list of correlations

Due to the askew distribution of correlations(see Fisher-Z-Transformation ), the mean of a list of correlations cannot simply be calculated by building the arithmetic mean. Usually, correlations are transformed into Fisher-Z-values and weighted by the number of cases before averaging and retransforming with an inverse Fisher-Z. While this is the usual approach, Eid et al. (2011, pp. 544) suggest using the correction of Olkin & Pratt (1958) instead, as simulations showed it to estimate the mean correlation more precisely. The following calculator computes both for you, the "traditional Fisher-Z-approach" and the algorithm of Olkin and Pratt.

Please fill in the correlations into column A and the number of cases into column B. You can as well copy the values from tables of your spreadsheet program. Finally click on "OK" to start the calculation. Some values already filled in for demonstration purposes.

9. Transformation of the effect sizes r , d , f , Odds Ratio and eta square

Correlations are an effect size measure. They quantify the magnitude of an empirical effect. There are a number of other effect size measures as well, with d Cohen probably being the most prominent one. The different effect size measures can be converted into another. Please have a look at the online calculators on the page Computation of Effect Sizes .

10. Calculation of Linear Correlations

The Online-Calculator computes linear pearson or product moment correlations of two variables. Please fill in the values of variable 1 in column A and the values of variable 2 in column B and press 'OK'. As a demonstration, values for a high positive correlation are already filled in by default.

Many hypothesis tests on this page are based on Eid et al. (2011). jStat is used to generate the Student's t-distribution for testing correlations against each other. The spreadsheet element is based on Handsontable.

• Bonett, D. G., & Wright, T. A. (2000). Sample size requirements for estimating Pearson, Kendall, and Spearman correlations. Psychometrika, 65(1), 23-28. doi: 10.1007/BF0229418
• Eid, M., Gollwitzer, M., & Schmitt, M. (2011). Statistik und Forschungsmethoden Lehrbuch . Weinheim: Beltz.
• Gnambs, T. (2022, April 6). A brief note on the standard error of the Pearson correlation. https://doi.org/10.31234/osf.io/uts98

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• Published: 27 August 2024

Hypothesis of an ancient northern ocean on Mars and insights from the Zhurong rover

• Le Wang   ORCID: orcid.org/0000-0002-1640-6844 1 &
• Jun Huang   ORCID: orcid.org/0000-0003-0168-6633 1  

Nature Astronomy ( 2024 ) Cite this article

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• Geomorphology

Various landforms suggest the past presence of liquid water on the surface of Mars. The putative coastal landforms, outflow channels and the hemisphere-wide Vastitas Borealis Formation sediments indicate that the northern lowlands may have housed an ancient ocean. Challenges to this hypothesis are from topography analysis, mineral formation environment and climate modelling. Determining whether there was a northern ocean on Mars is crucial for understanding its climate history, geological processes and potential for ancient life, and for guiding future explorations. Recently, China’s Zhurong rover has identified marine sedimentary structures and multiple subsurface sedimentary layers. The unique in situ perspective of the Zhurong rover, along with previous orbital observations, provides strong support for an episodic northern ocean during the early Hesperian and early Amazonian (about 3.6–2.5 billion years ago). The ground truth from future sample-return missions, such as China’s Tianwen-3 or the Mars sample-return programmes by NASA, ESA and other agencies, will be required for a more unambiguous confirmation.

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Acknowledgements

This study was supported by the National Natural Science Foundation of China (42273041). We thank L. Xiao and J. Zhao for the discussion on an Martian ancient northern ocean.

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J.H. designed this research. J.H. and L.W. discussed and analysed the results and their implications. L.W. prepared the figures and wrote the manuscript with edits from J.H.

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Wang, L., Huang, J. Hypothesis of an ancient northern ocean on Mars and insights from the Zhurong rover. Nat Astron (2024). https://doi.org/10.1038/s41550-024-02343-3

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DOI : https://doi.org/10.1038/s41550-024-02343-3

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