# Univariate Statistics: Tests & Examples

## What are Univariate Statistics?

The field of **univarite statistics** focuses on one variable at a time and does not involve any testing of variables against each other. Rather, it provides an opportunity for the researcher to describe individual variables. Thus, this category of statistics is also referred to as **descriptive statistics**. In this lesson, we will discuss some basic statistical tests under this category including mean, mode, median, standard deviation, variance and range.

## Mean & Mode

Let's say you were working as a psychologist and had 10 clients in your caseload. You might want to know what the average age of your client group is. In order to do this, you would add up their ages and then divide by the total number of clients to find their average age. By learning their average age, you are locating the **mean**, which is the same thing as the mathematical average. For example, let's consider we have 10 clients with the ages shown here:

Client #1 = 32 years old

Client #2 = 45 years old

Client #3 = 20 years old

Client #4 = 33 years old

Client #5 = 50 years old

Client #6 = 44 years old

Client #7 = 40 years old

Client #8 = 28 years old

Client #9 = 29 years old

Client #10 = 48 years old

When you divide the total of 369 by 10 clients, the mean age across all clients is 36.9 years old.

Another univariate statistic used in psychology is the **mode**, which simply means the most reported number. If you are wanting to know the most reported age across your clients from the list of ages from the previous example, you would see that there is no mode because everyone's age is different, and there is no one age that is reported more than any other age, or more than once, for that matter. If there was a client age that was reported more than once, then you would have a modal age.

## Median

The third test, the **median**, is calculated a bit differently than the mean. The median is when half the group's age falls below the middle value, and the other half of the group's age rests above the middle value. You can locate the median, or middle, value for any information that is presented in numerical format. In order to find the median value, the following statistical process must be used:

- Line up all ages from smallest to largest
- If the total of the set of numbers is even, then add up the middle two numbers and divide by 2
- If the total of the set of numbers is odd, then the middle number is the median.

For example, if you lined up all the clients' ages in order, your list would look like this:

Client #3 = 20 years old

Client #8 = 28 years old

Client #9 = 29 years old

Client #1 = 32 years old

Client #4 = 33 years old

Client #7 = 40 years old

Client #6 = 44 years old

Client #2 = 45 years old

Client #10 = 48 years old

Client #5 = 50 years old

Then you would take the middle two values, 33 and 40 years old, add them up, and then divide by 2 to find the median to get the median age of 36.5 years old.

Remember, the median represents the middle value, so that half of the group is below the middle value and half the group is above the middle value. Don't confuse the mean with the median or think they are the same just because they may come out to similar values! They provide different statistical purposes.

## Standard Deviation & Variance

Under the umbrella of univariate statistics are other statistical tests called standard deviation and variance. Why would someone be interested in using these additional univariate statistics? Well, to most precisely report the average age of your clients, you would also want to report the standard deviation. The **standard deviation** means that the average age could deviate a little bit one way or the other. There might be a large deviation or a small deviation in age. The standard deviation is simply the square root of the variance! What is the variance? In order to find the **variance**, the following statistical process must be used:

- Add up all the ages of your clients and then find the mean age
- Subtract the mean age from each individual age
- Square each result
- Add up all the new terms
- Divide the new number by the number of clients

The final number is your variance. Once the variance is established, then you would simply take the square root of the variance to find the standard deviation. Once the standard deviation is known, you can be more sure that the univariate statistical mean is reported more accurately, knowing that the age will only deviate one way or the other slightly. For example, perhaps you calculated the variance and standard deviation for the above group of clients, and your standard deviation is 1.50. Therefore, the mean client age of 36.5 years old could deviate by 1.5 years, meaning that the true average could very well be 38 years old. Note: the standard deviation simply provides a bit more accuracy in reporting the mean.

## Range

The final test we'll discuss is the **range**, which is simply the full span of numbers used in a sample. Using the same set of clients as previously, the range of ages would be 20 years old to 50 years old, producing a 30-year difference in ages from the youngest client to the oldest client. It is often helpful to know the range of variables you are working with as a psychologist. Why? Knowing the range of ages, for example, might assist you in developing a new program geared towards a certain age group of people.

## Lesson Summary

In this lesson, we learned about **univariate statistics**, which is the field of statistics that focuses on one variable at a time and doesn't involve testing variables against each other. We saw how the mean, mode, median, standard deviation, variance, and range tests are used to get different information. The **mean** shows you the mathematical average, while the **mode** shows the most reported number. The **median** locates the middle value in a set of information, while the **range** shows you the full span of numbers used in a sample. Finally, you use the **standard deviation & variance** to show how the average could deviate a bit one way or another.

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