Standard Deviation Calculator

Standard deviation is a number that tells you how far individual data points usually sit from the average of the group. If the number is low, it means the data points are clustered closely around the average. If the number is high, the data points are spread out over a wider range.

There are two different ways to calculate this number. You must choose the right one based on where your data came from.

When to Use Population Standard Deviation

You use the population formula when you have collected data from every single member of the group you want to study. For example, if you measure the heights of every single student in a small classroom, you have population data.

When to Use Sample Standard Deviation

You use the sample formula when you only have a small part of a larger group. This is the most common type in real-world scenarios. For example, if you survey 100 people to guess the opinion of an entire country, you have a sample.

The formula changes slightly depending on whether you are using population or sample data. Both formulas follow similar basic steps.

Population Formula

For a population, you take each number, subtract the mean, and square the result. Then, you add all those squared numbers together and divide by the total number of items in your group. Finally, you take the square root of that answer.

Sample Formula

The sample formula is almost exactly the same. The only difference is that you divide by the total number of items minus one. Subtracting one from the sample size makes the final result more accurate for the larger population you are trying to represent.

If you want to do the math by hand, follow these simple steps.

1. Find the mean, or average, of your data set.
2. Subtract the mean from each individual number in your set.
3. Square each of those differences. Squaring means multiplying the number by itself.
4. Add all of the squared numbers together.
5. Divide that total by the number of values in your set. If it is a sample, divide by the number of values minus one. This gives you the variance.
6. Take the square root of the variance to get the standard deviation.

Worked example

A help desk counts weekly tickets for five weeks: 12, 15, 18, 14, and 21. Treat this as a sample from a longer year.

1. Mean = (12 + 15 + 18 + 14 + 21) / 5 = 80 / 5 = 16
2. Deviations from 16: −4, −1, 2, −2, 5
3. Squared deviations: 16, 1, 4, 4, 25. Sum = 50
4. Sample variance = 50 / (5 − 1) = 12.5
5. Sample standard deviation = √12.5 ≈ 3.54

If those five weeks were the entire population, divide by 5 instead of 4. Population standard deviation = √10 ≈ 3.16. Paste 12, 15, 18, 14, 21 into the tool above and switch between sample and population to see both results.

Variance is the average of the squared differences from the mean. Standard deviation is simply the square root of the variance. People use standard deviation much more often because the result is in the same units as your original data. For instance, if your data is measured in meters, the variance will be in square meters, but the standard deviation will stay in meters.

There is no single "good" number. It depends entirely on what you are measuring. A standard deviation of 2 might be very low for a person's height in inches, but it would be incredibly high for a person's shoe size. To see how a specific data point compares to the average using your new standard deviation result, you can convert it into a z-score using our z-score calculator.