In this lesson, we’ll define the terms needed to calculate variance, and we’ll discuss the process for determining variance from the mean of a set of numbers.

You can also practice with an example that provides easy step-by-step calculations.

## Calculating Variance from the Mean

Calculating the variance from the mean can be quite simple; however, there are a few terms that need to be understood. In math, the **mean** is the *average* of a set of numbers – it represents the central tendency of a data set.

The **variance** is a measure of how spread out the numbers in a data set are amongst each other; the range between the highest and lowest numbers indicate variation. In general, the smaller the range, the lower the calculated variance will be. Likewise, the larger the range, the higher the calculated variance will be.

## Standard Deviation

There is one more word we should define, which is **standard deviation**. The standard deviation shows how much variation there is, or how the data disperses, around the mean or expected value. A low standard deviation from the mean will indicate that the data points are very close to the mean. A high standard deviation from the mean will indicate that the data points are spread out over a large range of values.

## Sample Problem

So, now that we have our definitions, let’s work on an example.

Let’s say we have a data set of three different numbers: 4, 5, 5, 6.The first number we need to calculate to get the variance is the mean. This is done by simply adding all the numbers and dividing the sum by how many numbers you have.

Look at the example below:

- Mean: (4 + 5 + 5 + 6) / 4 = (20) / 4 = 5

Now that we know that the mean is 5, we can calculate the variance. To calculate variance, we subtract the mean from each number in the data set and square the answer. We must then add together all the squared answers and divide by how many numbers we have in the data set.

Look at the example below:

- Variance: ((4 – 5) ² + (5 – 5) ² + (5 – 5) ² + (6 – 5) ²) / 4 = (1 + 0 + 0 + 1) / 4 = (2) / 4 = 0.5

Since we know the variance of the data set, we can also easily determine the standard deviation. The standard deviation is simply the square root of the variance, which is 0.5 in this example:

- Standard Deviation: Square root of 0.5 = .
707

## Lesson Summary

It’s important to remember that the mean is the key to calculating variation. Calculating variance requires that you subtract the mean from each number in your data set, square the answer, add all the squared answers together, and then divide that answer by the number of figures in your data set. The more numbers you have in a data set, the more complicated calculating variance can become.

This is when the calculator becomes your best friend!

## Learning Outcomes

Once you are done, you should be able to:

- Define mean
- Explain what the variance of a data set is
- Describe how to calculate the variance and standard deviation of a data set