it is natural to study its distribution. In the case that Yi are independent observations from a normal distribution, Cochran's theorem shows that S2 follows a scaled chi-squared distribution (see also: asymptotic properties): As a direct consequence, it follows that and If the Yi are independent and identically distributed, but not necessarily normally distributed, then where κ is the kurtosis of the distribution and μ4 is the fourth central moment. If the conditions of the law of large numbers hold for the squared observations, S2 is a consistent estimator of σ2. One can see indeed that the variance of the estimator tends asymptotically to zero. An asymptotically equivalent formula was given in Kenney and Keeping (1951:164), Rose and Smith (2002:264), and Weisstein (n.d.). Samuelson's inequality Samuelson's inequality is a result that states bounds on the values that individual observations in a sample can take, given that the sample mean and (biased) variance have been calculated. Values must lie within the limits Relations with the harmonic and arithmetic means It has been shown that for a sample {yi} of positive real numbers, where ymax is the maximum of the sample, A is the arithmetic mean, H is the harmonic mean of the sample and is the (biased) variance of the sample. This bound has been improved, and it is known that variance is bounded by where ymin is the minimum of the sample. Tests of equality of variances The F-test of equality of variances and the chi square tests are adequate when the sample is normally distributed. Non-normality makes testing for the equality of two or more variances more difficult. Several non parametric tests have been proposed: these include the Barton–David–Ansari–Freund–Siegel–Tukey test, the Capon test, Mood test, the Klotz test and the Sukhatme test. The Sukhatme test applies to two variances and requires that both medians be known and equal to zero. The Mood, Klotz, Capon and Barton–David–Ansari–Freund–Siegel–Tukey tests also apply to two variances. They allow the median to be unknown but do require that the two medians are equal. The Lehmann test is a parametric test of two variances. Of this test there are several variants known. Other tests of the equality of variances include the Box test, the Box–Anderson test and the Moses test. Resampling methods, which include the bootstrap and the jackknife, may be used to test the equality of variances. Moment of inertia The variance of a