An Upper Bound for the Sample Standard Deviation

keep our process in control, we can work to a tolerance of ? .005 and produce less than (say) 1% scrap. But they do not tell us where to put the limits. This is not to be expected, since the range is the difference of two observed values of the thing we are measuring, and would therefore remain unaltered however violently the mean value might fluctuate. The chart for means would have to be consulted to find where the nominal value should lie. The proofs of these inequalities are short. Let I be the length of the shortest interval which contains a fraction p of the distribution. Then any interval whose length is less than 1 contains less than p of the distribution. Suppose X1, X2, * . X, is a sample of n, and suppose X, is the least of these values of X. Then if the range Wn of this sample is less than 1, all the n 1 values other than Xr must lie in the interval (X,, X, + l). The probability of this happening is less than p~n-. The least value X, can be picked in n ways. Hence