I made an experiment where I measure certain parameter a number of times. The result is over a 100 samples which distribution is not really normal. Due to properties of my specimens, results tend to gather around 3 or 4 modes. The distribution is multimodal. I would like to find the type A uncertainty of the measurement. When the distribution is normal, unimodal, the standard deviation is easily calculated. How to proceed when the distribution is multimodal? I found the stdv, same way as for unimodal, but I am not sure that this is correct way. Are there any dedicated standard deviation formulas for multimodal distribution? Even if I split my results into 3 or 4 separate unimodal sets, each with its own stdv, how to find the overall deviation?
The standard deviation (SD) has a precise mathematical definition. There are no different formulas for different distributions. You calculate it just as for any other distribution. The only thing to consider is that the value of the SD is not so easy to interpret for distributions that are nor normal. For a normal distribution, it is well-known that about 68%/95%/99% of the values are within one/two/three SDs at either side of the mean. This is not true for other distributions. But in any case is the SD the square root of the variance, which is the expected squared difference of the values from the mean. This remains mathematically correct, no matter if we have or have not any nice or practical interpretation of such a value.*
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*btw: mean and variance of a distribution always have a coolphysical analogy: the mean is a kindof a "center of gravity" of the distribution, and the the variance is analoge to the "moment of inetrtia"of the distribution. The SD is then analoge to the square-rootof the moment of interia... whatever that means...
Many thanks for your answer Jochen Wilhelm . That helps me a lot. Actually I fully agree with what you wrote. There is only one definition of SD and there is no way to modify it. However my doubts came after I studied the “Guide to the expression of uncertainty in measurement”. In the chapter 4.2 the type A uncertainty is evaluated using assumption, that the measurements are of a normal distribution. But in chapter 4.3 where the type B uncertainty is evaluated, other distributions like triangular or rectangular are considered too and consequently different formulas for standard uncertainty are presented. It came to my mind, that for different distribution different formulas for standard deviation should be used. But indeed those different formulas are special cases formulas based on the same variance and SD equation. I hope I got it correctly now.
It's as always: one size doen not fit it all. There is not one descriptive statistic (like the SD, for instance) that is useful to describe relevant featuresof any distribution. If I know that the distribution is (at least approximately) rectangular, provinding the limits is much much more useful that listing some moments (like mean, variance, skew, kurtosis etc.). If I know that the distribution is exponential, it is much much more useful to simply give the rate parameter. In your case, if you know that the distribution is multimodal, it might be useful to deconvolute it into individual unimodal parts for which you can give the means and the SDs (google for"expectation maximization", a technique that works quite well to achieve such a deconvolution for a given sample).
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