How to Calculate Standard Deviation

Reviewed by Grace Imson, MA

Last Updated: September 11, 2023 Fact Checked

Standard deviation tells you how spread out the numbers are in a sample.^{[1]
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Part 1

Part 1 of 3:

Finding the Mean

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1
Look at your data set. This is a crucial step in any type of statistical calculation, even if it is a simple figure like the mean or median.^{[2]
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- Know how many numbers are in your sample.
- Do the numbers vary across a large range? Or are the differences between the numbers small, such as just a few decimal places?
- Know what type of data you are looking at. What do your numbers in your sample represent? this could be something like test scores, heart rate readings, height, weight etc.
- For example, a set of test scores is 10, 8, 10, 8, 8, and 4.
2
Gather all of your data. You will need every number in your sample to calculate the mean.^{[3]
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- The mean is the average of all your data points.
- This is calculated by adding all of the numbers in your sample, then dividing this figure by the how many numbers there are in your sample (n).
- In the sample of test scores (10, 8, 10, 8, 8, 4) there are 6 numbers in the sample. Therefore n = 6.
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3
Add the numbers in your sample together. This is the first part of calculating a mathematical average or mean.^{[4]
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- For example, use the data set of quiz scores: 10, 8, 10, 8, 8, and 4.
- 10 + 8 + 10 + 8 + 8 + 4 = 48. This is the sum of all the numbers in the data set or sample.
- Add the numbers a second time to check your answer.
4
Divide the sum by how many numbers there are in your sample (n). This will provide the average or mean of the data.^{[5]
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- In the sample of test scores (10, 8, 10, 8, 8, and 4) there are six numbers, so n = 6.
- The sum of the test scores in the example was 48. So you would divide 48 by n to figure out the mean.
- 48 / 6 = 8
- The mean test score in the sample is 8.

Part 2

Part 2 of 3:

Finding the Variance In Your Sample

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Find the variance. The variance is a figure that represents how far the data in your sample is clustered around the mean.^{[6]
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- This figure will give you an idea of how far your data is spread out.
- Samples with low variance have data that is clustered closely about the mean.
- Samples with high variance have data that is clustered far from the mean.
- Variance is often used to compare the distribution of two data sets.
2
Subtract the mean from each of your numbers in your sample. This will give you a figure of how much each data point differs from the mean.^{[7]
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- For example, in our sample of test scores (10, 8, 10, 8, 8, and 4) the mean or mathematical average was 8.
- 10 - 8 = 2; 8 - 8 = 0, 10 - 8 = 2, 8 - 8 = 0, 8 - 8 = 0, and 4 - 8 = -4.
- Do this procedure again to check each answer. It is very important you have each of these figures correct as you will need them for the next step.
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Square all of the numbers from each of the subtractions you just did. You will need each of these figures to find out the variance in your sample.^{[8]
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- Remember, in our sample we subtracted the mean (8) from each of the numbers in the sample (10, 8, 10, 8, 8, and 4) and came up with the following: 2, 0, 2, 0, 0 and -4.
- To do the next calculation in figuring out variance you would perform the following: 2², 0², 2², 0², 0², and (-4)² = 4, 0, 4, 0, 0, and 16.
- Check your answers before proceeding to the next step.
4
Add the squared numbers together. This figure is called the sum of squares.^{[9]
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- In our example of test scores, the squares were as follows: 4, 0, 4, 0, 0, and 16.
- Remember, in the example of test scores we started by subtracting the mean from each of the scores and squaring these figures: (10-8)^2 + (8-8)^2 + (10-8)^2 + (8-8)^2 + (8-8)^2 + (4-8)^2
- 4 + 0 + 4 + 0 + 0 + 16 = 24.
- The sum of squares is 24.
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Divide the sum of squares by (n-1). Remember, n is how many numbers are in your sample. Doing this step will provide the variance. The reason to use n-1 is to have sample variance and population variance unbiased. ^{[10]
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- In our sample of test scores (10, 8, 10, 8, 8, and 4) there are 6 numbers. Therefore, n = 6.
- n-1 = 5.
- Remember the sum of squares for this sample was 24.
- 24 / 5 = 4.8
- The variance in this sample is thus 4.8.

Part 3

Part 3 of 3:

Calculating the Standard Deviation

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1
Find your variance figure. You will need this to find the standard deviation for your sample.^{[11]
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- Remember, variance is how spread out your data is from the mean or mathematical average.
- Standard deviation is a similar figure, which represents how spread out your data is in your sample.
- In our example sample of test scores, the variance was 4.8.
2
Take the square root of the variance. This figure is the standard deviation.^{[12]
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- Usually, at least 68% of all the samples will fall inside one standard deviation from the mean.
- Remember in our sample of test scores, the variance was 4.8.
- √4.8 = 2.19. The standard deviation in our sample of test scores is therefore 2.19.
- 5 out of 6 (83%) of our sample of test scores (10, 8, 10, 8, 8, and 4) is within one standard deviation (2.19) from the mean (8).
3
Go through finding the mean, variance and standard deviation again. This will allow you to check your answer.^{[13]
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- It is important that you write down all steps to your problem when you are doing calculations by hand or with a calculator.
- If you come up with a different figure the second time around, check your work.
- If you cannot find where you made a mistake, start over a third time to compare your work.

Practice Problems and Answers

Practice Problems and Answers to Find Standard Deviation

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What is the standard deviation of 10 samples with a mean of 29.05?

Community Answer

Depends on the 10 samples of data. If all ten numbers were 29.05 then the standard deviation would be zero. Standard deviation is a measure of how much the data deviates from the mean.

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How do I calculate the standard deviation of 5 samples with the mean of 26?

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You take the average of 26 and 5, divide by b squared and multiply by deviation equation constant.

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How do I find the standard deviation of 10 samples with a mean of 29.05?

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Take each sample and subract the mean. Next, square each result, getting rid of the negative. Add the 10 results and divide the sun by 10 - 1 or 9. That is the standard deviation.

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