How to Find the Average of Several Standard Deviations


Occasionally you may be interested in finding the average of two or more standard deviations.

You can use one of two formulas to do so, depending on your data:

Method 1: Equal Sample Sizes

If you want to find the average standard deviation among k groups and each group has the same sample size, you can use the following formula:

Average S.D. = √ (s12 +  s22 + … + sk2) / k

where:

  • sk: Standard deviation for kth group
  • k: Total number of groups

Method 2: Unequal Sample Sizes

If you want to find the average standard deviation among k groups and each group does not have the same sample size, you can use the following formula:

Average S.D. = √ ((n1-1)s12 +  (n2-1)s22 + … +  (nk-1)sk2) /  (n1+n2 + … + nk – k)

where:

  • nk: Sample size for kth group
  • sk: Standard deviation for kth group
  • k: Total number of groups

The following examples show how to use each formula in practice.

Method 1: Averaging Standard Deviations for Equal Sample Sizes

Suppose we’d like to calculate the average standard deviation of sales during the following six sales periods:

Let’s assume that we made the same number of sales transactions in each sales period. We can use the following formula to calculate the average standard deviation of sales per period:

  • Average standard deviation = √ (s12 +  s22 + … + sk2) / k
  • Average standard deviation = √ (122 + 112 + 82 + 82 + 62 + 142) / 6
  • Average standard deviation = 10.21

The average standard deviation of sales per period is 10.21.

Method 2: Averaging Standard Deviations for Unequal Sample Sizes

Suppose we’d like to calculate the average standard deviation of sales during the following six sales periods:

Since the sample size (the total transactions) is not equal in each sales period, we’ll use the following formula to calculate the average standard deviation of sales per period:

  • Average S.D. = √ ((n1-1)s12 +  (n2-1)s22 + … +  (nk-1)sk2) /  (n1+n2 + … + nk – k)
  • Average S.D. = √ ((21)122 + (16)112 + (14)82 + (18)82 + (19)62 + (18)142) / 106
  • Average S.D. = 10.29

The average standard deviation of sales per period is 10.29.

Notice that the average standard deviation in both examples were quite similar. This is because the sample sizes (total transactions) in the second example were all fairly close together.

The two methods for calculating the average standard deviation will only differ greatly when the sample sizes differ greatly between the groups.

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