statistical problem solving process example

How to Solve Statistical Problems Efficiently [Master Your Data Analysis Skills]

• November 17, 2023

Are you tired of feeling overstimulated by statistical problems? Welcome – you have now found the perfect article.

We understand the frustration that comes with trying to make sense of complex data sets.

Let’s work hand-in-hand to unpack those statistical secrets and find clarity in the numbers.

Do you find yourself stuck, unable to move forward because of statistical roadblocks? We’ve been there too. Our skill in solving statistical problems will help you find the way in through the toughest tough difficulties with confidence. Let’s tackle these problems hand-in-hand and pave the way to success.

As experts in the field, we know what it takes to conquer statistical problems effectively. This article is adjusted to meet your needs and provide you with the solutions you’ve been searching for. Join us on this voyage towards mastering statistics and unpack a world of possibilities.

Key Takeaways

• Data collection is the foundation of statistical analysis and must be accurate.
• Understanding descriptive and inferential statistics is critical for looking at and interpreting data effectively.
• Probability quantifies uncertainty and helps in making smart decisionss during statistical analysis.
• Identifying common statistical roadblocks like misinterpreting data or selecting inappropriate tests is important for effective problem-solving.
• Strategies like understanding the problem, choosing the right tools, and practicing regularly are key to tackling statistical tough difficulties.
• Using tools such as statistical software, graphing calculators, and online resources can aid in solving statistical problems efficiently.

Understanding Statistical Problems

When exploring the world of statistics, it’s critical to assimilate the nature of statistical problems. These problems often involve interpreting data, looking at patterns, and drawing meaningful endings. Here are some key points to consider:

• Data Collection: The foundation of statistical analysis lies in accurate data collection. Whether it’s surveys, experiments, or observational studies, gathering relevant data is important.
• Descriptive Statistics: Understanding descriptive statistics helps in summarizing and interpreting data effectively. Measures such as mean, median, and standard deviation provide useful ideas.
• Inferential Statistics: This branch of statistics involves making predictions or inferences about a population based on sample data. It helps us understand patterns and trends past the observed data.
• Probability: Probability is huge in statistical analysis by quantifying uncertainty. It helps us assess the likelihood of events and make smart decisionss.

To solve statistical problems proficiently, one must have a solid grasp of these key concepts.

By honing our statistical literacy and analytical skills, we can find the way in through complex data sets with confidence.

Let’s investigate more into the area of statistics and unpack its secrets.

Identifying Common Statistical Roadblocks

When tackling statistical problems, identifying common roadblocks is important to effectively find the way in the problem-solving process.

Let’s investigate some key problems individuals often encounter:

• Misinterpretation of Data: One of the primary tough difficulties is misinterpreting the data, leading to erroneous endings and flawed analysis.
• Selection of Appropriate Statistical Tests: Choosing the right statistical test can be perplexing, impacting the accuracy of results. It’s critical to have a solid understanding of when to apply each test.
• Assumptions Violation: Many statistical methods are based on certain assumptions. Violating these assumptions can skew results and mislead interpretations.

To overcome these roadblocks, it’s necessary to acquire a solid foundation in statistical principles and methodologies.

By honing our analytical skills and continuously improving our statistical literacy, we can adeptly address these tough difficulties and excel in statistical problem-solving.

For more ideas on tackling statistical problems, refer to this full guide on Common Statistical Errors .

Strategies for Tackling Statistical Tough difficulties

When facing statistical tough difficulties, it’s critical to employ effective strategies to find the way in through complex data analysis.

Here are some key approaches to tackle statistical problems:

• Understand the Problem: Before exploring analysis, ensure a clear comprehension of the statistical problem at hand.
• Choose the Right Tools: Selecting appropriate statistical tests is important for accurate results.
• Check Assumptions: Verify that the data meets the assumptions of the chosen statistical method to avoid skewed outcomes.
• Consult Resources: Refer to reputable sources like textbooks or online statistical guides for assistance.
• Practice Regularly: Improve statistical skills through consistent practice and application in various scenarios.
• Seek Guidance: When in doubt, seek advice from experienced statisticians or mentors.

By adopting these strategies, individuals can improve their problem-solving abilities and overcome statistical problems with confidence.

For further ideas on statistical problem-solving, refer to a full guide on Common Statistical Errors .

Tools for Solving Statistical Problems

When it comes to tackling statistical tough difficulties effectively, having the right tools at our disposal is important.

Here are some key tools that can aid us in solving statistical problems:

• Statistical Software: Using software like R or Python can simplify complex calculations and streamline data analysis processes.
• Graphing Calculators: These tools are handy for visualizing data and identifying trends or patterns.
• Online Resources: Websites like Kaggle or Stack Overflow offer useful ideas, tutorials, and communities for statistical problem-solving.
• Textbooks and Guides: Referencing textbooks such as “Introduction to Statistical Learning” or online guides can provide in-depth explanations and step-by-step solutions.

By using these tools effectively, we can improve our problem-solving capabilities and approach statistical tough difficulties with confidence.

For further ideas on common statistical errors to avoid, we recommend checking out the full guide on Common Statistical Errors For useful tips and strategies.

Putting in place Effective Solutions

When approaching statistical problems, it’s critical to have a strategic plan in place.

Here are some key steps to consider for putting in place effective solutions:

• Define the Problem: Clearly outline the statistical problem at hand to understand its scope and requirements fully.
• Collect Data: Gather relevant data sets from credible sources or conduct surveys to acquire the necessary information for analysis.
• Choose the Right Model: Select the appropriate statistical model based on the nature of the data and the specific question being addressed.
• Use Advanced Tools: Use statistical software such as R or Python to perform complex analyses and generate accurate results.
• Validate Results: Verify the accuracy of the findings through strict testing and validation procedures to ensure the reliability of the endings.

By following these steps, we can streamline the statistical problem-solving process and arrive at well-informed and data-driven decisions.

For further ideas and strategies on tackling statistical tough difficulties, we recommend exploring resources such as DataCamp That offer interactive learning experiences and tutorials on statistical analysis.

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• Knowledge Base

The Beginner's Guide to Statistical Analysis | 5 Steps & Examples

Statistical analysis means investigating trends, patterns, and relationships using quantitative data . It is an important research tool used by scientists, governments, businesses, and other organizations.

To draw valid conclusions, statistical analysis requires careful planning from the very start of the research process . You need to specify your hypotheses and make decisions about your research design, sample size, and sampling procedure.

After collecting data from your sample, you can organize and summarize the data using descriptive statistics . Then, you can use inferential statistics to formally test hypotheses and make estimates about the population. Finally, you can interpret and generalize your findings.

This article is a practical introduction to statistical analysis for students and researchers. We’ll walk you through the steps using two research examples. The first investigates a potential cause-and-effect relationship, while the second investigates a potential correlation between variables.

Table of contents

Step 1: write your hypotheses and plan your research design, step 2: collect data from a sample, step 3: summarize your data with descriptive statistics, step 4: test hypotheses or make estimates with inferential statistics, step 5: interpret your results, other interesting articles.

To collect valid data for statistical analysis, you first need to specify your hypotheses and plan out your research design.

Writing statistical hypotheses

The goal of research is often to investigate a relationship between variables within a population . You start with a prediction, and use statistical analysis to test that prediction.

A statistical hypothesis is a formal way of writing a prediction about a population. Every research prediction is rephrased into null and alternative hypotheses that can be tested using sample data.

While the null hypothesis always predicts no effect or no relationship between variables, the alternative hypothesis states your research prediction of an effect or relationship.

• Null hypothesis: A 5-minute meditation exercise will have no effect on math test scores in teenagers.
• Alternative hypothesis: A 5-minute meditation exercise will improve math test scores in teenagers.
• Null hypothesis: Parental income and GPA have no relationship with each other in college students.
• Alternative hypothesis: Parental income and GPA are positively correlated in college students.

Planning your research design

A research design is your overall strategy for data collection and analysis. It determines the statistical tests you can use to test your hypothesis later on.

First, decide whether your research will use a descriptive, correlational, or experimental design. Experiments directly influence variables, whereas descriptive and correlational studies only measure variables.

• In an experimental design , you can assess a cause-and-effect relationship (e.g., the effect of meditation on test scores) using statistical tests of comparison or regression.
• In a correlational design , you can explore relationships between variables (e.g., parental income and GPA) without any assumption of causality using correlation coefficients and significance tests.
• In a descriptive design , you can study the characteristics of a population or phenomenon (e.g., the prevalence of anxiety in U.S. college students) using statistical tests to draw inferences from sample data.

Your research design also concerns whether you’ll compare participants at the group level or individual level, or both.

• In a between-subjects design , you compare the group-level outcomes of participants who have been exposed to different treatments (e.g., those who performed a meditation exercise vs those who didn’t).
• In a within-subjects design , you compare repeated measures from participants who have participated in all treatments of a study (e.g., scores from before and after performing a meditation exercise).
• In a mixed (factorial) design , one variable is altered between subjects and another is altered within subjects (e.g., pretest and posttest scores from participants who either did or didn’t do a meditation exercise).
• Experimental
• Correlational

First, you’ll take baseline test scores from participants. Then, your participants will undergo a 5-minute meditation exercise. Finally, you’ll record participants’ scores from a second math test.

In this experiment, the independent variable is the 5-minute meditation exercise, and the dependent variable is the math test score from before and after the intervention. Example: Correlational research design In a correlational study, you test whether there is a relationship between parental income and GPA in graduating college students. To collect your data, you will ask participants to fill in a survey and self-report their parents’ incomes and their own GPA.

Measuring variables

When planning a research design, you should operationalize your variables and decide exactly how you will measure them.

For statistical analysis, it’s important to consider the level of measurement of your variables, which tells you what kind of data they contain:

• Categorical data represents groupings. These may be nominal (e.g., gender) or ordinal (e.g. level of language ability).
• Quantitative data represents amounts. These may be on an interval scale (e.g. test score) or a ratio scale (e.g. age).

Many variables can be measured at different levels of precision. For example, age data can be quantitative (8 years old) or categorical (young). If a variable is coded numerically (e.g., level of agreement from 1–5), it doesn’t automatically mean that it’s quantitative instead of categorical.

Identifying the measurement level is important for choosing appropriate statistics and hypothesis tests. For example, you can calculate a mean score with quantitative data, but not with categorical data.

In a research study, along with measures of your variables of interest, you’ll often collect data on relevant participant characteristics.

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In most cases, it’s too difficult or expensive to collect data from every member of the population you’re interested in studying. Instead, you’ll collect data from a sample.

Statistical analysis allows you to apply your findings beyond your own sample as long as you use appropriate sampling procedures . You should aim for a sample that is representative of the population.

Sampling for statistical analysis

There are two main approaches to selecting a sample.

• Probability sampling: every member of the population has a chance of being selected for the study through random selection.
• Non-probability sampling: some members of the population are more likely than others to be selected for the study because of criteria such as convenience or voluntary self-selection.

In theory, for highly generalizable findings, you should use a probability sampling method. Random selection reduces several types of research bias , like sampling bias , and ensures that data from your sample is actually typical of the population. Parametric tests can be used to make strong statistical inferences when data are collected using probability sampling.

But in practice, it’s rarely possible to gather the ideal sample. While non-probability samples are more likely to at risk for biases like self-selection bias , they are much easier to recruit and collect data from. Non-parametric tests are more appropriate for non-probability samples, but they result in weaker inferences about the population.

If you want to use parametric tests for non-probability samples, you have to make the case that:

• your sample is representative of the population you’re generalizing your findings to.
• your sample lacks systematic bias.

Keep in mind that external validity means that you can only generalize your conclusions to others who share the characteristics of your sample. For instance, results from Western, Educated, Industrialized, Rich and Democratic samples (e.g., college students in the US) aren’t automatically applicable to all non-WEIRD populations.

If you apply parametric tests to data from non-probability samples, be sure to elaborate on the limitations of how far your results can be generalized in your discussion section .

Create an appropriate sampling procedure

Based on the resources available for your research, decide on how you’ll recruit participants.

• Will you have resources to advertise your study widely, including outside of your university setting?
• Will you have the means to recruit a diverse sample that represents a broad population?
• Do you have time to contact and follow up with members of hard-to-reach groups?

Your participants are self-selected by their schools. Although you’re using a non-probability sample, you aim for a diverse and representative sample. Example: Sampling (correlational study) Your main population of interest is male college students in the US. Using social media advertising, you recruit senior-year male college students from a smaller subpopulation: seven universities in the Boston area.

Calculate sufficient sample size

Before recruiting participants, decide on your sample size either by looking at other studies in your field or using statistics. A sample that’s too small may be unrepresentative of the sample, while a sample that’s too large will be more costly than necessary.

There are many sample size calculators online. Different formulas are used depending on whether you have subgroups or how rigorous your study should be (e.g., in clinical research). As a rule of thumb, a minimum of 30 units or more per subgroup is necessary.

To use these calculators, you have to understand and input these key components:

• Significance level (alpha): the risk of rejecting a true null hypothesis that you are willing to take, usually set at 5%.
• Statistical power : the probability of your study detecting an effect of a certain size if there is one, usually 80% or higher.
• Expected effect size : a standardized indication of how large the expected result of your study will be, usually based on other similar studies.
• Population standard deviation: an estimate of the population parameter based on a previous study or a pilot study of your own.

Once you’ve collected all of your data, you can inspect them and calculate descriptive statistics that summarize them.

Inspect your data

There are various ways to inspect your data, including the following:

• Organizing data from each variable in frequency distribution tables .
• Displaying data from a key variable in a bar chart to view the distribution of responses.
• Visualizing the relationship between two variables using a scatter plot .

By visualizing your data in tables and graphs, you can assess whether your data follow a skewed or normal distribution and whether there are any outliers or missing data.

A normal distribution means that your data are symmetrically distributed around a center where most values lie, with the values tapering off at the tail ends.

In contrast, a skewed distribution is asymmetric and has more values on one end than the other. The shape of the distribution is important to keep in mind because only some descriptive statistics should be used with skewed distributions.

Extreme outliers can also produce misleading statistics, so you may need a systematic approach to dealing with these values.

Calculate measures of central tendency

Measures of central tendency describe where most of the values in a data set lie. Three main measures of central tendency are often reported:

• Mode : the most popular response or value in the data set.
• Median : the value in the exact middle of the data set when ordered from low to high.
• Mean : the sum of all values divided by the number of values.

However, depending on the shape of the distribution and level of measurement, only one or two of these measures may be appropriate. For example, many demographic characteristics can only be described using the mode or proportions, while a variable like reaction time may not have a mode at all.

Calculate measures of variability

Measures of variability tell you how spread out the values in a data set are. Four main measures of variability are often reported:

• Range : the highest value minus the lowest value of the data set.
• Interquartile range : the range of the middle half of the data set.
• Standard deviation : the average distance between each value in your data set and the mean.
• Variance : the square of the standard deviation.

Once again, the shape of the distribution and level of measurement should guide your choice of variability statistics. The interquartile range is the best measure for skewed distributions, while standard deviation and variance provide the best information for normal distributions.

Using your table, you should check whether the units of the descriptive statistics are comparable for pretest and posttest scores. For example, are the variance levels similar across the groups? Are there any extreme values? If there are, you may need to identify and remove extreme outliers in your data set or transform your data before performing a statistical test.

From this table, we can see that the mean score increased after the meditation exercise, and the variances of the two scores are comparable. Next, we can perform a statistical test to find out if this improvement in test scores is statistically significant in the population. Example: Descriptive statistics (correlational study) After collecting data from 653 students, you tabulate descriptive statistics for annual parental income and GPA.

It’s important to check whether you have a broad range of data points. If you don’t, your data may be skewed towards some groups more than others (e.g., high academic achievers), and only limited inferences can be made about a relationship.

A number that describes a sample is called a statistic , while a number describing a population is called a parameter . Using inferential statistics , you can make conclusions about population parameters based on sample statistics.

Researchers often use two main methods (simultaneously) to make inferences in statistics.

• Estimation: calculating population parameters based on sample statistics.
• Hypothesis testing: a formal process for testing research predictions about the population using samples.

You can make two types of estimates of population parameters from sample statistics:

• A point estimate : a value that represents your best guess of the exact parameter.
• An interval estimate : a range of values that represent your best guess of where the parameter lies.

If your aim is to infer and report population characteristics from sample data, it’s best to use both point and interval estimates in your paper.

You can consider a sample statistic a point estimate for the population parameter when you have a representative sample (e.g., in a wide public opinion poll, the proportion of a sample that supports the current government is taken as the population proportion of government supporters).

There’s always error involved in estimation, so you should also provide a confidence interval as an interval estimate to show the variability around a point estimate.

A confidence interval uses the standard error and the z score from the standard normal distribution to convey where you’d generally expect to find the population parameter most of the time.

Hypothesis testing

Using data from a sample, you can test hypotheses about relationships between variables in the population. Hypothesis testing starts with the assumption that the null hypothesis is true in the population, and you use statistical tests to assess whether the null hypothesis can be rejected or not.

Statistical tests determine where your sample data would lie on an expected distribution of sample data if the null hypothesis were true. These tests give two main outputs:

• A test statistic tells you how much your data differs from the null hypothesis of the test.
• A p value tells you the likelihood of obtaining your results if the null hypothesis is actually true in the population.

Statistical tests come in three main varieties:

• Comparison tests assess group differences in outcomes.
• Regression tests assess cause-and-effect relationships between variables.
• Correlation tests assess relationships between variables without assuming causation.

Your choice of statistical test depends on your research questions, research design, sampling method, and data characteristics.

Parametric tests

Parametric tests make powerful inferences about the population based on sample data. But to use them, some assumptions must be met, and only some types of variables can be used. If your data violate these assumptions, you can perform appropriate data transformations or use alternative non-parametric tests instead.

A regression models the extent to which changes in a predictor variable results in changes in outcome variable(s).

• A simple linear regression includes one predictor variable and one outcome variable.
• A multiple linear regression includes two or more predictor variables and one outcome variable.

Comparison tests usually compare the means of groups. These may be the means of different groups within a sample (e.g., a treatment and control group), the means of one sample group taken at different times (e.g., pretest and posttest scores), or a sample mean and a population mean.

• A t test is for exactly 1 or 2 groups when the sample is small (30 or less).
• A z test is for exactly 1 or 2 groups when the sample is large.
• An ANOVA is for 3 or more groups.

The z and t tests have subtypes based on the number and types of samples and the hypotheses:

• If you have only one sample that you want to compare to a population mean, use a one-sample test .
• If you have paired measurements (within-subjects design), use a dependent (paired) samples test .
• If you have completely separate measurements from two unmatched groups (between-subjects design), use an independent (unpaired) samples test .
• If you expect a difference between groups in a specific direction, use a one-tailed test .
• If you don’t have any expectations for the direction of a difference between groups, use a two-tailed test .

The only parametric correlation test is Pearson’s r . The correlation coefficient ( r ) tells you the strength of a linear relationship between two quantitative variables.

However, to test whether the correlation in the sample is strong enough to be important in the population, you also need to perform a significance test of the correlation coefficient, usually a t test, to obtain a p value. This test uses your sample size to calculate how much the correlation coefficient differs from zero in the population.

You use a dependent-samples, one-tailed t test to assess whether the meditation exercise significantly improved math test scores. The test gives you:

• a t value (test statistic) of 3.00
• a p value of 0.0028

Although Pearson’s r is a test statistic, it doesn’t tell you anything about how significant the correlation is in the population. You also need to test whether this sample correlation coefficient is large enough to demonstrate a correlation in the population.

A t test can also determine how significantly a correlation coefficient differs from zero based on sample size. Since you expect a positive correlation between parental income and GPA, you use a one-sample, one-tailed t test. The t test gives you:

• a t value of 3.08
• a p value of 0.001

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The final step of statistical analysis is interpreting your results.

Statistical significance

In hypothesis testing, statistical significance is the main criterion for forming conclusions. You compare your p value to a set significance level (usually 0.05) to decide whether your results are statistically significant or non-significant.

Statistically significant results are considered unlikely to have arisen solely due to chance. There is only a very low chance of such a result occurring if the null hypothesis is true in the population.

This means that you believe the meditation intervention, rather than random factors, directly caused the increase in test scores. Example: Interpret your results (correlational study) You compare your p value of 0.001 to your significance threshold of 0.05. With a p value under this threshold, you can reject the null hypothesis. This indicates a statistically significant correlation between parental income and GPA in male college students.

Note that correlation doesn’t always mean causation, because there are often many underlying factors contributing to a complex variable like GPA. Even if one variable is related to another, this may be because of a third variable influencing both of them, or indirect links between the two variables.

Effect size

A statistically significant result doesn’t necessarily mean that there are important real life applications or clinical outcomes for a finding.

In contrast, the effect size indicates the practical significance of your results. It’s important to report effect sizes along with your inferential statistics for a complete picture of your results. You should also report interval estimates of effect sizes if you’re writing an APA style paper .

With a Cohen’s d of 0.72, there’s medium to high practical significance to your finding that the meditation exercise improved test scores. Example: Effect size (correlational study) To determine the effect size of the correlation coefficient, you compare your Pearson’s r value to Cohen’s effect size criteria.

Decision errors

Type I and Type II errors are mistakes made in research conclusions. A Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s false.

You can aim to minimize the risk of these errors by selecting an optimal significance level and ensuring high power . However, there’s a trade-off between the two errors, so a fine balance is necessary.

Frequentist versus Bayesian statistics

Traditionally, frequentist statistics emphasizes null hypothesis significance testing and always starts with the assumption of a true null hypothesis.

However, Bayesian statistics has grown in popularity as an alternative approach in the last few decades. In this approach, you use previous research to continually update your hypotheses based on your expectations and observations.

Bayes factor compares the relative strength of evidence for the null versus the alternative hypothesis rather than making a conclusion about rejecting the null hypothesis or not.

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.

• Student’s  t -distribution
• Normal distribution
• Null and Alternative Hypotheses
• Chi square tests
• Confidence interval

Methodology

• Cluster sampling
• Stratified sampling
• Data cleansing
• Reproducibility vs Replicability
• Peer review
• Likert scale

Research bias

• Implicit bias
• Framing effect
• Cognitive bias
• Placebo effect
• Hawthorne effect
• Hostile attribution bias
• Affect heuristic

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Statistical Problem Solving (SPS)

• Statistical Problem Solving

Problem solving in any organization is a problem. Nobody wants to own the responsibility for a problem and that is the reason, when a problem shows up fingers may be pointing at others rather than self.

This is a natural human instinctive defense mechanism and hence cannot hold it against any one. However, it is to be realized the problems in industry are real and cannot be wished away, solution must be sought either by hunch or by scientific methods. Only a systematic disciplined approach for defining and solving problems consistently and effectively reveal the real nature of a problem and the best possible solutions .

A Chinese proverb says, “ it is cheap to do guesswork for solution, but a wrong guess can be very expensive”. This is to emphasize that although occasional success is possible trough hunches gained through long years of experience in doing the same job, but a lasting solution is possible only through scientific methods.

One of the major scientific method for problem solving is through Statistical Problem Solving (SPS) this method is aimed at not only solving problems but may be used for improvement on existing situation. It involves a team armed with process and product knowledge, having willingness to work together as a team, can undertake selection of some statistical methods, have willingness to adhere to principles of economy and willingness to learn along the way.

Statistical Problem Solving (SPS) could be used for process control or product control. In many situations, the product would be customer dictated, tried, tested and standardized in the facility may involve testing at both internal to facility or external to facility may be complex and may require customer approval for changes which could be time consuming and complex. But if the problem warrants then this should be taken up. 

Process controls are lot simpler than product control where SPS may be used effectively for improving profitability of the industry, by reducing costs and possibly eliminating all 7 types of waste through use of Kaizen and lean management techniques.

The following could be used as 7 steps for Statistical Problem Solving (SPS)

• Defining the problem
• Listing variables
• Prioritizing variables
• Evaluating top few variables
• Optimizing variable settings
• Monitor and Measure results
• Reward/Recognize Team members

Defining the problem: Source for defining the problem could be from customer complaints, in-house rejections, observations by team lead or supervisor or QC personnel, levels of waste generated or such similar factors.

Listing and prioritizing variables involves all features associated with the processes. Example temperature, feed and speed of the machine, environmental factors, operator skills etc. It may be difficult to try and find solution for all variables together. Hence most probable variables are to be selected based on collective wisdom and experience of the team attempting to solve the problem.

Collection of data: Most common method in collecting data is the X bar and R charts.  Time is used as the variable in most cases and plotted on X axis, and other variables such as dimensions etc. are plotted graphically as shown in example below.

Once data is collected based on probable list of variables, then the data is brought to the attention of the team for brainstorming on what variables are to be controlled and how solution could be obtained. In other words , optimizing variables settings . Based on the brainstorming session process control variables are evaluated using popular techniques like “5 why”, “8D”, “Pareto Analysis”, “Ishikawa diagram”, “Histogram” etc. The techniques are used to limit variables and design the experiments and collect data again. Values of variables are identified from data which shows improvement. This would lead to narrowing down the variables and modify the processes, to achieve improvement continually. The solutions suggested are to be implemented and results are to be recorded. This data is to be measured at varying intervals to see the status of implementation and the progress of improvement is to be monitored till the suggested improvements become normal routine. When results indicate resolution of problem and the rsults are consistent then Team memebres are to be rewarded and recognized to keep up their morale for future projects.

Who Should Pursue SPS

• Statistical Problem Solving can be pursued by a senior leadership group for example group of quality executives meeting weekly to review quality issues, identify opportunities for costs saving and generate ideas for working smarter across the divisions
• Statistical Problem solving can equally be pursued by a staff work group within an institution that possesses a diversity of experience that can gather data on various product features and tabulate them statistically for drawing conclusions
• The staff work group proposes methods for rethinking and reworking models of collaboration and consultation at the facility
• The senior leadership group and staff work group work in partnership with university faculty and staff to identify research communications and solve problems across the organization

Benefits of Statistical Problem Solving

• Long term commitment to organizations and companies to work smarter.
• Reduces costs, enhances services and increases revenues.
• Mitigating the impact of budget reductions while at the same time reducing operational costs.
• Improving operations and processes, resulting in a more efficient, less redundant organization.
• Promotion of entrepreneurship intelligence, risk taking corporations and engagement across interactions with business and community partners.
• A culture change in a way a business or organization collaborates both internally and externally.
• Identification and solving of problems.
• Helps to repetition of problems
• Meets the mandatory requirement for using scientific methods for problem solving
• Savings in revenue by reducing quality costs
• Ultimate improvement in Bottom -Line
• Improvement in teamwork and morale in working
• Improvement in overall problem solving instead of harping on accountability

Business Impact

• Scientific data backed up problem solving techniques puts the business at higher pedestal in the eyes of the customer.
• Eradication of over consulting within businesses and organizations which may become a pitfall especially where it affects speed of information.
• Eradication of blame game

QSE’s Approach to Statistical Problem Solving

By leveraging vast experience, it has, QSE organizes the entire implementation process for Statistical Problem Solving in to Seven simple steps

• Define the Problem
• List Suspect Variables
• Prioritize Selected Variables
• Evaluate Critical Variables
• Optimize Critical Variables
• Monitor and Measure Results
• Reward/Recognize Team Members
• Define the Problem (Vital Few -Trivial Many):

List All the problems which may be hindering Operational Excellence . Place them in a Histogram under as many categories as required.

Select Problems based on a simple principle of Vital Few that is select few problems which contribute to most deficiencies within the facility

QSE advises on how to Use X and R Charts to gather process data.

• List Suspect Variables:

QSE Advises on how to gather data for the suspect variables involving cross functional teams and available past data

• Prioritize Selected Variables Using Cause and Effect Analysis:

QSE helps organizations to come up prioritization of select variables that are creating the problem and the effect that are caused by them. The details of this exercise are to be represented in the Fishbone Diagram or Ishikawa Diagram

• Evaluate Critical Variables:

Use Brain Storming method to use critical variables for collecting process data and Incremental Improvement for each selected critical variable

QSE with its vast experiences guides and conducts brain storming sessions in the facility to identify KAIZEN (Small Incremental projects) to bring in improvements. Create a bench mark to be achieved through the suggested improvement projects

• Optimize Critical Variable Through Implementing the Incremental Improvements:

QSE helps facilities to implement incremental improvements and gather data to see the results of the efforts in improvements

• Monitor and Measure to Collect Data on Consolidated incremental achievements :

Consolidate and make the major change incorporating all incremental improvements and then gather data again to see if the benchmarks have been reached

QSE educates and assists the teams on how these can be done in a scientific manner using lean and six sigma techniques

QSE organizes verification of Data to compare the results from the original results at the start of the projects. Verify if the suggestions incorporated are repeatable for same or better results as planned

              Validate the improvement project by multiple repetitions

• Reward and Recognize Team Members:

QSE will provide all kinds of support in identifying the great contributors to the success of the projects and make recommendation to the Management to recognize the efforts in a manner which befits the organization to keep up the morale of the contributors.

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ISO Standards

• ISO 9001:2015
• ISO 10993-1:2018
• ISO 13485:2016
• ISO 14001:2015
• ISO 15189:2018
• ISO 15190:2020
• ISO 15378:2017
• ISO/IEC 17020:2012
• ISO/IEC 17025:2017
• ISO 20000-1:2018
• ISO 22000:2018
• ISO 22301:2019
• ISO 27001:2015
• ISO 27701:2019
• ISO 28001:2007
• ISO 37001:2016
• ISO 45001:2018
• ISO 50001:2018
• ISO 55001:2014

Telecommunication Standards

• TL 9000 Version 6.1

Automotive Standards

• IATF 16949:2016
• ISO/SAE 21434:2021

Aerospace Standards

Forestry standards.

• FSC - Forest Stewardship Council
• PEFC - Program for the Endorsement of Forest Certification
• SFI - Sustainable Forest Initiative

Steel Construction Standards

Food safety standards.

• FDA Gluten Free Labeling & Certification
• Hygeine Excellence & Sanitation Excellence

GFSI Recognized Standards

• BRC Version 9
• FSSC 22000:2019
• Hygeine Excellent & Sanitation Excellence
• IFS Version 7
• SQF Edition 9
• All GFSI Recognized Standards for Packaging Industries

Problem Solving Tools

• Corrective & Preventative Actions
• Root Cause Analysis
• Supplier Development

Excellence Tools

• Bottom Line Improvement
• Customer Satisfaction Measurement
• Document Simplification
• Hygiene Excellence & Sanitation
• Lean & Six Sigma
• Malcom Baldridge National Quality Award
• Operational Excellence
• Safety (including STOP and OHSAS 45001)
• Sustainability (Reduce, Reuse, & Recycle)
• Total Productive Maintenance

Other Standards

• California Transparency Act
• Global Organic Textile Standard (GOTS)
• Hemp & Cannabis Management Systems
• Recycling & Re-Using Electronics
• ESG - Environmental, Social & Governance
• CDFA Proposition 12 Animal Welfare

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QSE has helped over 800 companies across North America achieve certification utilizing our unique 10-Step Approach ™ to management system consulting. Schedule a consultation and learn how we can help you achieve your goals as quickly, simply and easily as possible.

IndustryWired

Problem-solving in statistics: What You Need to Know

Mastering Problem-solving in Statistics: Techniques and Strategies for Effective Data Analysis

Statistics are not just statistics; it’s a powerful tool that helps us understand data and draw meaningful conclusions. Whether you are a student, a researcher, or an entrepreneur, it is important to understand how to solve problems mathematically. This article will teach you the basic concepts and techniques you need to know to solve mathematical problems effectively.

Understanding the Basics

It is important to have a solid understanding of basic mathematical concepts before engaging in problem-solving. These include:

• Types of data: Data can be qualitative (categorical) or quantitative (numeric). Understanding the type of data you are working with is the first step in any statistical analysis .
• Descriptive vs Inferential Statistics: Descriptive statistics summarize data (mean, median, mode), while inferential statistics help make predictions or inferences about a population based on a sample.
• Probability: The principle of probability is the basis for most statistics. Knowing how to estimate and interpret probability is key to understanding statistical results.

Forming the Problem

Effective problem-solving begins with a clear understanding of the problem at hand. This includes:

• Definition of the goal: What question are you trying to answer? It is important to clearly state your research question or hypothesis.
• Specify Variables: Display dependent and independent variables. Understanding how these variables interact will guide your research.
• Data Collection: Data collection methods must be appropriate for your research question. Make sure your data is reliable and accurate.

Selecting the Appropriate Accounting Method

Different problems require different mathematical methods. Common methods include:

• Regression analysis : Used to examine the relationship between a dependent variable and one or more independent variables.
• ANOVA (Analysis of Variance): It helps to compare means in different groups.
• Chi-Square Test: Used for categorical data to assess the likelihood of the observed distribution being random.
• T-test: Compare the means of the two groups to see if they are statistically different. The best method to choose depends on the nature of your data and the research question.

Data analysis

Once you’ve chosen the right statistical method, it’s time to analyze the data. This includes:

• Descriptive statistics: Begin with measures of central characteristics (mean, median, mode) and variance (range, variance, standard deviation).
• Running statistical tests: Use selected statistical tests to determine relationships, differences, or trends in the data.
• Interpretation of results: Understand the implications of the results in terms of your research question. Monitor p-values, confidence intervals, and effect sizes.

Solving common problems

Solving mathematical problems often requires solving difficulties such as:

• Outliers: Extreme values ​​can skew the results. Consider whether outliers should be eliminated or accounted for in your analysis.
• Missing data: Missing data can bias the results. Use imputation methods or sensitivity analysis to address this issue.
• Assumptions: Many statistical tests are based on assumptions (e.g., normality, homogeneity). Ensure that these assumptions are met before interpreting the results.

Communication of findings

The final step in solving statistical problems is to articulate your findings. This includes:

• Visualizing data : Use graphs and charts to make results more meaningful.
• Report Writing: Present your findings clearly and concisely, including a description of the methods used and the results.
• Make decisions: Based on your research, make appropriate decisions or recommendations. Make sure your conclusion is supported by strong evidence.

Continuous learning

Accounting is a dynamic field with continuous improvement. Keeping abreast of new methods, tools and techniques will enhance your problem-solving skills. Consider taking classes, attending workshops, or joining professional organizations to keep your skills up to date.

Conclusion: Problem solving in mathematics is a process that involves understanding the basics, formulating the problem, choosing the best method, analysing data , and discussing findings successful completion of these steps will ensure success in mathematics solve issues and make informed decisions.

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Guide: Statistical Process Control (SPC)

Author: Daniel Croft

Daniel Croft is an experienced continuous improvement manager with a Lean Six Sigma Black Belt and a Bachelor's degree in Business Management. With more than ten years of experience applying his skills across various industries, Daniel specializes in optimizing processes and improving efficiency. His approach combines practical experience with a deep understanding of business fundamentals to drive meaningful change.

Statistical Process Control is a key method used in making sure products are made to specification and efficiently, especially in making expensive products like cars or electronics where margins can be low and the cost of defects can eliminate profitaility. It uses a detailed, numbers-based way to monitor, manage, and improve processes where product are made or services are provided.

SPC focuses on carefully examining data to help businesses understand problems or issues in how they make products or provide services. It’s all about making decisions based on solid evidence rather than guesses or feelings, with the goal of finding and fixing any changes in the process that could affect the quality of the product.

What is Statistical Process Control?

Statistical Proces Control is a key method used in quality control, and Lean Six Sigma used to maintain and improve product quality and efficiency. It is used in various industries but is primarily used in manufacturing as a systematic, data-driven approach to uses statistical methods to monitor, control, and improve processes and process performance. 

SPC is a data-driven methodology that relies heavily on data to analyze process performance. By collecting and analyzing data from various stages of a manufacturing or service process, SPC enables organizations to identify trends, patterns, and anomalies. This data-driven approach helps in making informed decisions rather than relying on assumptions or estimations.

The main aim of SPC is to detect and reduce process variability. Variability is a natural aspect of any process, but excessive variability can lead to defects, inefficiency, and reduced product quality. By understanding and controlling this variability, organizations can ensure that their processes consistently produce items within desired specifications. SPC involves continuous monitoring of the process to quickly identify deviations from the norm. This systematic approach ensures that problems are detected early and can be rectified before they result in significant quality issues or production waste.

History and Background of the Development of SPC

Walter shewhart and control charts.

The foundations of SPC were laid in the early 20th century by Walter Shewhart working at Bell Laboratories. Shewhart’s primary contribution was the development of the control chart , a tool that graphically displays process data over time and helps in distinguishing between normal process variation and variation that signifies a problem. The control chart remains a cornerstone of SPC and is widely used in various industries to monitor process performance.

W. Edwards Deming and Post-War Japan

After World War II, W. Edwards Deming brought the concepts of SPC to Japan, where they played a key role in the country’s post-war industrial rebirth. Deming’s teachings emphasized not only statistical methods but also a broader philosophical approach to quality. He advocated for continuous improvement ( Kaizen ) and total quality management , integrating SPC into a more comprehensive quality management system.

Impact on Manufacturing and Beyond

The implementation of SPC led to significant improvements in manufacturing quality and efficiency. It allowed companies to produce goods more consistently and with fewer defects. The principles of SPC have since been adopted in various sectors beyond manufacturing, including healthcare, finance, and service industries, demonstrating its versatility and effectiveness in process improvement.

Fundamental Concepts of SPC

To understand and read what control charts are telling you it is first important to understand how variation and how it might be displayed on the chart. In everything, there is always a level of variation relative to what is being measured. But we must identify what acceptable variation and what is a variation that needs to be explored and addressed.

Understanding Process Variation

In SPC, process variation is categorized into two types: common causes and special causes.

• Common Cause Variation: These are inherent variations that occur naturally within a process. They are predictable and consistent over time. In the image below common cause variation is the variation within the control limits
• Special Cause Variation: These variations are due to external factors and are not part of the normal process. They are unpredictable and can indicate that the process is out of control. In the image below, the special cause variation is the data point outside the upper control limit 

Control Charts

Control charts are essential tools in SPC, used to monitor whether a process is in control.

• Graphical Representation: They graphically represent data over time, providing a visual means to monitor process performance.
• Control Limits: Control charts use upper and lower control limits, which are statistically derived boundaries. They help in distinguishing between normal process variation (within limits) and variations that require attention (outside limits).

Types of Control Charts

After understanding the types of variation you might find on a control chat, it is important to understand the types of control charts in SPC. This is crucial for effectively monitoring and improving various processes. These charts are broadly categorized based on the type of data they handle: variable data and attribute data. Additionally, the implementation of SPC in a process, from data collection to continuous improvement, is a systematic approach that requires diligence and precision. Let’s explore these aspects in more detail.

Variable Data Control Charts

Variable data control charts are used for data that can be measured on a continuous scale. This includes characteristics like weight, length, or time.

X-bar and R Chart

• Purpose: Used to monitor the mean (average) and range of a process.
• Application: Ideal for situations where sample measurements are taken at regular intervals and the mean and variability of the process need to be controlled.
• Structure: The X-bar chart shows how the mean changes over time, while the R chart displays the range (difference between the highest and lowest values) within each sample.

Individual-Moving Range (I-MR) Chart

• Purpose: Monitors individual observations and the moving range between observations.
• Application: Useful when measurements are not made in subgroups but as individual data points.
• Structure: The I chart tracks each individual data point, and the MR chart shows the range between consecutive measurements.

Attribute Data Control Charts

Attribute data control charts are used for data that are counted, such as defects or defective items.

P-chart (Proportion Chart)

• Purpose: Monitors the proportion of defective items in a sample.
• Application: Ideal for quality characteristics that are categorical (e.g., defective vs. non-defective) and when the sample size varies.
• Structure: It plots the proportion of defectives in each sample over time.

C-chart (Count Chart)

• Purpose: Tracks the count of defects per unit or item.
• Application: Used when the number of opportunities for defects is constant, and defects are counted per item or unit.
• Structure: It plots the number of defects in samples of a constant size.

Implementing SPC in a Process

Implementing SPC in a process is a structured approach that involves several key steps: data collection, establishing control limits, monitoring and interpretation, and continuous improvement. Each of these steps is critical to the successful application of SPC. Let’s explore these steps in more detail.

Step 1: Data Collection

First, data must be collected systematically to ensure it accurately represents the process. This involves deciding what data to collect, how often to collect it, and the methods used for collection. The selection of data is important. It should be relevant to the quality characteristics you want to control. For example, in a manufacturing process, this might include measurements of product dimensions, the time taken for a process step, or the number of defects.

The data should be representative of the actual operating conditions of the process. It means collecting data under various operating conditions and over a sufficient period.

The sample size and frequency of data collection should be adequate to capture the variability of the process. It’s a balance between collecting enough data for reliability and the practicality of data collection.

Step 2: Establishing Control Limits

Control limits are calculated using historical process data. They are statistical representations of the process variability and are usually set at ±3 standard deviations from the process mean.

These limits reflect what the process can achieve under current operating conditions.

To help you calculate your data control limits, you can use our Control Limits Calculator.

Control limits are not fixed forever. As process improvements are made, these limits may be recalculated to reflect the new level of process performance.

When significant changes are made to a process (like new machinery, materials, or methods), it might be necessary to recalculate the control limits based on new performance data.

Step 3: Monitoring and Interpretation

Regularly reviewing control charts is essential for timely detection of out-of-control conditions. Apart from individual points, it’s crucial to look for patterns or trends in the data, which could indicate potential issues.

When data points fall outside the control limits or exhibit non-random patterns, it triggers a need for investigation. The goal is to identify the root cause of the variation, whether it’s a common cause that requires a process change, or a special cause that might be addressed more immediately.

Step 4: Continuous Improvement

SPC is not just about maintaining control; it’s about continuous improvement. The insights gained from SPC should drive ongoing efforts to enhance process performance.

Based on SPC data, processes can be adjusted, improved, and refined over time. This might involve changes to equipment, materials, methods, or training.

In conclusion, SPC is a key tool in the aim for quality control and process improvement. Its strength lies in its ability to make process variability visible and manageable. From the seminal contributions of Walter Shewhart and W. Edwards Deming, SPC has evolved into a comprehensive approach that integrates seamlessly with various quality management systems.

By continuously monitoring processes through control charts and adapting to the insights these charts provide, SPC empowers organizations to maintain control over their processes and pursue relentless improvement. Thus, SPC not only sustains but also elevates the standards of quality, efficiency, and customer satisfaction in diverse industrial landscapes.

• Madanhire, I. and Mbohwa, C., 2016. Application of statistical process control (SPC) in manufacturing industry in a developing country .  Procedia Cirp ,  40 , pp.580-583.
• Gérard, K., Grandhaye, J.P., Marchesi, V., Kafrouni, H., Husson, F. and Aletti, P., 2009. A comprehensive analysis of the IMRT dose delivery process using statistical process control (SPC).   Medical physics ,  36 (4), pp.1275-1285.

Q: What is Statistical Process Control?

A : SPC is a method used to monitor, control, and improve processes by analyzing performance data to identify and eliminate unwanted variations.

Q: Why is SPC important?

A : SPC helps ensure processes are consistent and predictable. It aids in early detection of issues, reducing defects, and improving overall product or service quality.

Q: What is a control chart?

A : A control chart is a graphical representation used in SPC to plot process data over time, with control limits that help distinguish between common and special cause variations.

Q: How are control limits determined?

A : Control limits are typically set at three standard deviations above and below the process mean, based on historical data. However, these limits can be adjusted depending on the specific chart type and industry standards.

Q: What's the difference between common cause and special cause variation?

A : Common cause variation is the inherent variability in a process, while special cause variation arises from specific, unusual events and is not part of the normal process.

Daniel Croft

Hi im Daniel continuous improvement manager with a Black Belt in Lean Six Sigma and over 10 years of real-world experience across a range sectors, I have a passion for optimizing processes and creating a culture of efficiency. I wanted to create Learn Lean Siigma to be a platform dedicated to Lean Six Sigma and process improvement insights and provide all the guides, tools, techniques and templates I looked for in one place as someone new to the world of Lean Six Sigma and Continuous improvement.

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Statistical Thinking for Industrial Problem Solving

A free online statistics course.

Back to Course Overview

Statistical Thinking and Problem Solving

Statistical thinking is vital for solving real-world problems. At the heart of statistical thinking is making decisions based on data. This requires disciplined approaches to identifying problems and the ability to quantify and interpret the variation that you observe in your data.

In this module, you will learn how to clearly define your problem and gain an understanding of the underlying processes that you will improve. You will learn techniques for identifying potential root causes of the problem. Finally, you will learn about different types of data and different approaches to data collection.

Estimated time to complete this module: 2 to 3 hours

Statistical Thinking and Problem Solving Overview (0:36)

Specific topics covered in this module include:

Statistical thinking.

• What is Statistical Thinking

Problem Solving

• Overview of Problem Solving
• Statistical Problem Solving
• Types of Problems
• Defining the Problem
• Goals and Key Performance Indicators
• The White Polymer Case Study

Defining the Process

• What is a Process?
• Developing a SIPOC Map
• Developing an Input/Output Process Map
• Top-Down and Deployment Flowcharts

Identifying Potential Root Causes

• Tools for Identifying Potential Causes
• Brainstorming
• Multi-voting
• Using Affinity Diagrams
• Cause-and-Effect Diagrams
• The Five Whys
• Cause-and-Effect Matrices

Compiling and Collecting Data

• Data Collection for Problem Solving
• Types of Data
• Operational Definitions
• Data Collection Strategies
• Importing Data for Analysis

ORIGINAL RESEARCH article

Statistical analysis of complex problem-solving process data: an event history analysis approach.

• 1 Department of Statistics, London School of Economics and Political Science, London, United Kingdom
• 2 School of Statistics, University of Minnesota, Minneapolis, MN, United States
• 3 Department of Statistics, Columbia University, New York, NY, United States

Complex problem-solving (CPS) ability has been recognized as a central 21st century skill. Individuals' processes of solving crucial complex problems may contain substantial information about their CPS ability. In this paper, we consider the prediction of duration and final outcome (i.e., success/failure) of solving a complex problem during task completion process, by making use of process data recorded in computer log files. Solving this problem may help answer questions like “how much information about an individual's CPS ability is contained in the process data?,” “what CPS patterns will yield a higher chance of success?,” and “what CPS patterns predict the remaining time for task completion?” We propose an event history analysis model for this prediction problem. The trained prediction model may provide us a better understanding of individuals' problem-solving patterns, which may eventually lead to a good design of automated interventions (e.g., providing hints) for the training of CPS ability. A real data example from the 2012 Programme for International Student Assessment (PISA) is provided for illustration.

1. Introduction

Complex problem-solving (CPS) ability has been recognized as a central 21st century skill of high importance for several outcomes including academic achievement ( Wüstenberg et al., 2012 ) and workplace performance ( Danner et al., 2011 ). It encompasses a set of higher-order thinking skills that require strategic planning, carrying out multi-step sequences of actions, reacting to a dynamically changing system, testing hypotheses, and, if necessary, adaptively coming up with new hypotheses. Thus, there is almost no doubt that an individual's problem-solving process data contain substantial amount of information about his/her CPS ability and thus are worth analyzing. Meaningful information extracted from CPS process data may lead to better understanding, measurement, and even training of individuals' CPS ability.

Problem-solving process data typically have a more complex structure than that of panel data which are traditionally more commonly encountered in statistics. Specifically, individuals may take different strategies toward solving the same problem. Even for individuals who take the same strategy, their actions and time-stamps of the actions may be very different. Due to such heterogeneity and complexity, classical regression and multivariate data analysis methods cannot be straightforwardly applied to CPS process data.

Possibly due to the lack of suitable analytic tools, research on CPS process data is limited. Among the existing works, none took a prediction perspective. Specifically, Greiff et al. (2015) presented a case study, showcasing the strong association between a specific strategic behavior (identified by expert knowledge) in a CPS task from the 2012 Programme for International Student Assessment (PISA) and performance both in this specific task and in the overall PISA problem-solving score. He and von Davier (2015 , 2016) proposed an N-gram method from natural language processing for analyzing problem-solving items in technology-rich environments, focusing on identifying feature sequences that are important to task completion. Vista et al. (2017) developed methods for the visualization and exploratory analysis of students' behavioral pathways, aiming to detect action sequences that are potentially relevant for establishing particular paths as meaningful markers of complex behaviors. Halpin and De Boeck (2013) and Halpin et al. (2017) adopted a Hawkes process approach to analyzing collaborative problem-solving items, focusing on the psychological measurement of collaboration. Xu et al. (2018) proposed a latent class model that analyzes CPS patterns by classifying individuals into latent classes based on their problem-solving processes.

In this paper, we propose to analyze CPS process data from a prediction perspective. As suggested in Yarkoni and Westfall (2017) , an increased focus on prediction can ultimately lead us to greater understanding of human behavior. Specifically, we consider the simultaneous prediction of the duration and the final outcome (i.e., success/failure) of solving a complex problem based on CPS process data. Instead of a single prediction, we hope to predict at any time during the problem-solving process. Such a data-driven prediction model may bring us insights about individuals' CPS behavioral patterns. First, features that contribute most to the prediction may correspond to important strategic behaviors that are key to succeeding in a task. In this sense, the proposed method can be used as an exploratory data analysis tool for extracting important features from process data. Second, the prediction accuracy may also serve as a measure of the strength of the signal contained in process data that reflects one's CPS ability, which reflects the reliability of CPS tasks from a prediction perspective. Third, for low stake assessments, the predicted chance of success may be used to give partial credits when scoring task takers. Fourth, speed is another important dimension of complex problem solving that is closely associated with the final outcome of task completion ( MacKay, 1982 ). The prediction of the duration throughout the problem-solving process may provide us insights on the relationship between the CPS behavioral patterns and the CPS speed. Finally, the prediction model also enables us to design suitable interventions during their problem-solving processes. For example, a hint may be provided when a student is predicted having a high chance to fail after sufficient efforts.

More precisely, we model the conditional distribution of duration time and final outcome given the event history up to any time point. This model can be viewed as a special event history analysis model, a general statistical framework for analyzing the expected duration of time until one or more events happen (see e.g., Allison, 2014 ). The proposed model can be regarded as an extension to the classical regression approach. The major difference is that the current model is specified over a continuous-time domain. It consists of a family of conditional models indexed by time, while the classical regression approach does not deal with continuous-time information. As a result, the proposed model supports prediction at any time during one's problem-solving process, while the classical regression approach does not. The proposed model is also related to, but substantially different from response time models (e.g., van der Linden, 2007 ) which have received much attention in psychometrics in recent years. Specifically, response time models model the joint distribution of response time and responses to test items, while the proposed model focuses on the conditional distribution of CPS duration and final outcome given the event history.

Although the proposed method learns regression-type models from data, it is worth emphasizing that we do not try to make statistical inference, such as testing whether a specific regression coefficient is significantly different from zero. Rather, the selection and interpretation of the model are mainly justified from a prediction perspective. This is because statistical inference tends to draw strong conclusions based on strong assumptions on the data generation mechanism. Due to the complexity of CPS process data, a statistical model may be severely misspecified, making valid statistical inference a big challenge. On the other hand, the prediction framework requires less assumptions and thus is more suitable for exploratory analysis. More precisely, the prediction framework admits the discrepancy between the underlying complex data generation mechanism and the prediction model ( Yarkoni and Westfall, 2017 ). A prediction model aims at achieving a balance between the bias due to this discrepancy and the variance due to a limited sample size. As a price, findings from the predictive framework are preliminary and only suggest hypotheses for future confirmatory studies.

The rest of the paper is organized as follows. In Section 2, we describe the structure of complex problem-solving process data and then motivate our research questions, using a CPS item from PISA 2012 as an example. In Section 3, we formulate the research questions under a statistical framework, propose a model, and then provide details of estimation and prediction. The introduced model is illustrated through an application to an example item from PISA 2012 in Section 4. We discuss limitations and future directions in Section 5.

2. Complex Problem-Solving Process Data

2.1. a motivating example.

We use a specific CPS item, CLIMATE CONTROL (CC) 1 , to demonstrate the data structure and to motivate our research questions. It is part of a CPS unit in PISA 2012 that was designed under the “MicroDYN” framework ( Greiff et al., 2012 ; Wüstenberg et al., 2012 ), a framework for the development of small dynamic systems of causal relationships for assessing CPS.

In this item, students are instructed to manipulate the panel (i.e., to move the top, central, and bottom control sliders; left side of Figure 1A ) and to answer how the input variables (control sliders) are related to the output variables (temperature and humidity). Specifically, the initial position of each control slider is indicated by a triangle “▴.” The students can change the top, central and bottom controls on the left of Figure 1 by using the sliders. By clicking “APPLY,” they will see the corresponding changes in temperature and humidity. After exploration, the students are asked to draw lines in a diagram ( Figure 1B ) to answer what each slider controls. The item is considered correctly answered if the diagram is correctly completed. The problem-solving process for this item is that the students must experiment to determine which controls have an impact on temperature and which on humidity, and then represent the causal relations by drawing arrows between the three inputs (top, central, and bottom control sliders) and the two outputs (temperature and humidity).

Figure 1. (A) Simulation environment of CC item. (B) Answer diagram of CC item.

PISA 2012 collected students' problem-solving process data in computer log files, in the form of a sequence of time-stamped events. We illustrate the structure of data in Table 1 and Figure 2 , where Table 1 tabulates a sequence of time-stamped events from a student and Figure 2 visualizes the corresponding event time points on a time line. According to the data, 14 events were recorded between time 0 (start) and 61.5 s (success). The first event happened at 29.5 s that was clicking “APPLY” after the top, central, and bottom controls were set at 2, 0, and 0, respectively. A sequence of actions followed the first event and finally at 58, 59.1, and 59.6 s, a final answer was correctly given using the diagram. It is worth clarifying that this log file does not collect all the interactions between a student and the simulated system. That is, the status of the control sliders is only recorded in the log file, when the “APPLY” button is clicked.

Table 1 . An example of computer log file data from CC item in PISA 2012.

Figure 2 . Visualization of the structure of process data from CC item in PISA 2012.

The process data for solving a CPS item typically have two components, knowledge acquisition and knowledge application, respectively. This CC item mainly focuses the former, which includes learning the causal relationships between the inputs and the outputs and representing such relationships by drawing the diagram. Since data on representing the causal relationship is relatively straightforward, in the rest of the paper, we focus on the process data related to knowledge acquisition and only refer a student's problem-solving process to his/her process of exploring the air conditioner, excluding the actions involving the answer diagram.

Intuitively, students' problem-solving processes contain information about their complex problem-solving ability, whether in the context of the CC item or in a more general sense of dealing with complex tasks in practice. However, it remains a challenge to extract meaningful information from their process data, due to the complex data structure. In particular, the occurrences of events are heterogeneous (i.e., different people can have very different event histories) and unstructured (i.e., there is little restriction on the order and time of the occurrences). Different students tend to have different problem-solving trajectories, with different actions taken at different time points. Consequently, time series models, which are standard statistical tools for analyzing dynamic systems, are not suitable here.

2.2. Research Questions

We focus on two specific research questions. Consider an individual solving a complex problem. Given that the individual has spent t units of time and has not yet completed the task, we would like to ask the following two questions based on the information at time t : How much additional time does the individual need? And will the individual succeed or fail upon the time of task completion?

Suppose we index the individual by i and let T i be the total time of task completion and Y i be the final outcome. Moreover, we denote H i ( t ) = ( h i 1 ( t ) , ... , h i p ( t ) ) ⊤ as a p -vector function of time t , summarizing the event history of individual i from the beginning of task to time t . Each component of H i ( t ) is a feature constructed from the event history up to time t . Taking the above CC item as an example, components of H i ( t ) may be, the number of actions a student has taken, whether all three control sliders have been explored, the frequency of using the reset button, etc., up to time t . We refer to H i ( t ) as the event history process of individual i . The dimension p may be high, depending on the complexity of the log file.

With the above notation, the two questions become to simultaneously predict T i and Y i based on H i ( t ). Throughout this paper, we focus on the analysis of data from a single CPS item. Extensions of the current framework to multiple-item analysis are discussed in Section 5.

3. Proposed Method

3.1. a regression model.

We now propose a regression model to answer the two questions raised in Section 2.2. We specify the marginal conditional models of Y i and T i given H i ( t ) and T i > t , respectively. Specifically, we assume

where Φ is the cumulative distribution function of a standard normal distribution. That is, Y i is assumed to marginally follow a probit regression model. In addition, only the conditional mean and variance are assumed for log( T i − t ). Our model parameters include the regression coefficients B = ( b jk )2 × p and conditional variance σ 2 . Based on the above model specification, a pseudo-likelihood function will be devived in Section 3.3 for parameter estimation.

Although only marginal models are specified, we point out that the model specifications (1) through (3) impose quite strong assumptions. As a result, the model may not most closely approximate the data-generating process and thus a bias is likely to exist. On the other hand, however, it is a working model that leads to reasonable prediction and can be used as a benchmark model for this prediction problem in future investigations.

We further remark that the conditional variance of log( T i − t ) is time-invariant under the current specification, which can be further relaxed to be time-dependent. In addition, the regression model for response time is closely related to the log-normal model for response time analysis in psychometrics (e.g., van der Linden, 2007 ). The major difference is that the proposed model is not a measurement model disentangling item and person effects on T i and Y i .

3.2. Prediction

Under the model in Section 3.1, given the event history, we predict the final outcome based on the success probability Φ( b 11 h i 1 ( t ) + ⋯ + b 1 p h ip ( t )). In addition, based on the conditional mean of log( T i − t ), we predict the total time at time t by t + exp( b 21 h i 1 ( t ) + ⋯ + b 2 p h ip ( t )). Given estimates of B from training data, we can predict the problem-solving duration and final outcome at any t for an individual in the testing sample, throughout his/her entire problem-solving process.

3.3. Parameter Estimation

It remains to estimate the model parameters based on a training dataset. Let our data be (τ i , y i ) and { H i ( t ): t ≥ 0}, i = 1, …, N , where τ i and y i are realizations of T i and Y i , and { H i ( t ): t ≥ 0} is the entire event history.

We develop estimating equations based on a pseudo likelihood function. Specifically, the conditional distribution of Y i given H i ( t ) and T i > t can be written as

where b 2 = ( b 11 , ... , b 1 p ) ⊤ . In addition, using the log-normal model as a working model for T i − t , the corresponding conditional distribution of T i can be written as

where b 2 = ( b 21 , ... , b 2 p ) ⊤ . The pseudo-likelihood is then written as

where t 1 , …, t J are J pre-specified grid points that spread out over the entire time spectrum. The choice of the grid points will be discussed in the sequel. By specifying the pseudo-likelihood based on the sequence of time points, the prediction at different time is taken into accounting in the estimation. We estimate the model parameters by maximizing the pseudo-likelihood function L ( B , σ).

In fact, (5) can be factorized into

Therefore, b 1 is estimated by maximizing L 1 ( b 1 ), which takes the form of a likelihood function for probit regression. Similarly, b 2 and σ are estimated by maximizing L 2 ( b 2 , σ), which is equivalent to solving the following estimation equations,

The estimating equations (8) and (9) can also be derived directly based on the conditional mean and variance specification of log( T i − t ). Solving these equations is equivalent to solving a linear regression problem, and thus is computationally easy.

3.4. Some Remarks

We provide a few remarks. First, choosing suitable features into H i ( t ) is important. The inclusion of suitable features not only improves the prediction accuracy, but also facilitates the exploratory analysis and interpretation of how behavioral patterns affect CPS result. If substantive knowledge about a CPS task is available from cognition theory, one may choose features that indicate different strategies toward solving the task. Otherwise, a data-driven approach may be taken. That is, one may select a model from a candidate list based on certain cross-validation criteria, where, if possible, all reasonable features should be consider as candidates. Even when a set of features has been suggested by cognition theory, one can still take the data-driven approach to find additional features, which may lead to new findings.

Second, one possible extension of the proposed model is to allow the regression coefficients to be a function of time t , whereas they are independent of time under the current model. In that case, the regression coefficients become functions of time, b jk ( t ). The current model can be regarded as a special case of this more general model. In particular, if b jk ( t ) has high variation along time in the best predictive model, then simply applying the current model may yield a high bias. Specifically, in the current estimation procedure, a larger grid point tends to have a smaller sample size and thus contributes less to the pseudo-likelihood function. As a result, a larger bias may occur in the prediction at a larger time point. However, the estimation of the time-dependent coefficient is non-trivial. In particular, constraints should be imposed on the functional form of b jk ( t ) to ensure a certain level of smoothness over time. As a result, b jk ( t ) can be accurately estimated using information from a finite number of time points. Otherwise, without any smoothness assumptions, to predict at any time during one's problem-solving process, there are an infinite number of parameters to estimate. Moreover, when a regression coefficient is time-dependent, its interpretation becomes more difficult, especially if the sign changes over time.

Third, we remark on the selection of grid points in the estimation procedure. Our model is specified in a continuous time domain that supports prediction at any time point in a continuum during an individual's problem-solving process. The use of discretized grid points is a way to approximate the continuous-time system, so that estimation equations can be written down. In practice, we suggest to place the grid points based on the quantiles of the empirical distribution of duration based on the training set. See the analysis in Section 4 for an illustration. The number of grid points may be further selected by cross validation. We also point out that prediction can be made at any time point on the continuum, not limited to the grid points for parameter estimation.

4. An Example from PISA 2012

4.1. background.

In what follows, we illustrate the proposed method via an application to the above CC item 2 . This item was also analyzed in Greiff et al. (2015) and Xu et al. (2018) . The dataset was cleaned from the entire released dataset of PISA 2012. It contains 16,872 15-year-old students' problem-solving processes, where the students were from 42 countries and economies. Among these students, 54.5% answered correctly. On average, each student took 129.9 s and 17 actions solving the problem. Histograms of the students' problem-solving duration and number of actions are presented in Figure 3 .

Figure 3. (A) Histogram of problem-solving duration of the CC item. (B) Histogram of the number of actions for solving the CC item.

4.2. Analyses

The entire dataset was randomly split into training and testing sets, where the training set contains data from 13,498 students and the testing set contains data from 3,374 students. A predictive model was built solely based on the training set and then its performance was evaluated based on the testing set. We used J = 9 grid points for the parameter estimation, with t 1 through t 9 specified to be 64, 81, 94, 106, 118, 132, 149, 170, and 208 s, respectively, which are the 10% through 90% quantiles of the empirical distribution of duration. As discussed earlier, the number of grid points and their locations may be further engineered by cross validation.

4.2.1. Model Selection

We first build a model based on the training data, using a data-driven stepwise forward selection procedure. In each step, we add one feature into H i ( t ) that leads to maximum increase in a cross-validated log-pseudo-likelihood, which is calculated based on a five-fold cross validation. We stop adding features into H i ( t ) when the cross-validated log-pseudo-likelihood stops increasing. The order in which the features are added may serve as a measure of their contribution to predicting the CPS duration and final outcome.

The candidate features being considered for model selection are listed in Table 2 . These candidate features were chosen to reflect students' CPS behavioral patterns from different aspects. In what follows, we discuss some of them. For example, the feature I i ( t ) indicates whether or not all three control sliders have been explored by simple actions (i.e., moving one control slider at a time) up to time t . That is, I i ( t ) = 1 means that the vary-one-thing-at-a-time (VOTAT) strategy ( Greiff et al., 2015 ) has been taken. According to the design of the CC item, the VOTAT strategy is expected to be a strong predictor of task success. In addition, the feature N i ( t )/ t records a student's average number of actions per unit time. It may serve as a measure of the student's speed of taking actions. In experimental psychology, response time or equivalently speed has been a central source for inferences about the organization and structure of cognitive processes (e.g., Luce, 1986 ), and in educational psychology, joint analysis of speed and accuracy of item response has also received much attention in recent years (e.g., van der Linden, 2007 ; Klein Entink et al., 2009 ). However, little is known about the role of speed in CPS tasks. The current analysis may provide some initial result on the relation between a student's speed and his/her CPS performance. Moreover, the features defined by the repeating of previously taken actions may reflect students' need of verifying the derived hypothesis on the relation based on the previous action or may be related to students' attention if the same actions are repeated many times. We also include 1, t, t 2 , and t 3 in H i ( t ) as the initial set of features to capture the time effect. For simplicity, country information is not taken into account in the current analysis.

Table 2 . The list of candidate features to be incorporated into the model.

Our results on model selection are summarized in Figure 4 and Table 3 . The pseudo-likelihood stopped increasing after 11 steps, resulting a final model with 15 components in H i ( t ). As we can see from Figure 4 , the increase in the cross-validated log-pseudo-likelihood is mainly contributed by the inclusion of features in the first six steps, after which the increment is quite marginal. As we can see, the first, second, and sixth features entering into the model are all related to taking simple actions, a strategy known to be important to this task (e.g., Greiff et al., 2015 ). In particular, the first feature being selected is I i ( t ), which confirms the strong effect of the VOTAT strategy. In addition, the third and fourth features are both based on N i ( t ), the number of actions taken before time t . Roughly, the feature 1 { N i ( t )>0} reflects the initial planning behavior ( Eichmann et al., 2019 ). Thus, this feature tends to measure students' speed of reading the instruction of the item. As discussed earlier, the feature N i ( t )/ t measures students' speed of taking actions. Finally, the fifth feature is related to the use of the RESET button.

Figure 4 . The increase in the cross-validated log-pseudo-likelihood based on a stepwise forward selection procedure. (A–C) plot the cross-validated log-pseudo-likelihood, corresponding to L ( B , σ), L 1 ( b 1 ), L 2 ( b 2 , σ), respectively.

Table 3 . Results on model selection based on a stepwise forward selection procedure.

4.2.2. Prediction Performance on Testing Set

We now look at the prediction performance of the above model on the testing set. The prediction performance was evaluated at a larger set of time points from 19 to 281 s. Instead of reporting based on the pseudo-likelihood function, we adopted two measures that are more straightforward. Specifically, we measured the prediction of final outcome by the Area Under the Curve (AUC) of the predicted Receiver Operating Characteristic (ROC) curve. The value of AUC is between 0 and 1. A larger AUC value indicates better prediction of the binary final outcome, with AUC = 1 indicating perfect prediction. In addition, at each time point t , we measured the prediction of duration based on the root mean squared error (RMSE), defined as

where τ i , i = N + 1, …, N + n , denotes the duration of students in the testing set, and τ ^ i ( t ) denotes the prediction based on information up to time t according to the trained model.

Results are presented in Figure 5 , where the testing AUC and RMSE for the final outcome and duration are presented. In particular, results based on the model selected by cross validation ( p = 15) and the initial model ( p = 4, containing the initial covariates 1, t , t 2 , and t 3 ) are compared. First, based on the selected model, the AUC is never above 0.8 and the RMSE is between 53 and 64 s, indicating a low signal-to-noise ratio. Second, the students' event history does improve the prediction of final outcome and duration upon the initial model. Specifically, since the initial model does not take into account the event history, it predicts the students with duration longer than t to have the same success probability. Consequently, the test AUC is 0.5 at each value of t , which is always worse than the performance of the selected model. Moreover, the selected model always outperforms the initial model in terms of the prediction of duration. Third, the AUC for the prediction of the final outcome is low when t is small. It keeps increasing as time goes on and fluctuates around 0.72 after about 120 s.

Figure 5 . A comparison of prediction accuracy between the model selected by cross validation and a baseline model without using individual specific event history.

4.2.3. Interpretation of Parameter Estimates

To gain more insights into how the event history affects the final outcome and duration, we further look at the results of parameter estimation. We focus on a model whose event history H i ( t ) includes the initial features and the top six features selected by cross validation. This model has similar prediction accuracy as the selected model according to the cross-validation result in Figure 4 , but contains less features in the event history and thus is easier to interpret. Moreover, the parameter estimates under this model are close to those under the cross-validation selected model, and the signs of the regression coefficients remain the same.

The estimated regression coefficients are presented in Table 4 . First, the first selected feature I i ( t ), which indicates whether all three control sliders have been explored via simple actions, has a positive regression coefficient on final outcome and a negative coefficient on duration. It means that, controlling the rest of the parameters, a student who has taken the VOTAT strategy tends to be more likely to give a correct answer and to complete in a shorter period of time. This confirms the strong effect of VOTAT strategy in solving the current task.

Table 4 . Estimated regression coefficients for a model for which the event history process contains the initial features based on polynomials of t and the top six features selected by cross validation.

Second, besides I i ( t ), there are two features related to taking simple actions, 1 { S i ( t )>0} and S i ( t )/ t , which are the indicator of taking at least one simple action and the frequency of taking simple actions. Both features have positive regression coefficients on the final outcome, implying larger values of both features lead to a higher success rate. In addition, 1 { S i ( t )>0} has a negative coefficient on duration and S i ( t )/ t has a positive one. Under this estimated model, the overall simple action effect on duration is b ^ 25 I i ( t ) + b ^ 26 1 { S i ( t ) > 0 } + b ^ 2 , 10 S i ( t ) / t , which is negative for most students. It implies that, overall, taking simple actions leads to a shorter predicted duration. However, once all three types of simple actions have been taken, a higher frequency of taking simple actions leads to a weaker but sill negative simple action effect on the duration.

Third, as discussed earlier, 1 { N i ( t )>0} tends to measure the student's speed of reading the instruction of the task and N i ( t )/ t can be regarded as a measure of students' speed of taking actions. According to the estimated regression coefficients, the data suggest that a student who reads and acts faster tends to complete the task in a shorter period of time with a lower accuracy. Similar results have been seen in the literature of response time analysis in educational psychology (e.g., Klein Entink et al., 2009 ; Fox and Marianti, 2016 ; Zhan et al., 2018 ), where speed of item response was found to negatively correlated with accuracy. In particular, Zhan et al. (2018) found a moderate negative correlation between students' general mathematics ability and speed under a psychometric model for PISA 2012 computer-based mathematics data.

Finally, 1 { R i ( t )>0} , the use of the RESET button, has positive regression coefficients on both final outcome and duration. It implies that the use of RESET button leads to a higher predicted success probability and a longer duration time, given the other features controlled. The connection between the use of the RESET button and the underlying cognitive process of complex problem solving, if it exists, still remains to be investigated.

5. Discussions

5.1. summary.

As an early step toward understanding individuals' complex problem-solving processes, we proposed an event history analysis method for the prediction of the duration and the final outcome of solving a complex problem based on process data. This approach is able to predict at any time t during an individual's problem-solving process, which may be useful in dynamic assessment/learning systems (e.g., in a game-based assessment system). An illustrative example is provided that is based on a CPS item from PISA 2012.

5.2. Inference, Prediction, and Interpretability

As articulated previously, this paper focuses on a prediction problem, rather than a statistical inference problem. Comparing with a prediction framework, statistical inference tends to draw stronger conclusions under stronger assumptions on the data generation mechanism. Unfortunately, due to the complexity of CPS process data, such assumptions are not only hardly satisfied, but also difficult to verify. On the other hand, a prediction framework requires less assumptions and thus is more suitable for exploratory analysis. As a price, the findings from the predictive framework are preliminary and can only be used to generate hypotheses for future studies.

It may be useful to provide uncertainty measures for the prediction performance and for the parameter estimates, where the former indicates the replicability of the prediction performance and the later reflects the stability of the prediction model. In particular, patterns from a prediction model with low replicability and low stability should not be overly interpreted. Such uncertainty measures may be obtained from cross validation and bootstrapping (see Chapter 7, Friedman et al., 2001 ).

It is also worth distinguishing prediction methods based on a simple model like the one proposed above and those based on black-box machine learning algorithms (e.g., random forest). Decisions based on black-box algorithms can be very difficult to understood by human and thus do not provide us insights about the data, even though they may have a high prediction accuracy. On the other hand, a simple model can be regarded as a data dimension reduction tool that extracts interpretable information from data, which may facilitate our understanding of complex problem solving.

5.3. Extending the Current Model

The proposed model can be extended along multiple directions. First, as discussed earlier, we may extend the model by allowing the regression coefficients b jk to be time-dependent. In that case, nonparametric estimation methods (e.g., splines) need to be developed for parameter estimation. In fact, the idea of time-varying coefficients has been intensively investigated in the event history analysis literature (e.g., Fan et al., 1997 ). This extension will be useful if the effects of the features in H i ( t ) change substantially over time.

Second, when the dimension p of H i ( t ) is high, better interpretability and higher prediction power may be achieved by using Lasso-type sparse estimators (see e.g., Chapter 3 Friedman et al., 2001 ). These estimators perform simultaneous feature selection and regularization in order to enhance the prediction accuracy and interpretability.

Finally, outliers are likely to occur in the data due to the abnormal behavioral patterns of a small proportion of people. A better treatment of outliers will lead to better prediction performance. Thus, a more robust objective function will be developed for parameter estimation, by borrowing ideas from the literature of robust statistics (see e.g., Huber and Ronchetti, 2009 ).

5.4. Multiple-Task Analysis

The current analysis focuses on analyzing data from a single task. To study individuals' CPS ability, it may be of more interest to analyze multiple CPS tasks simultaneously and to investigate how an individual's process data from one or multiple tasks predict his/her performance on the other tasks. Generally speaking, one's CPS ability may be better measured by the information in the process data that is generalizable across a representative set of CPS tasks than only his/her final outcomes on these tasks. In this sense, this cross-task prediction problem is closely related to the measurement of CPS ability. This problem is also worth future investigation.

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

This research was funded by NAEd/Spencer postdoctoral fellowship, NSF grant DMS-1712657, NSF grant SES-1826540, NSF grant IIS-1633360, and NIH grant R01GM047845.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

1. ^ The item can be found on the OECD website ( http://www.oecd.org/pisa/test-2012/testquestions/question3/ )

2. ^ The log file data and code book for the CC item can be found online: http://www.oecd.org/pisa/pisaproducts/database-cbapisa2012.htm .

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Keywords: process data, complex problem solving, PISA data, response time, event history analysis

Citation: Chen Y, Li X, Liu J and Ying Z (2019) Statistical Analysis of Complex Problem-Solving Process Data: An Event History Analysis Approach. Front. Psychol . 10:486. doi: 10.3389/fpsyg.2019.00486

Received: 31 August 2018; Accepted: 19 February 2019; Published: 18 March 2019.

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Copyright © 2019 Chen, Li, Liu and Ying. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Yunxiao Chen, [email protected]

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

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In one state, 52% of the voters are Republicans, and 48% are Democrats. In a second state, 47% of the voters are Republicans, and 53% are Democrats. Suppose a simple random sample of 100 voters are surveyed from each state.

What is the probability that the survey will show a greater percentage of Republican voters in the second state than in the first state?

The correct answer is C. For this analysis, let P 1 = the proportion of Republican voters in the first state, P 2 = the proportion of Republican voters in the second state, p 1 = the proportion of Republican voters in the sample from the first state, and p 2 = the proportion of Republican voters in the sample from the second state. The number of voters sampled from the first state (n 1 ) = 100, and the number of voters sampled from the second state (n 2 ) = 100.

The solution involves four steps.

• Make sure the sample size is big enough to model differences with a normal population. Because n 1 P 1 = 100 * 0.52 = 52, n 1 (1 - P 1 ) = 100 * 0.48 = 48, n 2 P 2 = 100 * 0.47 = 47, and n 2 (1 - P 2 ) = 100 * 0.53 = 53 are each greater than 10, the sample size is large enough.
• Find the mean of the difference in sample proportions: E(p 1 - p 2 ) = P 1 - P 2 = 0.52 - 0.47 = 0.05.

σ d = sqrt{ [ P1( 1 - P 1 ) / n 1 ] + [ P 2 (1 - P 2 ) / n 2 ] }

σ d = sqrt{ [ (0.52)(0.48) / 100 ] + [ (0.47)(0.53) / 100 ] }

σ d = sqrt (0.002496 + 0.002491) = sqrt(0.004987) = 0.0706

z p 1 - p 2 = (x - μ p 1 - p 2 ) / σ d = (0 - 0.05)/0.0706 = -0.7082

Using Stat Trek's Normal Distribution Calculator , we find that the probability of a z-score being -0.7082 or less is 0.24.

Therefore, the probability that the survey will show a greater percentage of Republican voters in the second state than in the first state is 0.24.

See also: Difference Between Proportions