Difference Between Discrete Random Variables and Continuous Random Variables

You've probably heard of discrete vs continuous data. But what's the difference? What are discrete and continuous variables, and how can you distinguish between them? Read on to find out.

In fields like data analytics and data science, which often require advanced math, it's vital to understand the nature, structure, and characteristics of any dataset you're working with. Doing so helps you determine the best statistical techniques to apply (e.g. regression or classification analysis?) and to figure out which mathematical functions you might want to use for advanced analysis (such as, do you need differential or integral calculus?)

To help classify the different types of data, statisticians have long used a variety of complex yet elegant definitions. Is your data set qualitative or quantitative? Is it nominal or interval? Or is it something else entirely?

In this post, we focus on one of the most basic distinctions between different data types: discrete vs. continuous variables . Both types of quantitative data, we'll recap this before kicking off. By the time you've reached the end of this blog, you should be able to answer:

1. What are qualitative and quantitative data?
2. What are discrete variables?
3. What are continuous variables?
4. What are some examples of discrete and continuous variables?
5. Checklist: discrete vs continuous variables
6. Wrap-up and further reading

Ready? Then let's get started with a bit of background.

1. What are qualitative and quantitative data?

Of all the ways in which statisticians classify data, one of the most fundamental distinctions is that between qualitative and quantitative data. This is relevant for our current topic because, while discrete and continuous variables are distinct from each other, they are both types of quantitative data .

What is qualitative data?

The term 'qualitative' refers to anything which can be observed but not counted or measured. Qualitative data are primarily descriptive, tending to represent people's opinions or experiences. These types of data are generally collected through interviews and observations. Examples could include customer satisfaction surveys, pizza toppings, people's favorite brands, and so on.

What is quantitative data?

Quantitative data refers to anything that can be counted or measured. It is always numerical in nature. Numbers of things (e.g. lemons, melons, plants, cars, airplanes… you choose!) and measures of time, height, distance, volume, mass (and so on) are all types of quantitative data. Quantitative data can be further divided into two other types of data: discrete and continuous variables.

Since this post focuses purely on quantitative data , you can put qualitative data out of your mind for now. But if you're interested, you can learn more about the differences between qualitative and quantitative data in this post .

2. What are discrete variables?

Discrete data are a type of quantitative data that can take only fixed values. They are always numerical. These are data that can be counted , but not measured .

For example, if you conducted a household survey, you'd find that there are only certain numbers of individuals who can live under one roof. 1, 2, 3 people, and so on. While, theoretically, an infinite number of people could live in the house, the number will always be a distinct value, i.e. you cannot have 2.4 of a person living in a house.

While discrete variables are always fixed, this doesn't necessarily mean they're always whole numbers. What defines them as discrete is that there is a clear and consistent leap between variables and that these gaps don't take into account the difference. For instance, someone's shoe size might be 7.5 which is still a fixed number, but there is no shoe size of 7.7. For this reason, discrete data are, by their nature, relatively imprecise.

It's important to note here that you might find qualitative (descriptive) data described as discrete. This is probably because it can be categorized into separate groups, (e.g. cars that are blue, red, green, and so on). However, this is an inaccurate description because you cannot carry out mathematical functions on qualitative data. More accurately, they should be described as ordinal, categorical data .

Discrete data is most commonly represented using bar charts, pie charts, or scatterplots, which are excellent for comparing distinct and imprecise data points.

3. What are continuous variables?

Unlike discrete data, continuous data are not limited in the number of values they can take. If discrete data are values placed into separate boxes, you can think of continuous data as values placed along an infinite number line. Continuous variables, unlike discrete ones, can potentially be measured with an ever-increasing degree of precision. Temperature, weight, height, and length are all common examples of continuous variables.

For example, a child's birth weight can be measured to within a single gram or to within 10 grams. Or, with very accurate scales, you could measure the baby's weight to within a milligram. The point is, you can potentially measure the weight with ever-increasing degrees of accuracy because the measurement scale is continuous.

In general, continuous data is best represented using different types of visualizations like histograms or line charts, which are excellent for highlighting trends or patterns in data measured over time. And while we won't get into detail here, continuous variables can also be further subdivided into two additional data types: interval data and ratio data .

4. Discrete vs continuous data: Examples

Now we have a rough idea of the key differences between discrete vs continuous variables, let's look at some solid examples of the two.

Examples of discrete variables

• Shoe size
• Numbers of siblings
• Cars in a parking lot
• Days in the month with a temperature measuring above 30 degrees
• Number of students in a class
• A list of a baseball team's seasonal wins
• Number of different vegetables in a crate

Examples of continuous variables

• The volume of a gas tank in liters
• Wind speed in miles per hour
• The height of buildings in meters
• Length of a rope in inches
• Temperature (in degrees, on any measurement scale)
• The time it takes runners to complete a race in minutes
• The weight of a crate of vegetables in kilograms

And if you're still not clear on the difference, the next section should help.

5. Checklist: discrete vs continuous variables

Hopefully by now, you can tell the difference between discrete and continuous variables. Nevertheless, the different types can catch out even the most seasoned data analysts. The following checklist should help you distinguish between the different types at a glance.

Discrete variables:

• Have fixed values, with clear spaces between them.
• Can be counted in whole numbers, but cannot be measured.
• Cannot be divided into smaller values to add additional accuracy.
• Are most commonly represented using bar or pie charts.

Continuous variables:

• Can take on any value in a number line, and have no clear space between them.
• Can be measured but cannot be counted.
• Can be divided into an infinite number of smaller values that increase precision.
• Are most commonly represented using line graphs or histograms.

As we've seen, the distinction is not that tricky, but it's important to get right. It will, for example, determine the type of statistical analysis you carry out.

6. Wrap-up and further reading

In this post, we've explored the similarities and differences between two types of qualitative data: continuous and discrete variables. We've highlighted the importance of being able to distinguish between them and offered some examples to illustrate the differences.

Telling discrete vs continuous data apart might pose a challenge to begin with, but it'll soon become second nature once you've been working with data for a while.

To learn more about the importance of statistics in data analytics, try out afree introductory data analytics short course . For more introductory posts, you should also check out the following:

• Standard deviation vs standard error: What's the difference?
• What's the difference between covariance and correlation?
• The most useful data analysis methods and techniques

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Source: https://careerfoundry.com/en/blog/data-analytics/discrete-vs-continuous/

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