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I have data in the following form that correspond to data about sessions on a website: ID, # of actions (length), mean and standard deviation of time between actions:

#   ct  avg stdev
1   8   20  11.4371991243278
2   4   32  18
3   21  24  11.0142252516796
4   6   120 124.817867310734
5   54  12  14.7454698954735
6   4   12  2.08166599946613

What I want to compute is a "typical" standard deviation for a session, possibly weighted. I know that I can compute the standard deviation across all sessions by getting the aggregate second moment from the mean and stdev and subtracting the aggregate mean, but I'm not sure that's what I want.

I want to be able to compare the standard deviation in a particular session against others to see if it's higher or lower. Perhaps one way to do this would be to normalize standard deviations by their mean (giving a percentage measure) since the averages differ significantly and this also affects the magnitude of the standard deviation.

Does anyone have a principled way to approach this?

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I think your question needs further clarification. Do you want to compute kind of a weighted standard deviation of the values in the 'stdev' column? Or the statistics you want to estimate involves all the columns? –  emrea Aug 9 '12 at 20:33
    
Well, each standard deviation in the last column is calculated from a different number of values given in the second column. Whether to weight it or not, that's part of the question I'm asking. –  Andrew Mao Aug 9 '12 at 21:07
    
So your question is how to compute the standard deviation so that standard deviations of different sessions should be comparable against each other, right? Aren't the values in your stdev column comparable already? Can you elaborate on your problem? –  emrea Aug 9 '12 at 21:19
    
No, I want to compare the standard deviation for ONE session to an aggregated measure of standard deviation for SEVERAL sessions. For example, is the standard deviation for this session higher or lower than the typical standard deviation in past sessions? One simple way to do this would be to compare to the median. I'm looking for other suggestions too. –  Andrew Mao Aug 15 '12 at 19:47
    
Another suggestion might be the weighted average. The weights could come from number of actions. –  emrea Aug 15 '12 at 20:06

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