How to Calculate a Pooled Standard Deviation (With Example)


pooled standard deviation is simply a weighted average of standard deviations from two or more independent groups.

In statistics it appears most often in the two sample t-test, which is used to test whether or not the means of two populations are equal.

The formula to calculate a pooled standard deviation for two groups is as follows:

Pooled standard deviation = √ (n1-1)s12 +  (n2-1)s22 /  (n1+n2-2)

where:

  • n1, n2: Sample size for group 1 and group 2, respectively.
  • s1, s2: Standard deviation for group 1 and group 2, respectively.

Note that the pooled standard deviation should only be used when the standard deviations between the two groups can be assumed to be roughly equal.

Also note that because the pooled standard deviation is a weighted average, it will give more “weight” to the group with the larger sample size.

Example: Calculating the Pooled Standard Deviation

Suppose we have two different groups with the following information:

Group 1: 

  • Sample size (n1): 15
  • Sample standard deviation (s1): 6.4

Group 2: 

  • Sample size (n2): 19
  • Sample standard deviation (s2): 8.2

We can calculated the pooled standard deviation for these two groups as:

Pooled standard deviation = √ (15-1)6.42 +  (19-1)8.22 /  (15+19-2) = 7.466

Notice how the value for the pooled standard deviation (7.466) is between the values for the standard deviation of group 1 (6.4) and group 2 (8.2).

This should make sense considering the pooled standard deviation is just a weighted average between the two groups.

Bonus: Pooled Standard Deviation Calculator

You can also use the Pooled Standard Deviation Calculator to quickly calculate the pooled standard deviation between two groups.

For example, we could plug in the values from the previous example to come up with the same pooled standard deviation that we calculated by hand:

Pooled standard deviation calculator

Note that you can also use the “Enter raw data” option on the calculator to enter the raw data values for the two groups and calculate the pooled standard deviation in that manner.

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