If the random variable X represents samples generated by a continuous distribution with probability density function f ( x ), then the population variance is given by
22.
In other words, the expected value of the uncorrected sample variance does not equal the population variance " ? " 2, unless multiplied by a normalization factor.
23.
Where " S " 1 2 and " S " 2 2 are the usual unbiased ( Bessel-corrected ) estimates of the two population variances.
24.
For example, when " n " = 1 the variance of a single observation about the sample mean ( itself ) is obviously zero regardless of the population variance.
25.
Notice that the unknown population variance ? 2 does not appear in " T ", since it was in both the numerator and the denominator, so it canceled.
26.
When Rochelle became a founding member of the Western Sun, the NCIC began shifting members trying to keep together a conference that was built of schools with inordinately large population variances.
27.
Hence \ scriptstyle \ sigma _ y ^ 2 gives an estimate of the population variance that is biased by a factor of \ frac { n-1 } { n }.
28.
For example, robust estimators of scale are used to estimate the population variance or population standard deviation, generally by multiplying by a scale factor to make it an scale parameter : estimation.
29.
For example, the MAD of a sample from a standard Cauchy distribution is an estimator of the population MAD, which in this case is 1, whereas the population variance does not exist.
30.
Suppose it were known that the two population variances are equal, and samples of sizes " n " 1 and " n " 2 are taken from the two populations:
