**A discrete variable is any variable that can only take on a certain number of distinct values, typically represented by whole numbers (Norman & Streiner, 2008).**

These variables can be subdivided into dichtomous types, which only take two values, and polytomous types, taking three or more. Discrete variables are categorized as true or arbitrary based on the inherent nature of their categories.

True dichotomies and polytomies refer to variables where the categories are inherent divisions of the variable. In contrast, arbitrary dichotomies and polytomies refer to variables where the categories are artificially imposed (Babbie, 2011).

We may also want to have ranked discrete variables, such as nominal values on a likert scale (very likely, likely, unsure, unlikely, very unlikely) which would make them *ordinal discrete variables*.

**An Academic Definition:**“Discrete variables have a minimum-sized unit of measurement, which cannot be subdivided.” (Frankfort-Nachmias & Leon-Guerrero, 2006, p. 16)

**Contents**show

## Discrete Variables Examples

**1. Customer Satisfaction Survey (Arbitrary Polytomy)**

Companies often use a Likert Scale to measure customer satisfaction. Consumers rate their satisfaction with a product or service on a scale, such as: 1 (Extremely Dissatisfied), 2 (Somewhat Dissatisfied), 3 (Neutral), 4 (Somewhat Satisfied), 5 (Extremely Satisfied).

**2. Number of Cars Owned by an Individual (True Polytomy)**

The number of cars a person owns is a discrete variable. This number can range from zero to any other reasonable whole number. This constitutes a true polytomy, as the categories are naturally inherent to the variable.

**3. Number of Pencils in a Box (True Polytomy)**

Pencil boxes come packed with a clear, definite number of pencils, varying by brand and type. It’s considered a discrete variable as the count is an exact whole number. It’s a true polytomy with inherent numerical distinction.

**4. Employee Engagement Survey (Arbitrary Polytomy)**

Companies gauge employee engagement using a Likert Scale, asking employees to rate statements like “I feel valued at work”. The scale might look like this: 1 (Strongly Disagree), 2 (Disagree), 3 (Neutral), 4 (Agree), 5 (Strongly Agree).

**5. Products in Shopping Basket (Arbitrary Polytomy)**

A shopping basket can hold a varying number of products based on the shopper’s needs. This is an arbitrary polytomy as the range and categories can vary widely.

**6. Color of Traffic Lights (Arbitrary Polytomy)**

Traffic lights typically vary between three colors: red, yellow, and green. These categories, outlined by traffic rules, form an arbitrary polytomy.

**7. Teaching Evaluation Survey (Arbitrary Polytomy)**

Students can be asked to evaluate their teacher’s effectiveness using a Likert Scale, such as: 1 (Very Ineffective), 2 (Ineffective), 3 (Neither Ineffective nor Effective), 4 (Effective), 5 (Very Effective).

**8. Number of Basketball Team Members (Arbitrary Polytomy)**

Basketball teams typically include a limited number of players as determined by regulations. This forms an arbitrary polytomy, as roster sizes are usually predetermined.

**9. Presence of a Garage in a House: Yes or No (True Dichotomy)**

This parameter falls perfectly under true dichotomy, as a house either has a garage or it doesn’t. No guesswork, just a simple yes or no.

**10. Health Assessment Survey (Arbitrary Polytomy)**

Patients may rate their pain level on a Likert Scale to help physicians understand their current condition. The scale might look like this: 1 (No Pain), 2 (Mild Pain), 3 (Moderate Pain), 4 (Severe Pain), 5 (Worst Possible Pain).

**11. Number of Employees in a Company (True Polytomy)**

The workforce count of any company, large or small, is a concrete, whole number, making it a true polytomy, with divisions intrinsic to the variable.

**12. Number of Students in a Classroom (True Polytomy)**

The student count in each classroom during a particular class session is a whole number, falling perfectly into the true polytomy category.

**13. The Shoe Size of an Individual (Arbitrary Polytomy)**

Shoe sizes are discrete variables and can be broken down into various whole number sizes, quadrants, or styles. This categorization of sizes forms an arbitrary polytomy.

**14. Number of Languages Spoken by a Person (True Polytomy)**

This is a definite number and constitutes a true polytomy, as these categories are naturally occurring and inherent to language learning.

**15. Research Survey (Arbitrary Polytomy)**

Research surveys often use a Likert Scale to measure participants’ attitudes or responses to various statements. For example, in a study on dieting behaviors, participants might be asked to rate their agreement to a statement like “I regularly restrict my calorie intake for weight loss” on a scale from 1 (Strongly Disagree) to 5 (Strongly Agree).

**16. Gender: Male, Female, Others (Arbitrary Polytomy)**

By expanding beyond the binary gender categorization (male or female), we acknowledge the multitude of genders that exist, thus constituting an arbitrary polytomy.

**17. Number of Courses Taken in a Semester (Arbitrary Polytomy)**

The number of courses a student decides to take each semester varies and creates an arbitrary polytomy based on individual choices and course availability.

**18. Educational Level (Arbitrary Polytomy)**

People can be classified according to their highest completed education level—elementary, high school, bachelor’s degree, master’s degree, or doctorate. This categorization creates an arbitrary polytomy, artificially divided for analytical purposes.

**19. Number of Pets Owned (True Polytomy)**

Pet ownership is a discrete variable and can range from zero to any reasonable number, a natural division that makes it a true polytomy.

**20. Number of Children in a Family (True Polytomy)**

This is a true polytomy where the categories (number of children) are inherent and naturally divided.

**21. Stages of a Life Cycle (Arbitrary Polytomy)**

Life stages (infancy, childhood, adolescence, adulthood, old age) are discrete and form an arbitrary polytomy, as these divisions are artificially imposed for analysis purposes.

**22. Number of Laptops in an Office (True Polytomy)**

Each office will have a certain, defined number of laptops, making this a true polytomy. The categories are naturally occurring.

**23. Number of Visits to a Doctor in a Year (Arbitrary Polytomy)**

The frequency of doctor visits can vary widely among people, creating an arbitrary polytomy.

**24. Meal Preference: Vegetarian, Non-Vegetarian, Vegan (Arbitrary Polytomy)**

Meal preference is typically categorized into vegetarian, non-vegetarian, or vegan, creating an arbitrary polytomy as these categories are artificially imposed.

**25. Number of social media apps used (True Polytomy)**

The number of social media apps a person utilizes is a true polytomy. The numeric divisions are inherent and naturally occurring within this variable.

## Types of Discrete Variables (Compare and Contrast)

In the introduction, I noted that discrete variables consist of two major types: dichotomous variables (taking two distinct values) and polytomous variables (taking multiple distinct values).

Let’s look closer at both.

**True Dichotomies**are invariable with two inherent categories. For example, an individual is either pregnant or not pregnant (Orme & Combs-Orme, 2009).**True Polytomies**, on the other hand, possess three or more inherent categories. An example can be the number of books read in a month (Nemeroff & Craighead, 2002).**Arbitrary Dichotomies**are dichotomous variables, but the two divisions are not inherent and naturally occurring. An example may be categorizing people based on their height: tall and short (Mann & Lacke, 2010).**Arbitrary Polytomies**consist of three or more artificially created categories. Categorizing individuals based on their income levels into low income, middle income, and high income is a prime example of an arbitrary polytomy (Privitera, 2022).

Many others have noted that discrete variables can be classified into **ordinal or non-ordinal**. An ordinal variable refers to variables that can be placed in a *meaningful order*, such as first, second, third, and forth; or, oldest child, middle child, youngest child.

Note that many discrete variables *are *ordinal, ordinal variables are not exclusively discrete. For example, we can get a range of continuous variables and place them, too, into a meaningful order.

## Discrete vs Continuous Variables

**Discrete variables can only assume specific, separate values, while continuous variables can take on any value within a range.**

Discrete variables, by definition, can only take on a certain number of distinct values, typically whole numbers. They are used to classify and categorize data into unique bins or groups, representing units that can’t be further subdivided within the defined categories. (Mann & Lacke, 2010).

Continuous variables, on the other hand, can assume any values within a certain range. They exist on a continuum and afford infinite possibilities within a given interval. Examples may include height, weight, or time, where values exist along a continuum with infinite possibilities within a specified range (Norman & Streiner, 2008).

Unlike discrete variables that allow clear-cut classifications of data, continuous variables offer the feasibility of unlimited precision, making them ideal for instances demanding a high degree of specificity or when measuring in continuous units, such as temperature or currency.

Both discrete and continuous variables play critical roles in data analysis, statistical modeling, and research. The choice between these two largely depends on the depth of precision required and the nature of the data.

Discrete variables offer crisp delineation, making exploratory analysis and interpretation easier, while continuous variables excel at rendering a granular perspective on data, accommodating a greater depth of analysis (Privitera, 2022).

## Conclusion

Discrete variables, given their distinctive capacity to take on clearly defined, non-fractional values, serve as key players in statistical analysis and research. Unlike continuous variables, discrete variables cannot measure quantities to a very detailed degree; however, they offer the advantage of clear and distinct data categories. These variables provide a clear-cut view of the data making it easy to comprehend yet providing useful insights for data analysis, especially in cases where data precision is less of a concern. When dealing with discrete variables, it is crucial for researchers to correctly identify and categorize their data to ensure their research findings are precise and valid.

## References

Babbie, E. R. (2011). *Adventures in Social Research: Data Analysis Using IBM SPSS Statistics*. Pine Forge Press.

Frankfort-Nachmias, C., & Leon-Guerrero, A. (2006). *Social Statistics for a Diverse Society.* SAGE Publications.

Mann, P. S., & Lacke, C. J. (2010). *Introductory Statistics*. Wiley.

Nemeroff, C. B., & Craighead, W. E. (Eds.). (2002).* The Corsini Encyclopedia of Psychology and Behavioral Science, Volume 4*. Wiley.

Norman, G. R., & Streiner, D. L. (2008).* Biostatistics: The Bare Essentials*. New York: B.C. Decker.

Orme, J. G., & Combs-Orme, T. (2009). *Multiple Regression with Discrete Dependent Variables. *Oxford University Press.

Privitera, G. J. (2022). *Research Methods for the Behavioral Sciences*. New Jersey: SAGE Publications.

Weinberg, S. L., & Abramowitz, S. K. (2008). *Statistics Using SPSS: An Integrative Approach*. Cambridge: Cambridge University Press.

Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]