objectives in quantitative research example

21 Research Objectives Examples (Copy and Paste)

Research objectives refer to the definitive statements made by researchers at the beginning of a research project detailing exactly what a research project aims to achieve.

These objectives are explicit goals clearly and concisely projected by the researcher to present a clear intention or course of action for his or her qualitative or quantitative study. 

Research objectives are typically nested under one overarching research aim. The objectives are the steps you’ll need to take in order to achieve the aim (see the examples below, for example, which demonstrate an aim followed by 3 objectives, which is what I recommend to my research students).

Research Objectives vs Research Aims

Research aim and research objectives are fundamental constituents of any study, fitting together like two pieces of the same puzzle.

The ‘research aim’ describes the overarching goal or purpose of the study (Kumar, 2019). This is usually a broad, high-level purpose statement, summing up the central question that the research intends to answer.

Example of an Overarching Research Aim:

“The aim of this study is to explore the impact of climate change on crop productivity.” 

Comparatively, ‘research objectives’ are concrete goals that underpin the research aim, providing stepwise actions to achieve the aim.

Objectives break the primary aim into manageable, focused pieces, and are usually characterized as being more specific, measurable, achievable, relevant, and time-bound (SMART).

Examples of Specific Research Objectives:

1. “To examine the effects of rising temperatures on the yield of rice crops during the upcoming growth season.” 2. “To assess changes in rainfall patterns in major agricultural regions over the first decade of the twenty-first century (2000-2010).” 3. “To analyze the impact of changing weather patterns on crop diseases within the same timeframe.”

The distinction between these two terms, though subtle, is significant for successfully conducting a study. The research aim provides the study with direction, while the research objectives set the path to achieving this aim, thereby ensuring the study’s efficiency and effectiveness.

How to Write Research Objectives

I usually recommend to my students that they use the SMART framework to create their research objectives.

SMART is an acronym standing for Specific, Measurable, Achievable, Relevant, and Time-bound. It provides a clear method of defining solid research objectives and helps students know where to start in writing their objectives (Locke & Latham, 2013).

Each element of this acronym adds a distinct dimension to the framework, aiding in the creation of comprehensive, well-delineated objectives.

Here is each step:

• Specific : We need to avoid ambiguity in our objectives. They need to be clear and precise (Doran, 1981). For instance, rather than stating the objective as “to study the effects of social media,” a more focused detail would be “to examine the effects of social media use (Facebook, Instagram, and Twitter) on the academic performance of college students.”
• Measurable: The measurable attribute provides a clear criterion to determine if the objective has been met (Locke & Latham, 2013). A quantifiable element, such as a percentage or a number, adds a measurable quality. For example, “to increase response rate to the annual customer survey by 10%,” makes it easier to ascertain achievement.
• Achievable: The achievable aspect encourages researchers to craft realistic objectives, resembling a self-check mechanism to ensure the objectives align with the scope and resources at disposal (Doran, 1981). For example, “to interview 25 participants selected randomly from a population of 100” is an attainable objective as long as the researcher has access to these participants.
• Relevance : Relevance, the fourth element, compels the researcher to tailor the objectives in alignment with overarching goals of the study (Locke & Latham, 2013). This is extremely important – each objective must help you meet your overall one-sentence ‘aim’ in your study.
• Time-Bound: Lastly, the time-bound element fosters a sense of urgency and prioritization, preventing procrastination and enhancing productivity (Doran, 1981). “To analyze the effect of laptop use in lectures on student engagement over the course of two semesters this year” expresses a clear deadline, thus serving as a motivator for timely completion.

You’re not expected to fit every single element of the SMART framework in one objective, but across your objectives, try to touch on each of the five components.

Research Objectives Examples

1. Field: Psychology

Aim: To explore the impact of sleep deprivation on cognitive performance in college students.

• Objective 1: To compare cognitive test scores of students with less than six hours of sleep and those with 8 or more hours of sleep.
• Objective 2: To investigate the relationship between class grades and reported sleep duration.
• Objective 3: To survey student perceptions and experiences on how sleep deprivation affects their cognitive capabilities.

2. Field: Environmental Science

Aim: To understand the effects of urban green spaces on human well-being in a metropolitan city.

• Objective 1: To assess the physical and mental health benefits of regular exposure to urban green spaces.
• Objective 2: To evaluate the social impacts of urban green spaces on community interactions.
• Objective 3: To examine patterns of use for different types of urban green spaces. 

3. Field: Technology

Aim: To investigate the influence of using social media on productivity in the workplace.

• Objective 1: To measure the amount of time spent on social media during work hours.
• Objective 2: To evaluate the perceived impact of social media use on task completion and work efficiency.
• Objective 3: To explore whether company policies on social media usage correlate with different patterns of productivity.

4. Field: Education

Aim: To examine the effectiveness of online vs traditional face-to-face learning on student engagement and achievement.

• Objective 1: To compare student grades between the groups exposed to online and traditional face-to-face learning.
• Objective 2: To assess student engagement levels in both learning environments.
• Objective 3: To collate student perceptions and preferences regarding both learning methods.

5. Field: Health

Aim: To determine the impact of a Mediterranean diet on cardiac health among adults over 50.

• Objective 1: To assess changes in cardiovascular health metrics after following a Mediterranean diet for six months.
• Objective 2: To compare these health metrics with a similar group who follow their regular diet.
• Objective 3: To document participants’ experiences and adherence to the Mediterranean diet.

6. Field: Environmental Science

Aim: To analyze the impact of urban farming on community sustainability.

• Objective 1: To document the types and quantity of food produced through urban farming initiatives.
• Objective 2: To assess the effect of urban farming on local communities’ access to fresh produce.
• Objective 3: To examine the social dynamics and cooperative relationships in the creating and maintaining of urban farms.

7. Field: Sociology

Aim: To investigate the influence of home offices on work-life balance during remote work.

• Objective 1: To survey remote workers on their perceptions of work-life balance since setting up home offices.
• Objective 2: To conduct an observational study of daily work routines and family interactions in a home office setting.
• Objective 3: To assess the correlation, if any, between physical boundaries of workspaces and mental boundaries for work in the home setting.

8. Field: Economics

Aim: To evaluate the effects of minimum wage increases on small businesses.

• Objective 1: To analyze cost structures, pricing changes, and profitability of small businesses before and after minimum wage increases.
• Objective 2: To survey small business owners on the strategies they employ to navigate minimum wage increases.
• Objective 3: To examine employment trends in small businesses in response to wage increase legislation.

9. Field: Education

Aim: To explore the role of extracurricular activities in promoting soft skills among high school students.

• Objective 1: To assess the variety of soft skills developed through different types of extracurricular activities.
• Objective 2: To compare self-reported soft skills between students who participate in extracurricular activities and those who do not.
• Objective 3: To investigate the teachers’ perspectives on the contribution of extracurricular activities to students’ skill development.

10. Field: Technology

Aim: To assess the impact of virtual reality (VR) technology on the tourism industry.

• Objective 1: To document the types and popularity of VR experiences available in the tourism market.
• Objective 2: To survey tourists on their interest levels and satisfaction rates with VR tourism experiences.
• Objective 3: To determine whether VR tourism experiences correlate with increased interest in real-life travel to the simulated destinations.

11. Field: Biochemistry

Aim: To examine the role of antioxidants in preventing cellular damage.

• Objective 1: To identify the types and quantities of antioxidants in common fruits and vegetables.
• Objective 2: To determine the effects of various antioxidants on free radical neutralization in controlled lab tests.
• Objective 3: To investigate potential beneficial impacts of antioxidant-rich diets on long-term cellular health.

12. Field: Linguistics

Aim: To determine the influence of early exposure to multiple languages on cognitive development in children.

• Objective 1: To assess cognitive development milestones in monolingual and multilingual children.
• Objective 2: To document the number and intensity of language exposures for each group in the study.
• Objective 3: To investigate the specific cognitive advantages, if any, enjoyed by multilingual children.

13. Field: Art History

Aim: To explore the impact of the Renaissance period on modern-day art trends.

• Objective 1: To identify key characteristics and styles of Renaissance art.
• Objective 2: To analyze modern art pieces for the influence of the Renaissance style.
• Objective 3: To survey modern-day artists for their inspirations and the influence of historical art movements on their work.

14. Field: Cybersecurity

Aim: To assess the effectiveness of two-factor authentication (2FA) in preventing unauthorized system access.

• Objective 1: To measure the frequency of unauthorized access attempts before and after the introduction of 2FA.
• Objective 2: To survey users about their experiences and challenges with 2FA implementation.
• Objective 3: To evaluate the efficacy of different types of 2FA (SMS-based, authenticator apps, biometrics, etc.).

15. Field: Cultural Studies

Aim: To analyze the role of music in cultural identity formation among ethnic minorities.

• Objective 1: To document the types and frequency of traditional music practices within selected ethnic minority communities.
• Objective 2: To survey community members on the role of music in their personal and communal identity.
• Objective 3: To explore the resilience and transmission of traditional music practices in contemporary society.

16. Field: Astronomy

Aim: To explore the impact of solar activity on satellite communication.

• Objective 1: To categorize different types of solar activities and their frequencies of occurrence.
• Objective 2: To ascertain how variations in solar activity may influence satellite communication.
• Objective 3: To investigate preventative and damage-control measures currently in place during periods of high solar activity.

17. Field: Literature

Aim: To examine narrative techniques in contemporary graphic novels.

• Objective 1: To identify a range of narrative techniques employed in this genre.
• Objective 2: To analyze the ways in which these narrative techniques engage readers and affect story interpretation.
• Objective 3: To compare narrative techniques in graphic novels to those found in traditional printed novels.

18. Field: Renewable Energy

Aim: To investigate the feasibility of solar energy as a primary renewable resource within urban areas.

• Objective 1: To quantify the average sunlight hours across urban areas in different climatic zones. 
• Objective 2: To calculate the potential solar energy that could be harnessed within these areas.
• Objective 3: To identify barriers or challenges to widespread solar energy implementation in urban settings and potential solutions.

19. Field: Sports Science

Aim: To evaluate the role of pre-game rituals in athlete performance.

• Objective 1: To identify the variety and frequency of pre-game rituals among professional athletes in several sports.
• Objective 2: To measure the impact of pre-game rituals on individual athletes’ performance metrics.
• Objective 3: To examine the psychological mechanisms that might explain the effects (if any) of pre-game ritual on performance.

20. Field: Ecology

Aim: To investigate the effects of urban noise pollution on bird populations.

• Objective 1: To record and quantify urban noise levels in various bird habitats.
• Objective 2: To measure bird population densities in relation to noise levels.
• Objective 3: To determine any changes in bird behavior or vocalization linked to noise levels.

21. Field: Food Science

Aim: To examine the influence of cooking methods on the nutritional value of vegetables.

• Objective 1: To identify the nutrient content of various vegetables both raw and after different cooking processes.
• Objective 2: To compare the effect of various cooking methods on the nutrient retention of these vegetables.
• Objective 3: To propose cooking strategies that optimize nutrient retention.

The Importance of Research Objectives

The importance of research objectives cannot be overstated. In essence, these guideposts articulate what the researcher aims to discover, understand, or examine (Kothari, 2014).

When drafting research objectives, it’s essential to make them simple and comprehensible, specific to the point of being quantifiable where possible, achievable in a practical sense, relevant to the chosen research question, and time-constrained to ensure efficient progress (Kumar, 2019). 

Remember that a good research objective is integral to the success of your project, offering a clear path forward for setting out a research design , and serving as the bedrock of your study plan. Each objective must distinctly address a different dimension of your research question or problem (Kothari, 2014). Always bear in mind that the ultimate purpose of your research objectives is to succinctly encapsulate your aims in the clearest way possible, facilitating a coherent, comprehensive and rational approach to your planned study, and furnishing a scientific roadmap for your journey into the depths of knowledge and research (Kumar, 2019). 

Kothari, C.R (2014). Research Methodology: Methods and Techniques . New Delhi: New Age International.

Kumar, R. (2019). Research Methodology: A Step-by-Step Guide for Beginners .New York: SAGE Publications.

Doran, G. T. (1981). There’s a S.M.A.R.T. way to write management’s goals and objectives. Management review, 70 (11), 35-36.

Locke, E. A., & Latham, G. P. (2013). New Developments in Goal Setting and Task Performance . New York: Routledge.

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Research Aims, Objectives & Questions

The “Golden Thread” Explained Simply (+ Examples)

By: David Phair (PhD) and Alexandra Shaeffer (PhD) | June 2022

The research aims , objectives and research questions (collectively called the “golden thread”) are arguably the most important thing you need to get right when you’re crafting a research proposal , dissertation or thesis . We receive questions almost every day about this “holy trinity” of research and there’s certainly a lot of confusion out there, so we’ve crafted this post to help you navigate your way through the fog.

Overview: The Golden Thread

• What is the golden thread
• What are research aims ( examples )
• What are research objectives ( examples )
• What are research questions ( examples )
• The importance of alignment in the golden thread

What is the “golden thread”?  

The golden thread simply refers to the collective research aims , research objectives , and research questions for any given project (i.e., a dissertation, thesis, or research paper ). These three elements are bundled together because it’s extremely important that they align with each other, and that the entire research project aligns with them.

Importantly, the golden thread needs to weave its way through the entirety of any research project , from start to end. In other words, it needs to be very clearly defined right at the beginning of the project (the topic ideation and proposal stage) and it needs to inform almost every decision throughout the rest of the project. For example, your research design and methodology will be heavily influenced by the golden thread (we’ll explain this in more detail later), as well as your literature review.

The research aims, objectives and research questions (the golden thread) define the focus and scope ( the delimitations ) of your research project. In other words, they help ringfence your dissertation or thesis to a relatively narrow domain, so that you can “go deep” and really dig into a specific problem or opportunity. They also help keep you on track , as they act as a litmus test for relevance. In other words, if you’re ever unsure whether to include something in your document, simply ask yourself the question, “does this contribute toward my research aims, objectives or questions?”. If it doesn’t, chances are you can drop it.

Alright, enough of the fluffy, conceptual stuff. Let’s get down to business and look at what exactly the research aims, objectives and questions are and outline a few examples to bring these concepts to life.

Research Aims: What are they?

Simply put, the research aim(s) is a statement that reflects the broad overarching goal (s) of the research project. Research aims are fairly high-level (low resolution) as they outline the general direction of the research and what it’s trying to achieve .

Research Aims: Examples  

True to the name, research aims usually start with the wording “this research aims to…”, “this research seeks to…”, and so on. For example:

“This research aims to explore employee experiences of digital transformation in retail HR.”   “This study sets out to assess the interaction between student support and self-care on well-being in engineering graduate students”  

As you can see, these research aims provide a high-level description of what the study is about and what it seeks to achieve. They’re not hyper-specific or action-oriented, but they’re clear about what the study’s focus is and what is being investigated.

Need a helping hand?

Research Objectives: What are they?

The research objectives take the research aims and make them more practical and actionable . In other words, the research objectives showcase the steps that the researcher will take to achieve the research aims.

The research objectives need to be far more specific (higher resolution) and actionable than the research aims. In fact, it’s always a good idea to craft your research objectives using the “SMART” criteria. In other words, they should be specific, measurable, achievable, relevant and time-bound”.

Research Objectives: Examples  

Let’s look at two examples of research objectives. We’ll stick with the topic and research aims we mentioned previously.  

For the digital transformation topic:

To observe the retail HR employees throughout the digital transformation. To assess employee perceptions of digital transformation in retail HR. To identify the barriers and facilitators of digital transformation in retail HR.

And for the student wellness topic:

To determine whether student self-care predicts the well-being score of engineering graduate students. To determine whether student support predicts the well-being score of engineering students. To assess the interaction between student self-care and student support when predicting well-being in engineering graduate students.

  As you can see, these research objectives clearly align with the previously mentioned research aims and effectively translate the low-resolution aims into (comparatively) higher-resolution objectives and action points . They give the research project a clear focus and present something that resembles a research-based “to-do” list.

Research Questions: What are they?

Finally, we arrive at the all-important research questions. The research questions are, as the name suggests, the key questions that your study will seek to answer . Simply put, they are the core purpose of your dissertation, thesis, or research project. You’ll present them at the beginning of your document (either in the introduction chapter or literature review chapter) and you’ll answer them at the end of your document (typically in the discussion and conclusion chapters).  

The research questions will be the driving force throughout the research process. For example, in the literature review chapter, you’ll assess the relevance of any given resource based on whether it helps you move towards answering your research questions. Similarly, your methodology and research design will be heavily influenced by the nature of your research questions. For instance, research questions that are exploratory in nature will usually make use of a qualitative approach, whereas questions that relate to measurement or relationship testing will make use of a quantitative approach.  

Let’s look at some examples of research questions to make this more tangible.

Research Questions: Examples  

Again, we’ll stick with the research aims and research objectives we mentioned previously.  

For the digital transformation topic (which would be qualitative in nature):

How do employees perceive digital transformation in retail HR? What are the barriers and facilitators of digital transformation in retail HR?  

And for the student wellness topic (which would be quantitative in nature):

Does student self-care predict the well-being scores of engineering graduate students? Does student support predict the well-being scores of engineering students? Do student self-care and student support interact when predicting well-being in engineering graduate students?  

You’ll probably notice that there’s quite a formulaic approach to this. In other words, the research questions are basically the research objectives “converted” into question format. While that is true most of the time, it’s not always the case. For example, the first research objective for the digital transformation topic was more or less a step on the path toward the other objectives, and as such, it didn’t warrant its own research question.  

So, don’t rush your research questions and sloppily reword your objectives as questions. Carefully think about what exactly you’re trying to achieve (i.e. your research aim) and the objectives you’ve set out, then craft a set of well-aligned research questions . Also, keep in mind that this can be a somewhat iterative process , where you go back and tweak research objectives and aims to ensure tight alignment throughout the golden thread.

The importance of strong alignment 

Alignment is the keyword here and we have to stress its importance . Simply put, you need to make sure that there is a very tight alignment between all three pieces of the golden thread. If your research aims and research questions don’t align, for example, your project will be pulling in different directions and will lack focus . This is a common problem students face and can cause many headaches (and tears), so be warned.

Take the time to carefully craft your research aims, objectives and research questions before you run off down the research path. Ideally, get your research supervisor/advisor to review and comment on your golden thread before you invest significant time into your project, and certainly before you start collecting data .  

Recap: The golden thread

In this post, we unpacked the golden thread of research, consisting of the research aims , research objectives and research questions . You can jump back to any section using the links below.

As always, feel free to leave a comment below – we always love to hear from you. Also, if you’re interested in 1-on-1 support, take a look at our private coaching service here.

Psst... there’s more!

This post was based on one of our popular Research Bootcamps . If you're working on a research project, you'll definitely want to check this out ...

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39 Comments

Thank you very much for your great effort put. As an Undergraduate taking Demographic Research & Methodology, I’ve been trying so hard to understand clearly what is a Research Question, Research Aim and the Objectives in a research and the relationship between them etc. But as for now I’m thankful that you’ve solved my problem.

Well appreciated. This has helped me greatly in doing my dissertation.

An so delighted with this wonderful information thank you a lot.

so impressive i have benefited a lot looking forward to learn more on research.

I am very happy to have carefully gone through this well researched article.

Infact,I used to be phobia about anything research, because of my poor understanding of the concepts.

Now,I get to know that my research question is the same as my research objective(s) rephrased in question format.

I please I would need a follow up on the subject,as I intends to join the team of researchers. Thanks once again.

Thanks so much. This was really helpful.

I know you pepole have tried to break things into more understandable and easy format. And God bless you. Keep it up

i found this document so useful towards my study in research methods. thanks so much.

This is my 2nd read topic in your course and I should commend the simplified explanations of each part. I’m beginning to understand and absorb the use of each part of a dissertation/thesis. I’ll keep on reading your free course and might be able to avail the training course! Kudos!

Thank you! Better put that my lecture and helped to easily understand the basics which I feel often get brushed over when beginning dissertation work.

This is quite helpful. I like how the Golden thread has been explained and the needed alignment.

This is quite helpful. I really appreciate!

The article made it simple for researcher students to differentiate between three concepts.

Very innovative and educational in approach to conducting research.

I am very impressed with all these terminology, as I am a fresh student for post graduate, I am highly guided and I promised to continue making consultation when the need arise. Thanks a lot.

A very helpful piece. thanks, I really appreciate it .

Very well explained, and it might be helpful to many people like me.

Wish i had found this (and other) resource(s) at the beginning of my PhD journey… not in my writing up year… 😩 Anyways… just a quick question as i’m having some issues ordering my “golden thread”…. does it matter in what order you mention them? i.e., is it always first aims, then objectives, and finally the questions? or can you first mention the research questions and then the aims and objectives?

Thank you for a very simple explanation that builds upon the concepts in a very logical manner. Just prior to this, I read the research hypothesis article, which was equally very good. This met my primary objective.

My secondary objective was to understand the difference between research questions and research hypothesis, and in which context to use which one. However, I am still not clear on this. Can you kindly please guide?

In research, a research question is a clear and specific inquiry that the researcher wants to answer, while a research hypothesis is a tentative statement or prediction about the relationship between variables or the expected outcome of the study. Research questions are broader and guide the overall study, while hypotheses are specific and testable statements used in quantitative research. Research questions identify the problem, while hypotheses provide a focus for testing in the study.

Exactly what I need in this research journey, I look forward to more of your coaching videos.

This helped a lot. Thanks so much for the effort put into explaining it.

What data source in writing dissertation/Thesis requires?

What is data source covers when writing dessertation/thesis

This is quite useful thanks

I’m excited and thankful. I got so much value which will help me progress in my thesis.

where are the locations of the reserch statement, research objective and research question in a reserach paper? Can you write an ouline that defines their places in the researh paper?

Very helpful and important tips on Aims, Objectives and Questions.

Thank you so much for making research aim, research objectives and research question so clear. This will be helpful to me as i continue with my thesis.

Thanks much for this content. I learned a lot. And I am inspired to learn more. I am still struggling with my preparation for dissertation outline/proposal. But I consistently follow contents and tutorials and the new FB of GRAD Coach. Hope to really become confident in writing my dissertation and successfully defend it.

As a researcher and lecturer, I find splitting research goals into research aims, objectives, and questions is unnecessarily bureaucratic and confusing for students. For most biomedical research projects, including ‘real research’, 1-3 research questions will suffice (numbers may differ by discipline).

Awesome! Very important resources and presented in an informative way to easily understand the golden thread. Indeed, thank you so much.

Well explained

The blog article on research aims, objectives, and questions by Grad Coach is a clear and insightful guide that aligns with my experiences in academic research. The article effectively breaks down the often complex concepts of research aims and objectives, providing a straightforward and accessible explanation. Drawing from my own research endeavors, I appreciate the practical tips offered, such as the need for specificity and clarity when formulating research questions. The article serves as a valuable resource for students and researchers, offering a concise roadmap for crafting well-defined research goals and objectives. Whether you’re a novice or an experienced researcher, this article provides practical insights that contribute to the foundational aspects of a successful research endeavor.

A great thanks for you. it is really amazing explanation. I grasp a lot and one step up to research knowledge.

I really found these tips helpful. Thank you very much Grad Coach.

I found this article helpful. Thanks for sharing this.

thank you so much, the explanation and examples are really helpful

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Formulating Research Aims and Objectives

Formulating research aim and objectives in an appropriate manner is one of the most important aspects of your thesis. This is because research aim and objectives determine the scope, depth and the overall direction of the research. Research question is the central question of the study that has to be answered on the basis of research findings.

Research aim emphasizes what needs to be achieved within the scope of the research, by the end of the research process. Achievement of research aim provides answer to the research question.

Research objectives divide research aim into several parts and address each part separately. Research aim specifies WHAT needs to be studied and research objectives comprise a number of steps that address HOW research aim will be achieved.

As a rule of dumb, there would be one research aim and several research objectives. Achievement of each research objective will lead to the achievement of the research aim.

Consider the following as an example:

Research title: Effects of organizational culture on business profitability: a case study of Virgin Atlantic

Research aim: To assess the effects of Virgin Atlantic organizational culture on business profitability

Following research objectives would facilitate the achievement of this aim:

• Analyzing the nature of organizational culture at Virgin Atlantic by September 1, 2022
• Identifying factors impacting Virgin Atlantic organizational culture by September 16, 2022
• Analyzing impacts of Virgin Atlantic organizational culture on employee performances by September 30, 2022
• Providing recommendations to Virgin Atlantic strategic level management in terms of increasing the level of effectiveness of organizational culture by October 5, 2022

Figure below illustrates additional examples in formulating research aims and objectives:

Formulation of research question, aim and objectives

Common mistakes in the formulation of research aim relate to the following:

1. Choosing the topic too broadly . This is the most common mistake. For example, a research title of “an analysis of leadership practices” can be classified as too broad because the title fails to answer the following questions:

a) Which aspects of leadership practices? Leadership has many aspects such as employee motivation, ethical behaviour, strategic planning, change management etc. An attempt to cover all of these aspects of organizational leadership within a single research will result in an unfocused and poor work.

b) An analysis of leadership practices in which country? Leadership practices tend to be different in various countries due to cross-cultural differences, legislations and a range of other region-specific factors. Therefore, a study of leadership practices needs to be country-specific.

c) Analysis of leadership practices in which company or industry? Similar to the point above, analysis of leadership practices needs to take into account industry-specific and/or company-specific differences, and there is no way to conduct a leadership research that relates to all industries and organizations in an equal manner.

Accordingly, as an example “a study into the impacts of ethical behaviour of a leader on the level of employee motivation in US healthcare sector” would be a more appropriate title than simply “An analysis of leadership practices”.

2. Setting an unrealistic aim . Formulation of a research aim that involves in-depth interviews with Apple strategic level management by an undergraduate level student can be specified as a bit over-ambitious. This is because securing an interview with Apple CEO Tim Cook or members of Apple Board of Directors might not be easy. This is an extreme example of course, but you got the idea. Instead, you may aim to interview the manager of your local Apple store and adopt a more feasible strategy to get your dissertation completed.

3. Choosing research methods incompatible with the timeframe available . Conducting interviews with 20 sample group members and collecting primary data through 2 focus groups when only three months left until submission of your dissertation can be very difficult, if not impossible. Accordingly, timeframe available need to be taken into account when formulating research aims and objectives and selecting research methods.

Moreover, research objectives need to be formulated according to SMART principle,

 where the abbreviation stands for specific, measurable, achievable, realistic, and time-bound.

Examples of SMART research objectives

At the conclusion part of your research project you will need to reflect on the level of achievement of research aims and objectives. In case your research aims and objectives are not fully achieved by the end of the study, you will need to discuss the reasons. These may include initial inappropriate formulation of research aims and objectives, effects of other variables that were not considered at the beginning of the research or changes in some circumstances during the research process.

John Dudovskiy

Quantitative Research

• Reference work entry
• First Online: 13 January 2019
• Cite this reference work entry

• Leigh A. Wilson 2 , 3  

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Quantitative research methods are concerned with the planning, design, and implementation of strategies to collect and analyze data. Descartes, the seventeenth-century philosopher, suggested that how the results are achieved is often more important than the results themselves, as the journey taken along the research path is a journey of discovery. High-quality quantitative research is characterized by the attention given to the methods and the reliability of the tools used to collect the data. The ability to critique research in a systematic way is an essential component of a health professional’s role in order to deliver high quality, evidence-based healthcare. This chapter is intended to provide a simple overview of the way new researchers and health practitioners can understand and employ quantitative methods. The chapter offers practical, realistic guidance in a learner-friendly way and uses a logical sequence to understand the process of hypothesis development, study design, data collection and handling, and finally data analysis and interpretation.

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Home Market Research

Quantitative Research: What It Is, Practices & Methods

Quantitative research involves analyzing and gathering numerical data to uncover trends, calculate averages, evaluate relationships, and derive overarching insights. It’s used in various fields, including the natural and social sciences. Quantitative data analysis employs statistical techniques for processing and interpreting numeric data.

Research designs in the quantitative realm outline how data will be collected and analyzed with methods like experiments and surveys. Qualitative methods complement quantitative research by focusing on non-numerical data, adding depth to understanding. Data collection methods can be qualitative or quantitative, depending on research goals. Researchers often use a combination of both approaches to gain a comprehensive understanding of phenomena.

What is Quantitative Research?

Quantitative research is a systematic investigation of phenomena by gathering quantifiable data and performing statistical, mathematical, or computational techniques. Quantitative research collects statistically significant information from existing and potential customers using sampling methods and sending out online surveys , online polls , and questionnaires , for example.

One of the main characteristics of this type of research is that the results can be depicted in numerical form. After carefully collecting structured observations and understanding these numbers, it’s possible to predict the future of a product or service, establish causal relationships or Causal Research , and make changes accordingly. Quantitative research primarily centers on the analysis of numerical data and utilizes inferential statistics to derive conclusions that can be extrapolated to the broader population.

An example of a quantitative research study is the survey conducted to understand how long a doctor takes to tend to a patient when the patient walks into the hospital. A patient satisfaction survey can be administered to ask questions like how long a doctor takes to see a patient, how often a patient walks into a hospital, and other such questions, which are dependent variables in the research. This kind of research method is often employed in the social sciences, and it involves using mathematical frameworks and theories to effectively present data, ensuring that the results are logical, statistically sound, and unbiased.

Data collection in quantitative research uses a structured method and is typically conducted on larger samples representing the entire population. Researchers use quantitative methods to collect numerical data, which is then subjected to statistical analysis to determine statistically significant findings. This approach is valuable in both experimental research and social research, as it helps in making informed decisions and drawing reliable conclusions based on quantitative data.

Quantitative Research Characteristics

Quantitative research has several unique characteristics that make it well-suited for specific projects. Let’s explore the most crucial of these characteristics so that you can consider them when planning your next research project:

• Structured tools: Quantitative research relies on structured tools such as surveys, polls, or questionnaires to gather quantitative data . Using such structured methods helps collect in-depth and actionable numerical data from the survey respondents, making it easier to perform data analysis.
• Sample size: Quantitative research is conducted on a significant sample size  representing the target market . Appropriate Survey Sampling methods, a fundamental aspect of quantitative research methods, must be employed when deriving the sample to fortify the research objective and ensure the reliability of the results.
• Close-ended questions: Closed-ended questions , specifically designed to align with the research objectives, are a cornerstone of quantitative research. These questions facilitate the collection of quantitative data and are extensively used in data collection processes.
• Prior studies: Before collecting feedback from respondents, researchers often delve into previous studies related to the research topic. This preliminary research helps frame the study effectively and ensures the data collection process is well-informed.
• Quantitative data: Typically, quantitative data is represented using tables, charts, graphs, or other numerical forms. This visual representation aids in understanding the collected data and is essential for rigorous data analysis, a key component of quantitative research methods.
• Generalization of results: One of the strengths of quantitative research is its ability to generalize results to the entire population. It means that the findings derived from a sample can be extrapolated to make informed decisions and take appropriate actions for improvement based on numerical data analysis.

Quantitative Research Methods

Quantitative research methods are systematic approaches used to gather and analyze numerical data to understand and draw conclusions about a phenomenon or population. Here are the quantitative research methods:

• Primary quantitative research methods
• Secondary quantitative research methods

Primary Quantitative Research Methods

Primary quantitative research is the most widely used method of conducting market research. The distinct feature of primary research is that the researcher focuses on collecting data directly rather than depending on data collected from previously done research. Primary quantitative research design can be broken down into three further distinctive tracks and the process flow. They are:

A. Techniques and Types of Studies

There are multiple types of primary quantitative research. They can be distinguished into the four following distinctive methods, which are:

01. Survey Research

Survey Research is fundamental for all quantitative outcome research methodologies and studies. Surveys are used to ask questions to a sample of respondents, using various types such as online polls, online surveys, paper questionnaires, web-intercept surveys , etc. Every small and big organization intends to understand what their customers think about their products and services, how well new features are faring in the market, and other such details.

By conducting survey research, an organization can ask multiple survey questions , collect data from a pool of customers, and analyze this collected data to produce numerical results. It is the first step towards collecting data for any research. You can use single ease questions . A single-ease question is a straightforward query that elicits a concise and uncomplicated response.

This type of research can be conducted with a specific target audience group and also can be conducted across multiple groups along with comparative analysis . A prerequisite for this type of research is that the sample of respondents must have randomly selected members. This way, a researcher can easily maintain the accuracy of the obtained results as a huge variety of respondents will be addressed using random selection. 

Traditionally, survey research was conducted face-to-face or via phone calls. Still, with the progress made by online mediums such as email or social media, survey research has also spread to online mediums.There are two types of surveys , either of which can be chosen based on the time in hand and the kind of data required:

Cross-sectional surveys: Cross-sectional surveys are observational surveys conducted in situations where the researcher intends to collect data from a sample of the target population at a given point in time. Researchers can evaluate various variables at a particular time. Data gathered using this type of survey is from people who depict similarity in all variables except the variables which are considered for research . Throughout the survey, this one variable will stay constant.

• Cross-sectional surveys are popular with retail, SMEs, and healthcare industries. Information is garnered without modifying any parameters in the variable ecosystem.
• Multiple samples can be analyzed and compared using a cross-sectional survey research method.
• Multiple variables can be evaluated using this type of survey research.
• The only disadvantage of cross-sectional surveys is that the cause-effect relationship of variables cannot be established as it usually evaluates variables at a particular time and not across a continuous time frame.

Longitudinal surveys: Longitudinal surveys are also observational surveys , but unlike cross-sectional surveys, longitudinal surveys are conducted across various time durations to observe a change in respondent behavior and thought processes. This time can be days, months, years, or even decades. For instance, a researcher planning to analyze the change in buying habits of teenagers over 5 years will conduct longitudinal surveys.

• In cross-sectional surveys, the same variables were evaluated at a given time, and in longitudinal surveys, different variables can be analyzed at different intervals.
• Longitudinal surveys are extensively used in the field of medicine and applied sciences. Apart from these two fields, they are also used to observe a change in the market trend analysis , analyze customer satisfaction, or gain feedback on products/services.
• In situations where the sequence of events is highly essential, longitudinal surveys are used.
• Researchers say that when research subjects need to be thoroughly inspected before concluding, they rely on longitudinal surveys.

02. Correlational Research

A comparison between two entities is invariable. Correlation research is conducted to establish a relationship between two closely-knit entities and how one impacts the other, and what changes are eventually observed. This research method is carried out to give value to naturally occurring relationships, and a minimum of two different groups are required to conduct this quantitative research method successfully. Without assuming various aspects, a relationship between two groups or entities must be established.

Researchers use this quantitative research design to correlate two or more variables using mathematical analysis methods. Patterns, relationships, and trends between variables are concluded as they exist in their original setup. The impact of one of these variables on the other is observed, along with how it changes the relationship between the two variables. Researchers tend to manipulate one of the variables to attain the desired results.

Ideally, it is advised not to make conclusions merely based on correlational research. This is because it is not mandatory that if two variables are in sync that they are interrelated.

Example of Correlational Research Questions :

• The relationship between stress and depression.
• The equation between fame and money.
• The relation between activities in a third-grade class and its students.

03. Causal-comparative Research

This research method mainly depends on the factor of comparison. Also called quasi-experimental research , this quantitative research method is used by researchers to conclude the cause-effect equation between two or more variables, where one variable is dependent on the other independent variable. The independent variable is established but not manipulated, and its impact on the dependent variable is observed. These variables or groups must be formed as they exist in the natural setup. As the dependent and independent variables will always exist in a group, it is advised that the conclusions are carefully established by keeping all the factors in mind.

Causal-comparative research is not restricted to the statistical analysis of two variables but extends to analyzing how various variables or groups change under the influence of the same changes. This research is conducted irrespective of the type of relationship that exists between two or more variables. Statistical analysis plan is used to present the outcome using this quantitative research method.

Example of Causal-Comparative Research Questions:

• The impact of drugs on a teenager. The effect of good education on a freshman. The effect of substantial food provision in the villages of Africa.

04. Experimental Research

Also known as true experimentation, this research method relies on a theory. As the name suggests, experimental research is usually based on one or more theories. This theory has yet to be proven before and is merely a supposition. In experimental research, an analysis is done around proving or disproving the statement. This research method is used in natural sciences. Traditional research methods are more effective than modern techniques.

There can be multiple theories in experimental research. A theory is a statement that can be verified or refuted.

After establishing the statement, efforts are made to understand whether it is valid or invalid. This quantitative research method is mainly used in natural or social sciences as various statements must be proved right or wrong.

• Traditional research methods are more effective than modern techniques.
• Systematic teaching schedules help children who struggle to cope with the course.
• It is a boon to have responsible nursing staff for ailing parents.

B. Data Collection Methodologies

The second major step in primary quantitative research is data collection. Data collection can be divided into sampling methods and data collection using surveys and polls.

01. Data Collection Methodologies: Sampling Methods

There are two main sampling methods for quantitative research: Probability and Non-probability sampling .

Probability sampling: A theory of probability is used to filter individuals from a population and create samples in probability sampling . Participants of a sample are chosen by random selection processes. Each target audience member has an equal opportunity to be selected in the sample.

There are four main types of probability sampling:

• Simple random sampling: As the name indicates, simple random sampling is nothing but a random selection of elements for a sample. This sampling technique is implemented where the target population is considerably large.
• Stratified random sampling: In the stratified random sampling method , a large population is divided into groups (strata), and members of a sample are chosen randomly from these strata. The various segregated strata should ideally not overlap one another.
• Cluster sampling: Cluster sampling is a probability sampling method using which the main segment is divided into clusters, usually using geographic segmentation and demographic segmentation parameters.
• Systematic sampling: Systematic sampling is a technique where the starting point of the sample is chosen randomly, and all the other elements are chosen using a fixed interval. This interval is calculated by dividing the population size by the target sample size.

Non-probability sampling: Non-probability sampling is where the researcher’s knowledge and experience are used to create samples. Because of the researcher’s involvement, not all the target population members have an equal probability of being selected to be a part of a sample.

There are five non-probability sampling models:

• Convenience sampling: In convenience sampling , elements of a sample are chosen only due to one prime reason: their proximity to the researcher. These samples are quick and easy to implement as there is no other parameter of selection involved.
• Consecutive sampling: Consecutive sampling is quite similar to convenience sampling, except for the fact that researchers can choose a single element or a group of samples and conduct research consecutively over a significant period and then perform the same process with other samples.
• Quota sampling: Using quota sampling , researchers can select elements using their knowledge of target traits and personalities to form strata. Members of various strata can then be chosen to be a part of the sample as per the researcher’s understanding.
• Snowball sampling: Snowball sampling is conducted with target audiences who are difficult to contact and get information. It is popular in cases where the target audience for analysis research is rare to put together.
• Judgmental sampling: Judgmental sampling is a non-probability sampling method where samples are created only based on the researcher’s experience and research skill .

02. Data collection methodologies: Using surveys & polls

Once the sample is determined, then either surveys or polls can be distributed to collect the data for quantitative research.

Using surveys for primary quantitative research

A survey is defined as a research method used for collecting data from a pre-defined group of respondents to gain information and insights on various topics of interest. The ease of survey distribution and the wide number of people it can reach depending on the research time and objective makes it one of the most important aspects of conducting quantitative research.

Fundamental levels of measurement – nominal, ordinal, interval, and ratio scales

Four measurement scales are fundamental to creating a multiple-choice question in a survey. They are nominal, ordinal, interval, and ratio measurement scales without the fundamentals of which no multiple-choice questions can be created. Hence, it is crucial to understand these measurement levels to develop a robust survey.

Use of different question types

To conduct quantitative research, close-ended questions must be used in a survey. They can be a mix of multiple question types, including multiple-choice questions like semantic differential scale questions , rating scale questions , etc.

Survey Distribution and Survey Data Collection

In the above, we have seen the process of building a survey along with the research design to conduct primary quantitative research. Survey distribution to collect data is the other important aspect of the survey process. There are different ways of survey distribution. Some of the most commonly used methods are:

• Email: Sending a survey via email is the most widely used and effective survey distribution method. This method’s response rate is high because the respondents know your brand. You can use the QuestionPro email management feature to send out and collect survey responses.
• Buy respondents: Another effective way to distribute a survey and conduct primary quantitative research is to use a sample. Since the respondents are knowledgeable and are on the panel by their own will, responses are much higher.
• Embed survey on a website: Embedding a survey on a website increases a high number of responses as the respondent is already in close proximity to the brand when the survey pops up.
• Social distribution: Using social media to distribute the survey aids in collecting a higher number of responses from the people that are aware of the brand.
• QR code: QuestionPro QR codes store the URL for the survey. You can print/publish this code in magazines, signs, business cards, or on just about any object/medium.
• SMS survey: The SMS survey is a quick and time-effective way to collect a high number of responses.
• Offline Survey App: The QuestionPro App allows users to circulate surveys quickly, and the responses can be collected both online and offline.

Survey example

An example of a survey is a short customer satisfaction (CSAT) survey that can quickly be built and deployed to collect feedback about what the customer thinks about a brand and how satisfied and referenceable the brand is.

Using polls for primary quantitative research

Polls are a method to collect feedback using close-ended questions from a sample. The most commonly used types of polls are election polls and exit polls . Both of these are used to collect data from a large sample size but using basic question types like multiple-choice questions.

C. Data Analysis Techniques

The third aspect of primary quantitative research design is data analysis . After collecting raw data, there must be an analysis of this data to derive statistical inferences from this research. It is important to relate the results to the research objective and establish the statistical relevance of the results.

Remember to consider aspects of research that were not considered for the data collection process and report the difference between what was planned vs. what was actually executed.

It is then required to select precise Statistical Analysis Methods , such as SWOT, Conjoint, Cross-tabulation, etc., to analyze the quantitative data.

• SWOT analysis: SWOT Analysis stands for the acronym of Strengths, Weaknesses, Opportunities, and Threat analysis. Organizations use this statistical analysis technique to evaluate their performance internally and externally to develop effective strategies for improvement.
• Conjoint Analysis: Conjoint Analysis is a market analysis method to learn how individuals make complicated purchasing decisions. Trade-offs are involved in an individual’s daily activities, and these reflect their ability to decide from a complex list of product/service options.
• Cross-tabulation: Cross-tabulation is one of the preliminary statistical market analysis methods which establishes relationships, patterns, and trends within the various parameters of the research study.
• TURF Analysis: TURF Analysis , an acronym for Totally Unduplicated Reach and Frequency Analysis, is executed in situations where the reach of a favorable communication source is to be analyzed along with the frequency of this communication. It is used for understanding the potential of a target market.

Inferential statistics methods such as confidence interval, the margin of error, etc., can then be used to provide results.

Secondary Quantitative Research Methods

Secondary quantitative research or desk research is a research method that involves using already existing data or secondary data. Existing data is summarized and collated to increase the overall effectiveness of the research.

This research method involves collecting quantitative data from existing data sources like the internet, government resources, libraries, research reports, etc. Secondary quantitative research helps to validate the data collected from primary quantitative research and aid in strengthening or proving, or disproving previously collected data.

The following are five popularly used secondary quantitative research methods:

• Data available on the internet: With the high penetration of the internet and mobile devices, it has become increasingly easy to conduct quantitative research using the internet. Information about most research topics is available online, and this aids in boosting the validity of primary quantitative data.
• Government and non-government sources: Secondary quantitative research can also be conducted with the help of government and non-government sources that deal with market research reports. This data is highly reliable and in-depth and hence, can be used to increase the validity of quantitative research design.
• Public libraries: Now a sparingly used method of conducting quantitative research, it is still a reliable source of information, though. Public libraries have copies of important research that was conducted earlier. They are a storehouse of valuable information and documents from which information can be extracted.
• Educational institutions: Educational institutions conduct in-depth research on multiple topics, and hence, the reports that they publish are an important source of validation in quantitative research.
• Commercial information sources: Local newspapers, journals, magazines, radio, and TV stations are great sources to obtain data for secondary quantitative research. These commercial information sources have in-depth, first-hand information on market research, demographic segmentation, and similar subjects.

Quantitative Research Examples

Some examples of quantitative research are:

• A customer satisfaction template can be used if any organization would like to conduct a customer satisfaction (CSAT) survey . Through this kind of survey, an organization can collect quantitative data and metrics on the goodwill of the brand or organization in the customer’s mind based on multiple parameters such as product quality, pricing, customer experience, etc. This data can be collected by asking a net promoter score (NPS) question , matrix table questions, etc. that provide data in the form of numbers that can be analyzed and worked upon.
• Another example of quantitative research is an organization that conducts an event, collecting feedback from attendees about the value they see from the event. By using an event survey , the organization can collect actionable feedback about the satisfaction levels of customers during various phases of the event such as the sales, pre and post-event, the likelihood of recommending the organization to their friends and colleagues, hotel preferences for the future events and other such questions.

What are the Advantages of Quantitative Research?

There are many advantages to quantitative research. Some of the major advantages of why researchers use this method in market research are:

Collect Reliable and Accurate Data:

Quantitative research is a powerful method for collecting reliable and accurate quantitative data. Since data is collected, analyzed, and presented in numbers, the results obtained are incredibly reliable and objective. Numbers do not lie and offer an honest and precise picture of the conducted research without discrepancies. In situations where a researcher aims to eliminate bias and predict potential conflicts, quantitative research is the method of choice.

Quick Data Collection:

Quantitative research involves studying a group of people representing a larger population. Researchers use a survey or another quantitative research method to efficiently gather information from these participants, making the process of analyzing the data and identifying patterns faster and more manageable through the use of statistical analysis. This advantage makes quantitative research an attractive option for projects with time constraints.

Wider Scope of Data Analysis:

Quantitative research, thanks to its utilization of statistical methods, offers an extensive range of data collection and analysis. Researchers can delve into a broader spectrum of variables and relationships within the data, enabling a more thorough comprehension of the subject under investigation. This expanded scope is precious when dealing with complex research questions that require in-depth numerical analysis.

Eliminate Bias:

One of the significant advantages of quantitative research is its ability to eliminate bias. This research method leaves no room for personal comments or the biasing of results, as the findings are presented in numerical form. This objectivity makes the results fair and reliable in most cases, reducing the potential for researcher bias or subjectivity.

In summary, quantitative research involves collecting, analyzing, and presenting quantitative data using statistical analysis. It offers numerous advantages, including the collection of reliable and accurate data, quick data collection, a broader scope of data analysis, and the elimination of bias, making it a valuable approach in the field of research. When considering the benefits of quantitative research, it’s essential to recognize its strengths in contrast to qualitative methods and its role in collecting and analyzing numerical data for a more comprehensive understanding of research topics.

Best Practices to Conduct Quantitative Research

Here are some best practices for conducting quantitative research:

• Differentiate between quantitative and qualitative: Understand the difference between the two methodologies and apply the one that suits your needs best.
• Choose a suitable sample size: Ensure that you have a sample representative of your population and large enough to be statistically weighty.
• Keep your research goals clear and concise: Know your research goals before you begin data collection to ensure you collect the right amount and the right quantity of data.
• Keep the questions simple: Remember that you will be reaching out to a demographically wide audience. Pose simple questions for your respondents to understand easily.

Quantitative Research vs Qualitative Research

Quantitative research and qualitative research are two distinct approaches to conducting research, each with its own set of methods and objectives. Here’s a comparison of the two:

Quantitative Research

• Objective: The primary goal of quantitative research is to quantify and measure phenomena by collecting numerical data. It aims to test hypotheses, establish patterns, and generalize findings to a larger population.
• Data Collection: Quantitative research employs systematic and standardized approaches for data collection, including techniques like surveys, experiments, and observations that involve predefined variables. It is often collected from a large and representative sample.
• Data Analysis: Data is analyzed using statistical techniques, such as descriptive statistics, inferential statistics, and mathematical modeling. Researchers use statistical tests to draw conclusions and make generalizations based on numerical data.
• Sample Size: Quantitative research often involves larger sample sizes to ensure statistical significance and generalizability.
• Results: The results are typically presented in tables, charts, and statistical summaries, making them highly structured and objective.
• Generalizability: Researchers intentionally structure quantitative research to generate outcomes that can be helpful to a larger population, and they frequently seek to establish causative connections.
• Emphasis on Objectivity: Researchers aim to minimize bias and subjectivity, focusing on replicable and objective findings.

Qualitative Research

• Objective: Qualitative research seeks to gain a deeper understanding of the underlying motivations, behaviors, and experiences of individuals or groups. It explores the context and meaning of phenomena.
• Data Collection: Qualitative research employs adaptable and open-ended techniques for data collection, including methods like interviews, focus groups, observations, and content analysis. It allows participants to express their perspectives in their own words.
• Data Analysis: Data is analyzed through thematic analysis, content analysis, or grounded theory. Researchers focus on identifying patterns, themes, and insights in the data.
• Sample Size: Qualitative research typically involves smaller sample sizes due to the in-depth nature of data collection and analysis.
• Results: Findings are presented in narrative form, often in the participants’ own words. Results are subjective, context-dependent, and provide rich, detailed descriptions.
• Generalizability: Qualitative research does not aim for broad generalizability but focuses on in-depth exploration within a specific context. It provides a detailed understanding of a particular group or situation.
• Emphasis on Subjectivity: Researchers acknowledge the role of subjectivity and the researcher’s influence on the Research Process . Participant perspectives and experiences are central to the findings.

Researchers choose between quantitative and qualitative research methods based on their research objectives and the nature of the research question. Each approach has its advantages and drawbacks, and the decision between them hinges on the particular research objectives and the data needed to address research inquiries effectively.

Quantitative research is a structured way of collecting and analyzing data from various sources. Its purpose is to quantify the problem and understand its extent, seeking results that someone can project to a larger population.

Companies that use quantitative rather than qualitative research typically aim to measure magnitudes and seek objectively interpreted statistical results. So if you want to obtain quantitative data that helps you define the structured cause-and-effect relationship between the research problem and the factors, you should opt for this type of research.

At QuestionPro , we have various Best Data Collection Tools and features to conduct investigations of this type. You can create questionnaires and distribute them through our various methods. We also have sample services or various questions to guarantee the success of your study and the quality of the collected data.

Quantitative research is a systematic and structured approach to studying phenomena that involves the collection of measurable data and the application of statistical, mathematical, or computational techniques for analysis.

Quantitative research is characterized by structured tools like surveys, substantial sample sizes, closed-ended questions, reliance on prior studies, data presented numerically, and the ability to generalize findings to the broader population.

The two main methods of quantitative research are Primary quantitative research methods, involving data collection directly from sources, and Secondary quantitative research methods, which utilize existing data for analysis.

1.Surveying to measure employee engagement with numerical rating scales. 2.Analyzing sales data to identify trends in product demand and market share. 4.Examining test scores to assess the impact of a new teaching method on student performance. 4.Using website analytics to track user behavior and conversion rates for an online store.

1.Differentiate between quantitative and qualitative approaches. 2.Choose a representative sample size. 3.Define clear research goals before data collection. 4.Use simple and easily understandable survey questions.

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Home » Quantitative Research – Methods, Types and Analysis

Quantitative Research – Methods, Types and Analysis

Table of Contents

Quantitative Research

Quantitative research is a type of research that collects and analyzes numerical data to test hypotheses and answer research questions . This research typically involves a large sample size and uses statistical analysis to make inferences about a population based on the data collected. It often involves the use of surveys, experiments, or other structured data collection methods to gather quantitative data.

Quantitative Research Methods

Quantitative Research Methods are as follows:

Descriptive Research Design

Descriptive research design is used to describe the characteristics of a population or phenomenon being studied. This research method is used to answer the questions of what, where, when, and how. Descriptive research designs use a variety of methods such as observation, case studies, and surveys to collect data. The data is then analyzed using statistical tools to identify patterns and relationships.

Correlational Research Design

Correlational research design is used to investigate the relationship between two or more variables. Researchers use correlational research to determine whether a relationship exists between variables and to what extent they are related. This research method involves collecting data from a sample and analyzing it using statistical tools such as correlation coefficients.

Quasi-experimental Research Design

Quasi-experimental research design is used to investigate cause-and-effect relationships between variables. This research method is similar to experimental research design, but it lacks full control over the independent variable. Researchers use quasi-experimental research designs when it is not feasible or ethical to manipulate the independent variable.

Experimental Research Design

Experimental research design is used to investigate cause-and-effect relationships between variables. This research method involves manipulating the independent variable and observing the effects on the dependent variable. Researchers use experimental research designs to test hypotheses and establish cause-and-effect relationships.

Survey Research

Survey research involves collecting data from a sample of individuals using a standardized questionnaire. This research method is used to gather information on attitudes, beliefs, and behaviors of individuals. Researchers use survey research to collect data quickly and efficiently from a large sample size. Survey research can be conducted through various methods such as online, phone, mail, or in-person interviews.

Quantitative Research Analysis Methods

Here are some commonly used quantitative research analysis methods:

Statistical Analysis

Statistical analysis is the most common quantitative research analysis method. It involves using statistical tools and techniques to analyze the numerical data collected during the research process. Statistical analysis can be used to identify patterns, trends, and relationships between variables, and to test hypotheses and theories.

Regression Analysis

Regression analysis is a statistical technique used to analyze the relationship between one dependent variable and one or more independent variables. Researchers use regression analysis to identify and quantify the impact of independent variables on the dependent variable.

Factor Analysis

Factor analysis is a statistical technique used to identify underlying factors that explain the correlations among a set of variables. Researchers use factor analysis to reduce a large number of variables to a smaller set of factors that capture the most important information.

Structural Equation Modeling

Structural equation modeling is a statistical technique used to test complex relationships between variables. It involves specifying a model that includes both observed and unobserved variables, and then using statistical methods to test the fit of the model to the data.

Time Series Analysis

Time series analysis is a statistical technique used to analyze data that is collected over time. It involves identifying patterns and trends in the data, as well as any seasonal or cyclical variations.

Multilevel Modeling

Multilevel modeling is a statistical technique used to analyze data that is nested within multiple levels. For example, researchers might use multilevel modeling to analyze data that is collected from individuals who are nested within groups, such as students nested within schools.

Applications of Quantitative Research

Quantitative research has many applications across a wide range of fields. Here are some common examples:

• Market Research : Quantitative research is used extensively in market research to understand consumer behavior, preferences, and trends. Researchers use surveys, experiments, and other quantitative methods to collect data that can inform marketing strategies, product development, and pricing decisions.
• Health Research: Quantitative research is used in health research to study the effectiveness of medical treatments, identify risk factors for diseases, and track health outcomes over time. Researchers use statistical methods to analyze data from clinical trials, surveys, and other sources to inform medical practice and policy.
• Social Science Research: Quantitative research is used in social science research to study human behavior, attitudes, and social structures. Researchers use surveys, experiments, and other quantitative methods to collect data that can inform social policies, educational programs, and community interventions.
• Education Research: Quantitative research is used in education research to study the effectiveness of teaching methods, assess student learning outcomes, and identify factors that influence student success. Researchers use experimental and quasi-experimental designs, as well as surveys and other quantitative methods, to collect and analyze data.
• Environmental Research: Quantitative research is used in environmental research to study the impact of human activities on the environment, assess the effectiveness of conservation strategies, and identify ways to reduce environmental risks. Researchers use statistical methods to analyze data from field studies, experiments, and other sources.

Characteristics of Quantitative Research

Here are some key characteristics of quantitative research:

• Numerical data : Quantitative research involves collecting numerical data through standardized methods such as surveys, experiments, and observational studies. This data is analyzed using statistical methods to identify patterns and relationships.
• Large sample size: Quantitative research often involves collecting data from a large sample of individuals or groups in order to increase the reliability and generalizability of the findings.
• Objective approach: Quantitative research aims to be objective and impartial in its approach, focusing on the collection and analysis of data rather than personal beliefs, opinions, or experiences.
• Control over variables: Quantitative research often involves manipulating variables to test hypotheses and establish cause-and-effect relationships. Researchers aim to control for extraneous variables that may impact the results.
• Replicable : Quantitative research aims to be replicable, meaning that other researchers should be able to conduct similar studies and obtain similar results using the same methods.
• Statistical analysis: Quantitative research involves using statistical tools and techniques to analyze the numerical data collected during the research process. Statistical analysis allows researchers to identify patterns, trends, and relationships between variables, and to test hypotheses and theories.
• Generalizability: Quantitative research aims to produce findings that can be generalized to larger populations beyond the specific sample studied. This is achieved through the use of random sampling methods and statistical inference.

Examples of Quantitative Research

Here are some examples of quantitative research in different fields:

• Market Research: A company conducts a survey of 1000 consumers to determine their brand awareness and preferences. The data is analyzed using statistical methods to identify trends and patterns that can inform marketing strategies.
• Health Research : A researcher conducts a randomized controlled trial to test the effectiveness of a new drug for treating a particular medical condition. The study involves collecting data from a large sample of patients and analyzing the results using statistical methods.
• Social Science Research : A sociologist conducts a survey of 500 people to study attitudes toward immigration in a particular country. The data is analyzed using statistical methods to identify factors that influence these attitudes.
• Education Research: A researcher conducts an experiment to compare the effectiveness of two different teaching methods for improving student learning outcomes. The study involves randomly assigning students to different groups and collecting data on their performance on standardized tests.
• Environmental Research : A team of researchers conduct a study to investigate the impact of climate change on the distribution and abundance of a particular species of plant or animal. The study involves collecting data on environmental factors and population sizes over time and analyzing the results using statistical methods.
• Psychology : A researcher conducts a survey of 500 college students to investigate the relationship between social media use and mental health. The data is analyzed using statistical methods to identify correlations and potential causal relationships.
• Political Science: A team of researchers conducts a study to investigate voter behavior during an election. They use survey methods to collect data on voting patterns, demographics, and political attitudes, and analyze the results using statistical methods.

How to Conduct Quantitative Research

Here is a general overview of how to conduct quantitative research:

• Develop a research question: The first step in conducting quantitative research is to develop a clear and specific research question. This question should be based on a gap in existing knowledge, and should be answerable using quantitative methods.
• Develop a research design: Once you have a research question, you will need to develop a research design. This involves deciding on the appropriate methods to collect data, such as surveys, experiments, or observational studies. You will also need to determine the appropriate sample size, data collection instruments, and data analysis techniques.
• Collect data: The next step is to collect data. This may involve administering surveys or questionnaires, conducting experiments, or gathering data from existing sources. It is important to use standardized methods to ensure that the data is reliable and valid.
• Analyze data : Once the data has been collected, it is time to analyze it. This involves using statistical methods to identify patterns, trends, and relationships between variables. Common statistical techniques include correlation analysis, regression analysis, and hypothesis testing.
• Interpret results: After analyzing the data, you will need to interpret the results. This involves identifying the key findings, determining their significance, and drawing conclusions based on the data.
• Communicate findings: Finally, you will need to communicate your findings. This may involve writing a research report, presenting at a conference, or publishing in a peer-reviewed journal. It is important to clearly communicate the research question, methods, results, and conclusions to ensure that others can understand and replicate your research.

When to use Quantitative Research

Here are some situations when quantitative research can be appropriate:

• To test a hypothesis: Quantitative research is often used to test a hypothesis or a theory. It involves collecting numerical data and using statistical analysis to determine if the data supports or refutes the hypothesis.
• To generalize findings: If you want to generalize the findings of your study to a larger population, quantitative research can be useful. This is because it allows you to collect numerical data from a representative sample of the population and use statistical analysis to make inferences about the population as a whole.
• To measure relationships between variables: If you want to measure the relationship between two or more variables, such as the relationship between age and income, or between education level and job satisfaction, quantitative research can be useful. It allows you to collect numerical data on both variables and use statistical analysis to determine the strength and direction of the relationship.
• To identify patterns or trends: Quantitative research can be useful for identifying patterns or trends in data. For example, you can use quantitative research to identify trends in consumer behavior or to identify patterns in stock market data.
• To quantify attitudes or opinions : If you want to measure attitudes or opinions on a particular topic, quantitative research can be useful. It allows you to collect numerical data using surveys or questionnaires and analyze the data using statistical methods to determine the prevalence of certain attitudes or opinions.

Purpose of Quantitative Research

The purpose of quantitative research is to systematically investigate and measure the relationships between variables or phenomena using numerical data and statistical analysis. The main objectives of quantitative research include:

• Description : To provide a detailed and accurate description of a particular phenomenon or population.
• Explanation : To explain the reasons for the occurrence of a particular phenomenon, such as identifying the factors that influence a behavior or attitude.
• Prediction : To predict future trends or behaviors based on past patterns and relationships between variables.
• Control : To identify the best strategies for controlling or influencing a particular outcome or behavior.

Quantitative research is used in many different fields, including social sciences, business, engineering, and health sciences. It can be used to investigate a wide range of phenomena, from human behavior and attitudes to physical and biological processes. The purpose of quantitative research is to provide reliable and valid data that can be used to inform decision-making and improve understanding of the world around us.

Advantages of Quantitative Research

There are several advantages of quantitative research, including:

• Objectivity : Quantitative research is based on objective data and statistical analysis, which reduces the potential for bias or subjectivity in the research process.
• Reproducibility : Because quantitative research involves standardized methods and measurements, it is more likely to be reproducible and reliable.
• Generalizability : Quantitative research allows for generalizations to be made about a population based on a representative sample, which can inform decision-making and policy development.
• Precision : Quantitative research allows for precise measurement and analysis of data, which can provide a more accurate understanding of phenomena and relationships between variables.
• Efficiency : Quantitative research can be conducted relatively quickly and efficiently, especially when compared to qualitative research, which may involve lengthy data collection and analysis.
• Large sample sizes : Quantitative research can accommodate large sample sizes, which can increase the representativeness and generalizability of the results.

Limitations of Quantitative Research

There are several limitations of quantitative research, including:

• Limited understanding of context: Quantitative research typically focuses on numerical data and statistical analysis, which may not provide a comprehensive understanding of the context or underlying factors that influence a phenomenon.
• Simplification of complex phenomena: Quantitative research often involves simplifying complex phenomena into measurable variables, which may not capture the full complexity of the phenomenon being studied.
• Potential for researcher bias: Although quantitative research aims to be objective, there is still the potential for researcher bias in areas such as sampling, data collection, and data analysis.
• Limited ability to explore new ideas: Quantitative research is often based on pre-determined research questions and hypotheses, which may limit the ability to explore new ideas or unexpected findings.
• Limited ability to capture subjective experiences : Quantitative research is typically focused on objective data and may not capture the subjective experiences of individuals or groups being studied.
• Ethical concerns : Quantitative research may raise ethical concerns, such as invasion of privacy or the potential for harm to participants.

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

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Quantitative methods emphasize objective measurements and the statistical, mathematical, or numerical analysis of data collected through polls, questionnaires, and surveys, or by manipulating pre-existing statistical data using computational techniques . Quantitative research focuses on gathering numerical data and generalizing it across groups of people or to explain a particular phenomenon.

Babbie, Earl R. The Practice of Social Research . 12th ed. Belmont, CA: Wadsworth Cengage, 2010; Muijs, Daniel. Doing Quantitative Research in Education with SPSS . 2nd edition. London: SAGE Publications, 2010.

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Characteristics of Quantitative Research

Your goal in conducting quantitative research study is to determine the relationship between one thing [an independent variable] and another [a dependent or outcome variable] within a population. Quantitative research designs are either descriptive [subjects usually measured once] or experimental [subjects measured before and after a treatment]. A descriptive study establishes only associations between variables; an experimental study establishes causality.

Quantitative research deals in numbers, logic, and an objective stance. Quantitative research focuses on numeric and unchanging data and detailed, convergent reasoning rather than divergent reasoning [i.e., the generation of a variety of ideas about a research problem in a spontaneous, free-flowing manner].

Its main characteristics are :

• The data is usually gathered using structured research instruments.
• The results are based on larger sample sizes that are representative of the population.
• The research study can usually be replicated or repeated, given its high reliability.
• Researcher has a clearly defined research question to which objective answers are sought.
• All aspects of the study are carefully designed before data is collected.
• Data are in the form of numbers and statistics, often arranged in tables, charts, figures, or other non-textual forms.
• Project can be used to generalize concepts more widely, predict future results, or investigate causal relationships.
• Researcher uses tools, such as questionnaires or computer software, to collect numerical data.

The overarching aim of a quantitative research study is to classify features, count them, and construct statistical models in an attempt to explain what is observed.

  Things to keep in mind when reporting the results of a study using quantitative methods :

• Explain the data collected and their statistical treatment as well as all relevant results in relation to the research problem you are investigating. Interpretation of results is not appropriate in this section.
• Report unanticipated events that occurred during your data collection. Explain how the actual analysis differs from the planned analysis. Explain your handling of missing data and why any missing data does not undermine the validity of your analysis.
• Explain the techniques you used to "clean" your data set.
• Choose a minimally sufficient statistical procedure ; provide a rationale for its use and a reference for it. Specify any computer programs used.
• Describe the assumptions for each procedure and the steps you took to ensure that they were not violated.
• When using inferential statistics , provide the descriptive statistics, confidence intervals, and sample sizes for each variable as well as the value of the test statistic, its direction, the degrees of freedom, and the significance level [report the actual p value].
• Avoid inferring causality , particularly in nonrandomized designs or without further experimentation.
• Use tables to provide exact values ; use figures to convey global effects. Keep figures small in size; include graphic representations of confidence intervals whenever possible.
• Always tell the reader what to look for in tables and figures .

NOTE:   When using pre-existing statistical data gathered and made available by anyone other than yourself [e.g., government agency], you still must report on the methods that were used to gather the data and describe any missing data that exists and, if there is any, provide a clear explanation why the missing data does not undermine the validity of your final analysis.

Babbie, Earl R. The Practice of Social Research . 12th ed. Belmont, CA: Wadsworth Cengage, 2010; Brians, Craig Leonard et al. Empirical Political Analysis: Quantitative and Qualitative Research Methods . 8th ed. Boston, MA: Longman, 2011; McNabb, David E. Research Methods in Public Administration and Nonprofit Management: Quantitative and Qualitative Approaches . 2nd ed. Armonk, NY: M.E. Sharpe, 2008; Quantitative Research Methods. Writing@CSU. Colorado State University; Singh, Kultar. Quantitative Social Research Methods . Los Angeles, CA: Sage, 2007.

Basic Research Design for Quantitative Studies

Before designing a quantitative research study, you must decide whether it will be descriptive or experimental because this will dictate how you gather, analyze, and interpret the results. A descriptive study is governed by the following rules: subjects are generally measured once; the intention is to only establish associations between variables; and, the study may include a sample population of hundreds or thousands of subjects to ensure that a valid estimate of a generalized relationship between variables has been obtained. An experimental design includes subjects measured before and after a particular treatment, the sample population may be very small and purposefully chosen, and it is intended to establish causality between variables. Introduction The introduction to a quantitative study is usually written in the present tense and from the third person point of view. It covers the following information:

• Identifies the research problem -- as with any academic study, you must state clearly and concisely the research problem being investigated.
• Reviews the literature -- review scholarship on the topic, synthesizing key themes and, if necessary, noting studies that have used similar methods of inquiry and analysis. Note where key gaps exist and how your study helps to fill these gaps or clarifies existing knowledge.
• Describes the theoretical framework -- provide an outline of the theory or hypothesis underpinning your study. If necessary, define unfamiliar or complex terms, concepts, or ideas and provide the appropriate background information to place the research problem in proper context [e.g., historical, cultural, economic, etc.].

Methodology The methods section of a quantitative study should describe how each objective of your study will be achieved. Be sure to provide enough detail to enable the reader can make an informed assessment of the methods being used to obtain results associated with the research problem. The methods section should be presented in the past tense.

• Study population and sampling -- where did the data come from; how robust is it; note where gaps exist or what was excluded. Note the procedures used for their selection;
• Data collection – describe the tools and methods used to collect information and identify the variables being measured; describe the methods used to obtain the data; and, note if the data was pre-existing [i.e., government data] or you gathered it yourself. If you gathered it yourself, describe what type of instrument you used and why. Note that no data set is perfect--describe any limitations in methods of gathering data.
• Data analysis -- describe the procedures for processing and analyzing the data. If appropriate, describe the specific instruments of analysis used to study each research objective, including mathematical techniques and the type of computer software used to manipulate the data.

Results The finding of your study should be written objectively and in a succinct and precise format. In quantitative studies, it is common to use graphs, tables, charts, and other non-textual elements to help the reader understand the data. Make sure that non-textual elements do not stand in isolation from the text but are being used to supplement the overall description of the results and to help clarify key points being made. Further information about how to effectively present data using charts and graphs can be found here .

• Statistical analysis -- how did you analyze the data? What were the key findings from the data? The findings should be present in a logical, sequential order. Describe but do not interpret these trends or negative results; save that for the discussion section. The results should be presented in the past tense.

Discussion Discussions should be analytic, logical, and comprehensive. The discussion should meld together your findings in relation to those identified in the literature review, and placed within the context of the theoretical framework underpinning the study. The discussion should be presented in the present tense.

• Interpretation of results -- reiterate the research problem being investigated and compare and contrast the findings with the research questions underlying the study. Did they affirm predicted outcomes or did the data refute it?
• Description of trends, comparison of groups, or relationships among variables -- describe any trends that emerged from your analysis and explain all unanticipated and statistical insignificant findings.
• Discussion of implications – what is the meaning of your results? Highlight key findings based on the overall results and note findings that you believe are important. How have the results helped fill gaps in understanding the research problem?
• Limitations -- describe any limitations or unavoidable bias in your study and, if necessary, note why these limitations did not inhibit effective interpretation of the results.

Conclusion End your study by to summarizing the topic and provide a final comment and assessment of the study.

• Summary of findings – synthesize the answers to your research questions. Do not report any statistical data here; just provide a narrative summary of the key findings and describe what was learned that you did not know before conducting the study.
• Recommendations – if appropriate to the aim of the assignment, tie key findings with policy recommendations or actions to be taken in practice.
• Future research – note the need for future research linked to your study’s limitations or to any remaining gaps in the literature that were not addressed in your study.

Black, Thomas R. Doing Quantitative Research in the Social Sciences: An Integrated Approach to Research Design, Measurement and Statistics . London: Sage, 1999; Gay,L. R. and Peter Airasain. Educational Research: Competencies for Analysis and Applications . 7th edition. Upper Saddle River, NJ: Merril Prentice Hall, 2003; Hector, Anestine. An Overview of Quantitative Research in Composition and TESOL . Department of English, Indiana University of Pennsylvania; Hopkins, Will G. “Quantitative Research Design.” Sportscience 4, 1 (2000); "A Strategy for Writing Up Research Results. The Structure, Format, Content, and Style of a Journal-Style Scientific Paper." Department of Biology. Bates College; Nenty, H. Johnson. "Writing a Quantitative Research Thesis." International Journal of Educational Science 1 (2009): 19-32; Ouyang, Ronghua (John). Basic Inquiry of Quantitative Research . Kennesaw State University.

Strengths of Using Quantitative Methods

Quantitative researchers try to recognize and isolate specific variables contained within the study framework, seek correlation, relationships and causality, and attempt to control the environment in which the data is collected to avoid the risk of variables, other than the one being studied, accounting for the relationships identified.

Among the specific strengths of using quantitative methods to study social science research problems:

• Allows for a broader study, involving a greater number of subjects, and enhancing the generalization of the results;
• Allows for greater objectivity and accuracy of results. Generally, quantitative methods are designed to provide summaries of data that support generalizations about the phenomenon under study. In order to accomplish this, quantitative research usually involves few variables and many cases, and employs prescribed procedures to ensure validity and reliability;
• Applying well established standards means that the research can be replicated, and then analyzed and compared with similar studies;
• You can summarize vast sources of information and make comparisons across categories and over time; and,
• Personal bias can be avoided by keeping a 'distance' from participating subjects and using accepted computational techniques .

Babbie, Earl R. The Practice of Social Research . 12th ed. Belmont, CA: Wadsworth Cengage, 2010; Brians, Craig Leonard et al. Empirical Political Analysis: Quantitative and Qualitative Research Methods . 8th ed. Boston, MA: Longman, 2011; McNabb, David E. Research Methods in Public Administration and Nonprofit Management: Quantitative and Qualitative Approaches . 2nd ed. Armonk, NY: M.E. Sharpe, 2008; Singh, Kultar. Quantitative Social Research Methods . Los Angeles, CA: Sage, 2007.

Limitations of Using Quantitative Methods

Quantitative methods presume to have an objective approach to studying research problems, where data is controlled and measured, to address the accumulation of facts, and to determine the causes of behavior. As a consequence, the results of quantitative research may be statistically significant but are often humanly insignificant.

Some specific limitations associated with using quantitative methods to study research problems in the social sciences include:

• Quantitative data is more efficient and able to test hypotheses, but may miss contextual detail;
• Uses a static and rigid approach and so employs an inflexible process of discovery;
• The development of standard questions by researchers can lead to "structural bias" and false representation, where the data actually reflects the view of the researcher instead of the participating subject;
• Results provide less detail on behavior, attitudes, and motivation;
• Researcher may collect a much narrower and sometimes superficial dataset;
• Results are limited as they provide numerical descriptions rather than detailed narrative and generally provide less elaborate accounts of human perception;
• The research is often carried out in an unnatural, artificial environment so that a level of control can be applied to the exercise. This level of control might not normally be in place in the real world thus yielding "laboratory results" as opposed to "real world results"; and,
• Preset answers will not necessarily reflect how people really feel about a subject and, in some cases, might just be the closest match to the preconceived hypothesis.

Research Tip

Finding Examples of How to Apply Different Types of Research Methods

SAGE publications is a major publisher of studies about how to design and conduct research in the social and behavioral sciences. Their SAGE Research Methods Online and Cases database includes contents from books, articles, encyclopedias, handbooks, and videos covering social science research design and methods including the complete Little Green Book Series of Quantitative Applications in the Social Sciences and the Little Blue Book Series of Qualitative Research techniques. The database also includes case studies outlining the research methods used in real research projects. This is an excellent source for finding definitions of key terms and descriptions of research design and practice, techniques of data gathering, analysis, and reporting, and information about theories of research [e.g., grounded theory]. The database covers both qualitative and quantitative research methods as well as mixed methods approaches to conducting research.

SAGE Research Methods Online and Cases

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Research questions, hypotheses and objectives

Patricia farrugia.

* Michael G. DeGroote School of Medicine, the

Bradley A. Petrisor

† Division of Orthopaedic Surgery and the

Forough Farrokhyar

‡ Departments of Surgery and

§ Clinical Epidemiology and Biostatistics, McMaster University, Hamilton, Ont

Mohit Bhandari

There is an increasing familiarity with the principles of evidence-based medicine in the surgical community. As surgeons become more aware of the hierarchy of evidence, grades of recommendations and the principles of critical appraisal, they develop an increasing familiarity with research design. Surgeons and clinicians are looking more and more to the literature and clinical trials to guide their practice; as such, it is becoming a responsibility of the clinical research community to attempt to answer questions that are not only well thought out but also clinically relevant. The development of the research question, including a supportive hypothesis and objectives, is a necessary key step in producing clinically relevant results to be used in evidence-based practice. A well-defined and specific research question is more likely to help guide us in making decisions about study design and population and subsequently what data will be collected and analyzed. 1

Objectives of this article

In this article, we discuss important considerations in the development of a research question and hypothesis and in defining objectives for research. By the end of this article, the reader will be able to appreciate the significance of constructing a good research question and developing hypotheses and research objectives for the successful design of a research study. The following article is divided into 3 sections: research question, research hypothesis and research objectives.

Research question

Interest in a particular topic usually begins the research process, but it is the familiarity with the subject that helps define an appropriate research question for a study. 1 Questions then arise out of a perceived knowledge deficit within a subject area or field of study. 2 Indeed, Haynes suggests that it is important to know “where the boundary between current knowledge and ignorance lies.” 1 The challenge in developing an appropriate research question is in determining which clinical uncertainties could or should be studied and also rationalizing the need for their investigation.

Increasing one’s knowledge about the subject of interest can be accomplished in many ways. Appropriate methods include systematically searching the literature, in-depth interviews and focus groups with patients (and proxies) and interviews with experts in the field. In addition, awareness of current trends and technological advances can assist with the development of research questions. 2 It is imperative to understand what has been studied about a topic to date in order to further the knowledge that has been previously gathered on a topic. Indeed, some granting institutions (e.g., Canadian Institute for Health Research) encourage applicants to conduct a systematic review of the available evidence if a recent review does not already exist and preferably a pilot or feasibility study before applying for a grant for a full trial.

In-depth knowledge about a subject may generate a number of questions. It then becomes necessary to ask whether these questions can be answered through one study or if more than one study needed. 1 Additional research questions can be developed, but several basic principles should be taken into consideration. 1 All questions, primary and secondary, should be developed at the beginning and planning stages of a study. Any additional questions should never compromise the primary question because it is the primary research question that forms the basis of the hypothesis and study objectives. It must be kept in mind that within the scope of one study, the presence of a number of research questions will affect and potentially increase the complexity of both the study design and subsequent statistical analyses, not to mention the actual feasibility of answering every question. 1 A sensible strategy is to establish a single primary research question around which to focus the study plan. 3 In a study, the primary research question should be clearly stated at the end of the introduction of the grant proposal, and it usually specifies the population to be studied, the intervention to be implemented and other circumstantial factors. 4

Hulley and colleagues 2 have suggested the use of the FINER criteria in the development of a good research question ( Box 1 ). The FINER criteria highlight useful points that may increase the chances of developing a successful research project. A good research question should specify the population of interest, be of interest to the scientific community and potentially to the public, have clinical relevance and further current knowledge in the field (and of course be compliant with the standards of ethical boards and national research standards).

FINER criteria for a good research question

Adapted with permission from Wolters Kluwer Health. 2

Whereas the FINER criteria outline the important aspects of the question in general, a useful format to use in the development of a specific research question is the PICO format — consider the population (P) of interest, the intervention (I) being studied, the comparison (C) group (or to what is the intervention being compared) and the outcome of interest (O). 3 , 5 , 6 Often timing (T) is added to PICO ( Box 2 ) — that is, “Over what time frame will the study take place?” 1 The PICOT approach helps generate a question that aids in constructing the framework of the study and subsequently in protocol development by alluding to the inclusion and exclusion criteria and identifying the groups of patients to be included. Knowing the specific population of interest, intervention (and comparator) and outcome of interest may also help the researcher identify an appropriate outcome measurement tool. 7 The more defined the population of interest, and thus the more stringent the inclusion and exclusion criteria, the greater the effect on the interpretation and subsequent applicability and generalizability of the research findings. 1 , 2 A restricted study population (and exclusion criteria) may limit bias and increase the internal validity of the study; however, this approach will limit external validity of the study and, thus, the generalizability of the findings to the practical clinical setting. Conversely, a broadly defined study population and inclusion criteria may be representative of practical clinical practice but may increase bias and reduce the internal validity of the study.

PICOT criteria 1

A poorly devised research question may affect the choice of study design, potentially lead to futile situations and, thus, hamper the chance of determining anything of clinical significance, which will then affect the potential for publication. Without devoting appropriate resources to developing the research question, the quality of the study and subsequent results may be compromised. During the initial stages of any research study, it is therefore imperative to formulate a research question that is both clinically relevant and answerable.

Research hypothesis

The primary research question should be driven by the hypothesis rather than the data. 1 , 2 That is, the research question and hypothesis should be developed before the start of the study. This sounds intuitive; however, if we take, for example, a database of information, it is potentially possible to perform multiple statistical comparisons of groups within the database to find a statistically significant association. This could then lead one to work backward from the data and develop the “question.” This is counterintuitive to the process because the question is asked specifically to then find the answer, thus collecting data along the way (i.e., in a prospective manner). Multiple statistical testing of associations from data previously collected could potentially lead to spuriously positive findings of association through chance alone. 2 Therefore, a good hypothesis must be based on a good research question at the start of a trial and, indeed, drive data collection for the study.

The research or clinical hypothesis is developed from the research question and then the main elements of the study — sampling strategy, intervention (if applicable), comparison and outcome variables — are summarized in a form that establishes the basis for testing, statistical and ultimately clinical significance. 3 For example, in a research study comparing computer-assisted acetabular component insertion versus freehand acetabular component placement in patients in need of total hip arthroplasty, the experimental group would be computer-assisted insertion and the control/conventional group would be free-hand placement. The investigative team would first state a research hypothesis. This could be expressed as a single outcome (e.g., computer-assisted acetabular component placement leads to improved functional outcome) or potentially as a complex/composite outcome; that is, more than one outcome (e.g., computer-assisted acetabular component placement leads to both improved radiographic cup placement and improved functional outcome).

However, when formally testing statistical significance, the hypothesis should be stated as a “null” hypothesis. 2 The purpose of hypothesis testing is to make an inference about the population of interest on the basis of a random sample taken from that population. The null hypothesis for the preceding research hypothesis then would be that there is no difference in mean functional outcome between the computer-assisted insertion and free-hand placement techniques. After forming the null hypothesis, the researchers would form an alternate hypothesis stating the nature of the difference, if it should appear. The alternate hypothesis would be that there is a difference in mean functional outcome between these techniques. At the end of the study, the null hypothesis is then tested statistically. If the findings of the study are not statistically significant (i.e., there is no difference in functional outcome between the groups in a statistical sense), we cannot reject the null hypothesis, whereas if the findings were significant, we can reject the null hypothesis and accept the alternate hypothesis (i.e., there is a difference in mean functional outcome between the study groups), errors in testing notwithstanding. In other words, hypothesis testing confirms or refutes the statement that the observed findings did not occur by chance alone but rather occurred because there was a true difference in outcomes between these surgical procedures. The concept of statistical hypothesis testing is complex, and the details are beyond the scope of this article.

Another important concept inherent in hypothesis testing is whether the hypotheses will be 1-sided or 2-sided. A 2-sided hypothesis states that there is a difference between the experimental group and the control group, but it does not specify in advance the expected direction of the difference. For example, we asked whether there is there an improvement in outcomes with computer-assisted surgery or whether the outcomes worse with computer-assisted surgery. We presented a 2-sided test in the above example because we did not specify the direction of the difference. A 1-sided hypothesis states a specific direction (e.g., there is an improvement in outcomes with computer-assisted surgery). A 2-sided hypothesis should be used unless there is a good justification for using a 1-sided hypothesis. As Bland and Atlman 8 stated, “One-sided hypothesis testing should never be used as a device to make a conventionally nonsignificant difference significant.”

The research hypothesis should be stated at the beginning of the study to guide the objectives for research. Whereas the investigators may state the hypothesis as being 1-sided (there is an improvement with treatment), the study and investigators must adhere to the concept of clinical equipoise. According to this principle, a clinical (or surgical) trial is ethical only if the expert community is uncertain about the relative therapeutic merits of the experimental and control groups being evaluated. 9 It means there must exist an honest and professional disagreement among expert clinicians about the preferred treatment. 9

Designing a research hypothesis is supported by a good research question and will influence the type of research design for the study. Acting on the principles of appropriate hypothesis development, the study can then confidently proceed to the development of the research objective.

Research objective

The primary objective should be coupled with the hypothesis of the study. Study objectives define the specific aims of the study and should be clearly stated in the introduction of the research protocol. 7 From our previous example and using the investigative hypothesis that there is a difference in functional outcomes between computer-assisted acetabular component placement and free-hand placement, the primary objective can be stated as follows: this study will compare the functional outcomes of computer-assisted acetabular component insertion versus free-hand placement in patients undergoing total hip arthroplasty. Note that the study objective is an active statement about how the study is going to answer the specific research question. Objectives can (and often do) state exactly which outcome measures are going to be used within their statements. They are important because they not only help guide the development of the protocol and design of study but also play a role in sample size calculations and determining the power of the study. 7 These concepts will be discussed in other articles in this series.

From the surgeon’s point of view, it is important for the study objectives to be focused on outcomes that are important to patients and clinically relevant. For example, the most methodologically sound randomized controlled trial comparing 2 techniques of distal radial fixation would have little or no clinical impact if the primary objective was to determine the effect of treatment A as compared to treatment B on intraoperative fluoroscopy time. However, if the objective was to determine the effect of treatment A as compared to treatment B on patient functional outcome at 1 year, this would have a much more significant impact on clinical decision-making. Second, more meaningful surgeon–patient discussions could ensue, incorporating patient values and preferences with the results from this study. 6 , 7 It is the precise objective and what the investigator is trying to measure that is of clinical relevance in the practical setting.

The following is an example from the literature about the relation between the research question, hypothesis and study objectives:

Study: Warden SJ, Metcalf BR, Kiss ZS, et al. Low-intensity pulsed ultrasound for chronic patellar tendinopathy: a randomized, double-blind, placebo-controlled trial. Rheumatology 2008;47:467–71.

Research question: How does low-intensity pulsed ultrasound (LIPUS) compare with a placebo device in managing the symptoms of skeletally mature patients with patellar tendinopathy?

Research hypothesis: Pain levels are reduced in patients who receive daily active-LIPUS (treatment) for 12 weeks compared with individuals who receive inactive-LIPUS (placebo).

Objective: To investigate the clinical efficacy of LIPUS in the management of patellar tendinopathy symptoms.

The development of the research question is the most important aspect of a research project. A research project can fail if the objectives and hypothesis are poorly focused and underdeveloped. Useful tips for surgical researchers are provided in Box 3 . Designing and developing an appropriate and relevant research question, hypothesis and objectives can be a difficult task. The critical appraisal of the research question used in a study is vital to the application of the findings to clinical practice. Focusing resources, time and dedication to these 3 very important tasks will help to guide a successful research project, influence interpretation of the results and affect future publication efforts.

Tips for developing research questions, hypotheses and objectives for research studies

• Perform a systematic literature review (if one has not been done) to increase knowledge and familiarity with the topic and to assist with research development.
• Learn about current trends and technological advances on the topic.
• Seek careful input from experts, mentors, colleagues and collaborators to refine your research question as this will aid in developing the research question and guide the research study.
• Use the FINER criteria in the development of the research question.
• Ensure that the research question follows PICOT format.
• Develop a research hypothesis from the research question.
• Develop clear and well-defined primary and secondary (if needed) objectives.
• Ensure that the research question and objectives are answerable, feasible and clinically relevant.

FINER = feasible, interesting, novel, ethical, relevant; PICOT = population (patients), intervention (for intervention studies only), comparison group, outcome of interest, time.

Competing interests: No funding was received in preparation of this paper. Dr. Bhandari was funded, in part, by a Canada Research Chair, McMaster University.

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Tailoring bistability in optical tweezers with vortex beams and spherical aberration

Arthur l. da fonseca, kainã diniz, paula b. monteiro, luís b. pires, guilherme t. moura, mateus borges, rafael s. dutra, diney s. ether, jr., nathan b. viana, and paulo a. maia neto, phys. rev. research 6 , 023226 – published 3 june 2024.

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We demonstrate a bistable optical trap by tightly focusing a vortex laser beam. The optical potential has the form of a Mexican hat with an additional minimum at the center. The bistable trapping corresponds to a nonequilibrium steady state, where the microsphere continually hops, due to thermal activation, between an axial equilibrium state and an orbital state driven by the optical torque. We develop a theoretical model for the optical force field, based entirely on experimentally accessible parameters, combining a Debye-type nonparaxial description of the focused vortex beam with Mie scattering by the microsphere. The theoretical prediction that the microsphere and the annular laser focal spot should have comparable sizes is confirmed experimentally by taking different values for the topological charge of the vortex beam. Spherical aberration introduced by refraction at the interface between the glass slide and the sample is considered and allows us to finetune between axial, bistable, and orbital states as the sample is shifted with respect to the objective focal plane. We find overall agreement between theory and experiment for a rather broad range of topological charges. Our results open the way for applications in stochastic thermodynamics, as they establish a control parameter—the height of the objective focal plane with respect to the glass slide—that allows us to shape the optical force field in real time and in a controllable way.

• Received 1 November 2023
• Accepted 29 April 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.023226

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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• 1 Instituto de Física, Universidade Federal do Rio de Janeiro Caixa Postal 68528, Rio de Janeiro, Rio de Janeiro 21941-972, Brazil
• 2 CENABIO—Centro Nacional de Biologia Estrutural e Bioimagem, Universidade Federal do Rio de Janeiro , Rio de Janeiro, Rio de Janeiro 21941-902, Brazil
• 3 Instituto Federal de Santa Catarina , Campus Florianopolis, Av. Mauro Ramos, 950, Florianópolis, Santa Catarina 88020-300, Brazil
• 4 Université de Strasbourg , CNRS, Institut de Science et d'Ingénierie Supramoléculaires, UMR 7006, F-67000 Strasbourg, France
• 5 LISComp-IFRJ, Instituto Federal de Educação , Ciência e Tecnologia, Rua Sebastão de Lacerda, Paracambi, Rio de Janeiro 26600-000, Brasil
• * [email protected]

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Illustration of the experimental setup. The laser beam goes through a polarizing beam splitter (PBS) and is directed to the spatial light modulator (SLM). The first order of diffraction propagates through a quarter-wave plate (QWP) and, for values of ℓ up to 5, through a beam expander (lenses L1 and L2) toward the microscope (dotted frame). Inside the microscope, the beam is reflected by a dichroic mirror (DM) and then focused by an oil-immersion objective. The inset shows a magnified view of the sample region with the glass slide lying at its bottom. The refractive index mismatch between the glass slide (index n i ) and the aqueous medium filling the sample (index n 1 ) leads to the introduction of a spherical aberration phase upon refraction at the interface between the slide and the sample. Such a phase is proportional to the height L of the objective focal plane with respect to the slide, whose position is controlled by a piezoelectric nanopositioning stage (not shown). The resulting nonparaxial focused beam is indicated by the density plot of the electric energy density for L = 10 µ m (red). The inset also depicts a trapped microsphere of refractive index n 2 .

Transverse trap stiffness per unit power κ ρ / P as a function of the sphere radius for topological charges ℓ = 1 (blue), 5 (red), and 10 (green). We consider an aberration-free trapping beam. As the particle size increases, κ ρ changes sign from negative to positive at the critical radius R on .

For an aberration-free trapping beam, the parameter space spanned by the microsphere radius and the topological charge ℓ is divided into three different regions. For radii smaller than the critical radius R on ( ℓ ) (green), on-axis trapping is excluded, and the particle rotates around the optical axis along a circular trajectory. For radii larger than the critical radius R off ( ℓ ) (blue), orbital trapping is excluded, and the particle is trapped at a position along the optical axis. On-axis and orbital states coexist in the (colored) bistable stripe bounded by the plots of R on ( ℓ ) and R off ( ℓ ) . Such a region corresponds to radii close to the maximum of electric energy density r ̃ ℓ (circle). When an oil-immersion objective is employed, the spherical aberration introduced by refraction at the glass slide opens the way to switch between trapping regimes by changing the height L of the objective focal plane. However, experimental bistability is possible only for radii and topological charges (red stars) close to or within the orange colored stripe.

Radial optical force component Q ρ as a function of radial position ρ for ℓ = 11 and radii a = 1.5 µ m (blue), a = 2.25 µ m (red), and a = 3.5 µ m (green). The fill plot indicates the electric energy density variation with distance to the optical axis. Its maximum is located at r ̃ ℓ = 2.1 µ m

Experimental realization of bistability with a polystyrene microsphere of radius a = 1.5 µ m and a vortex beam with ℓ = 8 . (a)–(d) show frames of the trapped microsphere at times t ( a ) = 85 s ,   t ( b ) = 90 s ,   t ( c ) = 95 s , and t ( d ) = 100 s . The outline (dash) of the trapped microsphere of diameter a = 3 µ m is superimposed on each image to represent the scale. When the microsphere hops to an off-axis location (b) and (c), it circulates around the axis. The black arrows indicate the direction of motion in (b) and (c). The difference between the image patterns in (a) and (d) and in (c) and (d) indicates that the trapping height changes when hoping from axial equilibrium to orbital motion. (e) Time series for the radial coordinate. The vertical green lines indicate the times corresponding to the frames shown in (a)–(d). The horizontal dashed line represents the mean radial coordinate ρ exp = ( 1.04 ± 0.05 ) µ m for the orbital state. (f) Color map of the energy distribution U ( x , y ) / ( k B T ) across the x y plane as derived from the position distribution density p ( x , y ) . (g) Power spectrum density (PSD) of the x coordinate showing a peak at f = 1.82 Hz , which corresponds to an orbital period T exp = ( 0.55 ± 0.03 ) s .

Theoretical results for the variation of the optical force field with spherical aberration. The distance L between the objective focal plane (for paraxial rays) and the glass slide is increased from left to right, thus enhancing the spherical aberration introduced by refraction at the interface between the slide and the sample. We consider a vortex beam with ℓ = 8 and left-handed circular polarization ( σ = + 1 ) at the objective entrance port and a polystyrene microsphere with radius a = 1.5 µ m . (a) From top to bottom, density plots representing the electric energy density E 2 and the axial and radial force components Q z and Q ρ , respectively, on a meridional plane. The force components are normalized by Eq. ( 6 ) and their values are indicated by the color bar. The roots of Q z = 0 (blue line) and Q ρ = 0 (green line) as well as the positions of axial (brown dot) and orbital (magenta dot) stable equilibria are also indicated. Optical potential (in units of the thermal energy k B T ) vs radial distance ρ for (b)  L = 0 µ m , (c)  L = 1.75 µ m , and (d)  L = 3.5 µ m .

Radius r ̃ ℓ of the focal annular spot (solid, left axis) and filling factor A ℓ representing the fraction of total power transmitted to the sample chamber (dot, right axis) as functions of the radius r ℓ of the paraxial vortex beam (in units of the objective entrance radius R p = 2.8 mm ). The topological charges are ℓ = 1 , 5 , 10 , and 15 . We consider an oil-immersion objective lens with numerical aperture NA = 1.4 .

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Origin of symmetry breaking in the grasshopper model

David llamas, jaron kent-dobias, kun chen, adrian kent, and olga goulko, phys. rev. research 6 , 023235 – published 3 june 2024.

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The planar grasshopper problem, originally introduced by Goulko and Kent [ Proc. R. Soc. A 473 , 20170494 (2017) ], is a striking example of a model with long-range isotropic interactions whose ground states break rotational symmetry. In this paper we analyze and explain the nature of this symmetry breaking with emphasis on the importance of dimensionality. Interestingly, rotational symmetry is recovered in three dimensions for small jumps, which correspond to the nonisotropic cogwheel regime of the two-dimensional problem. We discuss simplified models that reproduce the symmetry properties of the original system in N dimensions. For the full grasshopper model in two dimensions we obtain quantitative predictions for optimal perturbations of the disk. Our analytical results are confirmed by numerical simulations.

• Received 20 November 2023
• Accepted 18 March 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.023235

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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• 1 Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
• 2 Istituto Nazionale di Fisica Nucleare, Sezione di Roma I, 00185 Rome, Italy
• 3 Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, New York 10010, USA
• 4 Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
• 5 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada
• * [email protected]
• † [email protected]
• ‡ [email protected]

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The stability of flat half-space in N dimensions to plane-wave perturbations of wavenumber k . All values are negative except at k = 0 (which corresponds to translation of the interface and is not a probability-conserving perturbation) and for k d that are multiples of 2 π in the case N = 2 . These values are zero, meaning that there is a marginal stability to perturbations with wavenumber commensurate with d in two dimensions but not an instability. For larger dimension N of space, the result becomes increasingly insensitive to the commensurability of k and d .

The coefficient of stability for the disk to small perturbations of n -fold symmetry. A large dot is drawn at the smallest value of d where each curve first becomes positive. The black bar shows the point in d at which the transition to disconnected shapes occurs as measured in [ 1 ].

The scaled coefficient δ p n for increasing n on the disk as a function of k d = π n d , along with the result for the semi-infinite plane ( n = ∞ ) from ( 14 ). As n is increased, the disk result asymptotically approaches that of the semi-infinite plane. Since the finite- n curves tend to the zeros of the limit curve from above, the marginally stable points of the half-plane are destabilized at finite n .

First and second most unstable modes n at a given d for n ≤ 18 . The blue and yellow lines show the location of the first and second peak in δ p n ( d ) , respectively. The location of the first peak is extremely similar to (3.4) of [ 1 ] but not identical. If the cutoff in n is increased, more bands appear at higher n . Black markers denote the corresponding numbers of cogs for the optimal solutions found through numerical simulation in Ref. [ 1 ] (the same data are shown in the left panel of Fig. 5 in Ref. [ 1 ]). The numerical results from [ 1 ] are in very close agreement with the current prediction.

(Left) Grasshopper probability for lawns with fixed cog number for two values of d , obtained numerically for discrete lawns with 10 000 cells. The maxima corresponding to the two leading unstable modes are clearly visible ( n = 9 and n = 17 for d = 0.4 ; n = 7 and n = 15 for d = 0.46 ) and are marked with thin vertical lines. These are indeed (local) maxima of the grasshopper problem. Horizontal-dashed lines denote the corresponding disk probabilities given by Eq. ( 3 ). (Right) The same probability rescaled by the squared amplitude ε of the cogs (symbols with error bars). Solid lines show δ p n ( d ) from ( 25 ), which corresponds roughly with the finite- ε numerical data.

Study of discretization effects. (Left) The exact continuous probability functional p μ ( d ) for the solid 3-ball of unit volume given by Eq. ( 4 ) (red solid line) compared with the corresponding discrete P { s } ( d ) (black dots) as function of the grasshopper jump distance d . For d ≤ R 0 , 3 the 3-ball configuration is the optimal lawn shape. (Right) Relative deviation of P { s } ( d ) for the solid 3-ball configuration from the corresponding p μ ( d ) as function of the lattice spacing h for two representative values of the grasshopper jump: d = 0.2 ≈ 0.3 R 0 , 3 (blue dots and line) and d = 0.5 ≈ 0.8 R 0 , 3 (red dots and line). For the highest resolutions considered ( M ≈ 113 000 ) the discretization error is well below 0.3 % . Lines are to guide the eye.

Study of discretization effects. (Left) The exact continuous probability functional p μ ( d ) for the 3-shell where the inner radius is selected as R i , 3 = d − R 0 , 3 (solid blue line) compared with the corresponding discrete P { s } ( d ) (black dots) as function of the grasshopper jump distance d ≥ R 0 , 3 . The corresponding probability for the solid 3-ball (red line) is also shown for comparison. For d > R 0 , 3 the 3-shell has a higher success probability than the 3-ball. (Right) Relative deviation of P { s } ( d ) for the 3-shell configuration from the corresponding p μ ( d ) as function of the lattice spacing h for two representative values of the grasshopper jump: d = 0.94 ≈ 1.5 R 0 , 3 (blue dots and line) and d = 0.77 ≈ 1.25 R 0 , 3 (red dots and line). The inner radius for each d is R i , 3 = d − R 0 , 3 , as before. For the highest resolutions considered ( M ≈ 162 000 ) the discretization error is below 1 % . Lines are to guide the eye.

(Left) Cross section of the optimal configuration for d = 0.64 R 0 , 3 found numerically for a system with M = 160 000 spins. The configuration has the shape of a solid 3-ball. This configuration was found to be optimal for all d ≤ R 0 , 3 . (Right) Cross section of the optimal configuration for d = 1.32 R 0 , 3 found numerically for a system with M = 160 , 000 spins. If the jump length exceeds R 0 , 3 the configurations remain isotropic (for d ≲ 1.4 R 0 , 3 ) but develop a spherical hole in the center; the radius of the hole grows with increasing d . Note that the outer radius of the configuration is slightly larger than R 0 , 3 to ensure that it has unit volume.

(Left) The optimal inner radius R i , 3 of the 3-shell vs grasshopper jump. The numerically found inner radii (black dots) match very well the analytical result (red solid line). The value d − R 0 , 3 (blue-dashed lines) is shown for comparison. (Right) The corresponding optimal grasshopper probabilities for the optimal inner radius (red-solid line for analytical value and black dots for the numerical value) and for d − R 0 , 3 (blue-dashed lines). The isotropic 3-shell ceases to be optimal for jumps exceeding a critical value of approximately 1.4 R 0 , 3 .

Histograms of the radial coordinates of the configuration boundary points for, from left to right, d = 1.39 R 0 , 3 (isotropic 3-shell), d = 1.42 R 0 , 3 (eightfold perturbation), and d = 1.58 R 0 , 3 (sixfold perturbation). Results shown were obtained for systems with M = 160 000 spins. The vertical-red lines denote the theoretical values for the optimal (for the respective value of d ) inner and outer radii of the corresponding 3-shell configurations. The shaded regions mark the ± h interval around the optimal radii.

Maximal values found numerically for the discrete grasshopper lawn probability P { s } ( d ) . Results shown were obtained for systems with M = 40 000 spins. Vertical lines denote boundaries between the different regimes. From left to right these are: isotropic solid ball, isotropic shells, eightfold shell perturbations, sixfold shell perturbations, ring with caps, nested crescents. Insets show examples of representative configurations. 3d animations displaying in full the features summarized here are given within the Supplemental Material [ 14 ].

The points give the difference in probability between the perturbed disk and the disk as a function of perturbation size ε for n = 2 and d = 0.98 > d 0 . They were computed using numeric integration on the full expression. The solid line gives δ p 2 ( d ) ε 2 . The results agree at small ε , as expected.

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