So you’ve taken a bunch of samples (or you’re borrowing somebody else’s sample data), and you want to know the overall standard deviation for the combined sample, but you don’t have the raw data?

As long as you have the original sample sizes, standard deviations, and means, you can still get the combined standard deviation.

It’s not as easy as you might think, though, especially if you’re combining more than two samples. For starters:

It’s Not The Average Standard Deviation

Most people will ask for the “average standard deviation” for a set of samples, when what they really want to do is find the combined standard deviation for a combined sample. These are two different things and you shouldn’t confuse them. Worse, the average standard deviation doesn’t even really have many practical applications.

What you actually want is this formula:

To make matters worse, this formula only works for combining two samples. To combine more, you have to combine the first two, then use that as a new sample and combine it with the third, and so on.

If you want to skip all that, you can just use a handy little spreadsheet I put together instead.

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