Probability and Statistics > Descriptive Statistics > Sample Variance

Watch the video on how to find sample variance, or read below for an explanation of what the variance means:

The sample variance, s^{2}, is used to calculate how varied a sample is. A sample is a select number of items taken from a population. For example, if you are measuring American people’s weights, it wouldn’t be feasible (from either a time or a monetary standpoint)for you to measure the weights of every person in the population. The solution is to take a sample of the population, say 1000 people, and use that sample size to estimate the actual weights of the whole population. The variance helps you to figure out how spread out your weights are.

## Definition of Sample Variance

The variance is mathematically defined as **the average of the squared differences from the mean**. But what does that actually mean in English? In order to understand what you are calculating with the variance, break it down into steps:

Step 1: Calculate the mean (the average weight).

Step 2: Subtract the mean and square the result.

Step 3: Work out the average of those differences.

- Use the sample variance and standard deviation calculator
- Or see: how to calculate the sample variance (by hand).

## What does the sample variance mean?

While the variance is useful in a mathematical sense, it won’t actually give you any information that you can use. For example, if you take a sample population of weights, you might end up with a variance of 9801. That might leave you scratching your head about why you’re calculating it in the first place! The answer is, you can use the variance to figure out the standard deviation — a much better measure of how spread out your weights are. In order to get the standard deviation, take the square root of the sample variance:

√9801 = 99.

The standard deviation, in combination with the mean, will tell you what the majority of people weigh. For example, if your mean is 150 pounds and your variance is 99 pounds, the majority of people weigh between 51 pounds (mean-99) and 249 pounds (mean+99).

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