Can I take an average of many SDs ? i.e. is it mathematically acceptable to use the mean of a number of standard deviations to show changes in spread over time.
Essentially I have repeated measures observations, each generating a standard deviation (or could be any other measure of spread) for each time point. Some time points show wild variation across all participants within one set of measurements (and have a high SD), other time points do not and show more stable measurements (generating lower SDs). I would like to show this mathematically (and visually using error bars) that there is this change in variance over time across the whole cohort. Is the mean of many SDs an OK way to do this ?
My limited google searching suggests that I should be adding variances, dividing by number of measurements and taking Square root of that. Seems to get a bit more tricky when participants have differing numbers of measurements (which mine do).
It depends on what you are trying to do. For averaging SDs you should probably use the rules on pooling variances - and there are methods for this when n is unequal, however its not clear to me that this solves your problem. (Also note that it gets more complicated if the variances aren't independent).
For comparing variation in different samples I'd probably take a different approach. Two spring to mind - one would involve a multilevel model of some kind and allowing the variances to be constrained or not. This can be quite tricky to set up in practice and depends on your software (e.g., if it allows you to constrain parameters). The other is probably much simpler and also potentially much more robust. that would be adapt the Levene test for equality of variances. despite the name its really a test for assessing differences in variability - its just that if you choose the right version it has good frequentist properties for testing differences in variances (or SDs).
The Levene test isn't really a single test but a procedure for generating a test. In the general case you compare the absolute deviations from a statistic (e.g., mean or median) between groups using one-way ANOVA. If you use absolute deviations from the median this is more robust than absolute deviations from the mean. You can extend this fairly easily to more complex situations. You can also plot the absolute deviations from the median as if for a standard ANOVA design and compute CIs. It should be more robust than working with the SDs and should be fairly easy to explain. As (at heart) it's an ANOVA it also opens up two-way designs, post hoc tests and so on if desired.
Here is a simple way to take the mean of equal and unequal n SD:
look it up and be happy:)
How to Find the Average of Several Standard Deviations - Statology https://www.statology.org/averaging-standard-deviations
You may also want to calculate the SE, the standard error of measurement.
The standard error of measurement (SEm) estimates how repeated measures of a person on the same instrument tend to be distributed around his or her “true” score. The true score is always an unknown because no measure can be constructed that provides a perfect reflection of the true score.
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