# how to average standard deviations of independent groups

I have means and standard deviations on a given variable, $X$, for six independent samples of unequal sample sizes:

Group 1: $SD = 0.96 , n = 290$ Group 2: $SD = 1.082, n = 250$ Group 3: $SD = 0.70, n = 250$ Group 4: $SD = 0.794, n = 310$ Group 5: $SD = 0.819, n = 290$ Group 6: $SD = 0.691, n = 190$

total sample size = $1,580$

I am asked to report the the average standard deviation based on these values, so, in essence, I am looking to compute the average standard deviation based on six independent datasets - that is, an average of six separate standard deviations (not the combined standard deviation for the total sample).

Can I just calculate the weigthed mean of all six standard deviations, with the weights for each sample given its sample size divided by total sample size of all six samples combined?

So, in essence, producing the following: average standard deviation = $\frac{[(\frac{290}{1,580}\cdot0.96) + (\frac{250}{1,580}\cdot1.082)+(\frac{250}{1,580}\cdot0.70)+..continued for all six groups..)]}{6}?$

Thank you!

• Exactly what is this "average standard deviation" intended to represent? The reason I ask is that there are (infinitely) many kinds of averages, so the solution is not determined solely from the need to average these values: it is determined primarily by how the average needs to be related to the quantities that contribute to it. For instance, why shouldn't we take (say) the weighted mean of the variances (the squares of the SDs) and then take the square root of that? – whuber Jun 2 '16 at 14:20
• You're welcome, Kristi--but it doesn't count as an answer; not here, anyway: it's a request that you clarify or amplify your question. Our hope is that you will thereby eventually get a more appropriate answer for your needs and future readers will have better information, too. – whuber Jun 2 '16 at 14:27
• I hit "enter" too fast :) I actually meant to write you wrote in your last sentences - I am asked to calculate the weighted mean of the variances and then take the square root of those. Literally, I want to calculate an average variance that represents the mean variation of the target variable based on the variances of all six samples (they can be considered independent subsamples). So, in essence, how does variable vary, on average, within the six samples. – Kristi Lo Jun 2 '16 at 14:31
• In other words, it sounds like you want to combine the samples in order to obtain a standard error of the mean, assuming the samples are independent. However you wish to ask it, you need to edit your question to reflect this need--comments can disappear at any time and not everybody reads them. – whuber Jun 2 '16 at 14:41
• Perhaps the way forward is to go back to the person who asked you to do this and ask them to clarify what they meant because it is not clear to any of us what purpose they had in mind. – mdewey Jun 2 '16 at 15:45