Raul Miller wrote: > For one thing, the reasoning for using N-1 in the denominator of > standard deviation seems specious. > > For another, in most cases where it really matters (correlation, for > example), it just cancels out. > As I indicate below, it is necessary for estimation.
> For another, "standard deviation" jumps from a manageable quantity to > an unknowable quantity when the result is determined, and it seems > to me that it should be zero in that case. > I think here you are confusing the random variable sample standard deviation, which has a distribution, with the value of the sample standard deviation on a specific sample, which does not. > To my mind, the reasoning for the N-1 factor in standard deviation > may be validly applied to the mean -- if I include all the deviation > terms AND the deviation of the mean itself from itself, I should > exclude the count of the mean in the divisor. But no one bothers > to express it that way, so this seems an exercise in futility. > The denominator of the sample mean really should be n, and for the sample variance it really should be n-1 if you are estimating the population mean from the sample. Suppose you have a population with distribution f, mean mu and standard deviation sigma. Let X1,...Xn be independent random variables with distribution f (these correspond to how to select a sample of size n). You want to use the sample to estimate mu and sigma, that is, find statistics whose expected values are mu and sigma. Let M=(X1+...+Xn)/n be the sample mean. Then E(M)=mu. Now let S^2=((X1-mu)^2+...+(Xn-mu)^2)/n. Then E(S^2)=sigma^2. The problem in the latter is that you do not know mu: you have to estimate it from the sample. If you write (Xi-mu)^2=((Xi-M)+(M-mu))^2 and expand it out, you will be able to eliminate mu from the sum: However you will find S^2=((X1-M)^2+...+(Xn-M)^2)/(n-1). As before, E(S^2)=sigma^2. Best wishes, John ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
