How to Calculate Standard Deviation for a Set of Data

Standard Deviation, simply put, is a measure of the dispersion of a set of data. The standard deviation statistic tells you how tightly or how loosely the data are clustered around the mean (the mean is the average) in the set. If the examples are bunched tightly together, the standard deviation is small. If the examples are spread far apart, the standard deviation is large. Learn how to calculate the standard deviation for a set of data by following these steps.

Difficulty: Moderate

Instructions

1. The first step in calculating a standard deviation is to find the mean in a set of data. For example, if your data includes 23, 92, 46, 55, 63, 94, 77, 38, 84, and 26, you would simply add these numbers together and divide by 10, which is the actual total number of numbers.
   
   23+92+46+55+63+94+77+38+84+26=598
   
   Now divide by 10.
   
   598/10 = 59.8
   
   The mean of this set of numbers is 59.8.

2. The next step is to get the deviations in these numbers. Do this by subtracting the mean from each of the numbers in the set of data.
   
   23-59.6 = -36.8
   92-59.6 = 32.2
   46-59.6 = -13.8
   55-59.6 = -4.8
   63-59.6 = 3.2
   94-59.6 = 34.2
   77-59.6 = 17.2
   38-59.6 = -21.8
   84-59.6 = 24.2
   26-59.6 = -33.8

3. 
    

   Now, square the deviations calculated in Step 2. To "square" a number simply means to multiply the number by itself. Remember that when you square a negative number, the result is a positive number.
   
   -36.8 squared = 1354.24
   32.2 squared = 1036.84
   -13.8 squared = 190.44
   -4.8 squared = 23.04
   3.2 squared = 10.24
   34.2 squared = 1169.64
   17.2 squared = 295.84
   -21.8 squared = 475.24
   24.2 squared = 585.64
   -33.8 squared = 1142.44

4. Next, add all of the squares in Step 3 together.
   
   1354.24 + 1036.84 + 190.44 + 23.04 + 10.24 + 1169.64 + 295.84 + 475.24 + 585.64 + 1142.44 = 6,283.60

5. Now divide the sum in Step 4 by the total number of numbers, less one. You started with a total of 10 numbers, so one less than 10 is 9. So, divide the sum by 9.
   
   6283.60 / 9 = 698.18

6. 
    

   Next take the square root of the result in Step 5. The square root is the number, when multiplied by itself, equals the original number, in this case, 698.18. The easiest way to calculate square root is to use a calculator with the square root function. The proper key on the calculator is a picture of a check mark. This will be the Standard Deviation of the original set of the 10 numbers.
   
   The square root of 698.18 is 26.4.
   
   Thus, the standard deviation of this set of numbers is 26.4.

[gloribe]

on 10/10/2009 Thanks! I was really having diffiuclty understanding how to get the standard deviation. You really helped!

[Fuller1972]

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[sonni57]

on 3/26/2009 Thanks I didn't know how to calculate standard deviation.

[sonni57]

on 3/26/2009 Thanks I didn't know how to calculate standard deviation.

[eclecticprof]

on 3/26/2009 FANTASTIC! I always wandered about this. I may be doing some empirical research for my phd program soon. This is very helpful. Thanks for the clear and easy to follow steps. 5*