How to Calculate Standard Deviation: 12 Steps (with Pictures) (2024)

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Reviewed byGrace Imson, MA

Last Updated: September 11, 2023Fact Checked

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1. 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]

   • 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. 2

   Gather all of your data. You will need every number in your sample to calculate the mean.^[3]

   • 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. 3

   Add the numbers in your sample together. This is the first part of calculating a mathematical average or mean.^[4]

   • 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. 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]

   • 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.

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1. See Also

   Standard Deviation CalculatorStandard Deviation CalculatorStandard deviation: calculating step by step (article) | Khan AcademyStandard Deviation Formulas

   1

   Find the variance. The variance is a figure that represents how far the data in your sample is clustered around the mean.^[6]

   • 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. 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]

   • 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.
3. 3

   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]

   • 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. 4

   Add the squared numbers together. This figure is called the sum of squares.^[9]

   • 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.
5. 5

   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]

   • 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.

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1. 1

   Find your variance figure. You will need this to find the standard deviation for your sample.^[11]

   • 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. 2

   Take the square root of the variance. This figure is the standard deviation.^[12]

   • 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. 3

   Go through finding the mean, variance and standard deviation again. This will allow you to check your answer.^[13]

   • 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.

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