A key difference perhaps is that on one situation (all mice, all electron density values) we are dealing with the entire population. There is no doubt about what the rmsd is, no expectations and uncertainty. Of that population 'sigma' is per definition the same. Whether it is a good idea to call it standard error/deviation or use expectation value for the mean in his case is another question.
In contrast, if we sample a population, we do not know what all members of the population look like. We have only per CLT an expectation value of the sample distribution mean, with an associated variance/sigma. Here, the term expectation value and standard error/sigma do convey exactly that meaning. So I think the distinction between an entire population and its measures and the measures of its/a sampling distribution is important. Some well-known and not remotely pathological looking distributions do not even have a defined variance/sigma, cf. Cauchy but that is another theme. Best, BR -----Original Message----- From: Mailing list for users of COOT Crystallographic Software <[email protected]> On Behalf Of Edward A. Berry Sent: Thursday, April 19, 2018 09:44 To: [email protected] Subject: Re: Calculating sigma value On 04/19/2018 08:57 AM, Ian Tickle wrote: > > Hi, first maps are produced by Refmac, not Coot, and second it shouldn't be > called sigma because it's not an uncertainty, it's a root-mean-square > deviation from the mean. The equation for the RMSD can be found in any basic > text on statistics, e.g. just type 'RMSD' in Wikipedia. > > Cheers > > -- Ian With all due respect, and I may be misunderstanding something here, but I think that that is an unnecessarily restrictive definition of sigma! I'm assuming sigma stands for the standard deviation. Although standard deviation is often associated with a probability distribution, it is defined for (any?) kind of distribution. From the Wikipedia page on standard deviation, "the standard deviation (SD, also represented by the Greek letter sigma σ or the Latin letter s, is a measure that is used to quantify the amount of variation or dispersion of a set of data values", and "There are also other measures of deviation from the norm . . .". That together with the formula for population standard deviation suggests standard deviation is exactly the RMS Deviation from the mean. For an analogy, suppose a dietician weighs a dozen mice that have undergone the same regimen, and calculates a certain mean value with a standard deviation deviation of 1.2 g. Now he weighed the mice on a scale reading to the tenth of a gram, so the standard deviation of the measurement is around 0.1 g or less. Nonetheless he is going to report the deviation of his population, which is 1.2 g. Likewise even if we knew precisely the electron density at every point in the unit cell of a crystal, that density would still have a distribution, and that distribution would have a standard deviation. The important thing, and I think this was the main point of Ian's remark, is that that standard deviation would have nothing to do with the uncertainty of our estimate of the density. You could make a probability distribution out of the weight distribution of the mice. Say if I pick a random mouse and weigh it, or if I repeat the experiment with only a single mouse, that standard deviation tells me something about how likely my result is to be close to the population mean. In the latter case, this could also be viewed as a measure of the error in the experiment. But in the same way, you could say if I pick a random point in the asymmetric unit and sample the density there, the RMSD tells me something about the probability that my result will be close to the mean value for the map. However, in keeping with the main point mentioned above, it may be a good convention to use sigma only for standard deviation of a probability function such as normally (or otherwise) distributed error of a measurement, and RMSD for standard deviation in all other cases. I think the way most people use coot nowadays, refmac (or other) is producing map coefficients, and coot is calculating the map (presumably using the FFT alogorithm as mentioned) and contouring it for us to see. eab > > > On 19 April 2018 at 13:20, Mohamed Ibrahim <[email protected] > <mailto:[email protected]>> wrote: > > Dear COOT users, > > Do you know how to extract the equations that COOT uses for generating > the maps and calculating the sigma values? > > Best regards, > Mohamed > > -- > > -- > /* > > ----------------------------------*/ > /*Mohamed Ibrahim > *//**//* > */ > /*Humboldt University > */ > /*Berlin, Germany > */ > /*Tel: +49 30 209347931 > > */ > >
