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I have a series of repeated measurements of light intensity at different points on a grid. I have made a histogram for each point showing the relative difference between modelling and measurement. My histograms are fairly close to normal with a mean of zero. I am using the standard deviation of this distribution to quantify the width of this distribution for each point on the grid. Ideally I would now like to combine all of these standard deviations into one figure to describe the entire grid.

Is there a better way of doing this than simply finding the mean of all standard deviations? I don't want to assume that each point is ultimately a sample of the same population distribution (although in practice if there are significant advantages to doing so then pragmatically its probbaly not far from true). There is an equation at the bottom of the wiki page on standard deviations but I have never seen it before and no references are given.

My question may be a repeat of Determining true mean from noisy observations, but I wasn't sure if that was assuming that each point was a sample from the same population.

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    $\begingroup$ See en.wikipedia.org/wiki/Pooled_variance $\endgroup$ – onestop Jun 21 '12 at 11:34
  • $\begingroup$ Does that not assume different means, same std. dev. I sort have the opposite, although if nothing else is available they aren't that different. $\endgroup$ – Bowler Jun 21 '12 at 11:53
  • $\begingroup$ If you don't think that there is one true underlying standard deviation, then it does not make sense to report only one number. $\endgroup$ – Aniko Jun 21 '12 at 13:34
  • $\begingroup$ I thought that was probably the case, just wondered if there was something I didn't know about. They may well have the same SD they seem as likely to have the same SD as the pooled variance example on wiki. $\endgroup$ – Bowler Jun 21 '12 at 14:07
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"The mean of all standard deviations" is not the "composite" standard deviation (the standard deviation for all measurements).

But if you still have the full set of measurements, then it's easy to calculate the composite standard deviation: just calculate the quantity of interest at each point (the difference between measured value and modeled value), then calculate the mean and standard deviation of all those quantities.

If you no longer have the full set of measurements, then it's harder, but still possible to correctly calculate the composite standard deviation. You need three numbers for each point: the mean of the quantity of interest from the measurements taken for that point, the standard deviation calculated for the quantity of interest from the measurements taken for that point, and the number of samples which were taken for that point. This web page describes how you can then correctly calculate the standard deviation of the whole dataset: http://www.burtonsys.com/climate/composite_standard_deviations.html

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