Standard deviation tells you how spread out a set of numbers is. If the numbers are close together, the standard deviation is small. If they are far apart, it is large.

It sounds like a hard topic, but the steps are simple. And if you just want the answer, our standard deviation calculator gives you the mean, variance and standard deviation from a list of numbers.

A quick example of why it matters

Two classes both have an average test score of 70.

• Class A scores: 68, 70, 72, 70

• Class B scores: 40, 100, 55, 85

The average is the same, but the classes are very different. In class A almost everyone is near 70. In class B scores are all over the place. Standard deviation puts a number on that difference.

Steps to find standard deviation

Let's use this data: 2, 4, 4, 4, 5, 5, 7, 9

Step 1: Find the mean (average). Add the numbers: 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40 Divide by how many there are: 40 ÷ 8 = 5

Step 2: Subtract the mean from each number, then square the result.

Step 3: Add up the squares. 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32

Step 4: Divide to get the variance. For a whole population, divide by the number of values: 32 ÷ 8 = 4

Step 5: Take the square root. √4 = 2

So the standard deviation is 2.

What is variance?

Variance is the number you get in Step 4. It is the average of the squared distances from the mean. Standard deviation is just the square root of the variance. The reason we use the square root is that it puts the answer back into the same units as your data.

Population vs sample

This is the part that confuses most people.

• Population: You have data for every member of the group. Divide by n (the count).

• Sample: You only have some of the group, and you are using it to guess about the whole group. Divide by n − 1.

Using the same data as a sample:

• Variance = 32 ÷ 7 = about 4.57

• Standard deviation = about 2.14

A sample divides by a slightly smaller number, because a sample tends to underestimate how spread out the full group is. If you aren't sure which one to use, ask yourself: do I have everyone, or just some? Most real-world data is a sample.

What does the number tell you?

For data that follows a bell curve:

• About 68% of values fall within 1 standard deviation of the mean.

• About 95% fall within 2 standard deviations.

• About 99.7% fall within 3.

So if the mean height in a group is 170 cm with a standard deviation of 7 cm, most people (about 68%) are between 163 and 177 cm.

Where people use it

• School. Statistics homework and exams.

• Science. Showing how much measurements vary.

• Finance. Measuring how much an investment goes up and down.

• Quality control. Checking if a product is made to a steady size.

• Sports. Seeing which player is more consistent.

How to use the calculators

1. Type or paste your numbers, separated by commas or spaces.

2. Choose population or sample.

3. Read the mean, variance and standard deviation.

Common questions

Can standard deviation be negative? No. It is a distance, so it is zero or higher.

What does a standard deviation of 0 mean? Every number in the list is the same.

Is a high standard deviation bad? Not on its own. It only says the data is spread out. Whether that is good or bad depends on what you are measuring.

More tools

Visit our free online calculators home page to see the full list. You'll find related tools in our math calculators section.