The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector \ textstyle \ mathbf { X }, a row vector whose " j " th element ( " j " = 1, . . ., " K " ) is one of the random variables.
42.
If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space "'x "', an unbiased estimate of the true standard error of the mean ( actually a correction on the standard deviation part ) may be obtained by multiplying the calculated standard error of the sample by the factor " f ":
43.
When only a sample of data from a population is available, the term "'standard deviation of the sample "'or "'sample standard deviation "'can refer to either the above-mentioned quantity as applied to those data or to a modified quantity that is an unbiased estimate of the "'population standard deviation "'( the standard deviation of the entire population ).
44.
Similarly, the sample variance ( estimator ), denoted " S " 2, can be used to estimate the " variance " parameter ( estimand ), denoted " ? " 2, of the population from which the sample was drawn . ( Note that the sample standard deviation ( " S " ) is not an unbiased estimate of the population standard deviation ( " ? " ) : see Unbiased estimation of standard deviation .)
45.
Where " k " 4 is the unique symmetric unbiased estimator of the fourth cumulant, " k " 2 is the unbiased estimate of the second cumulant ( identical to the unbiased estimate of the sample variance ), " m " 4 is the fourth sample moment about the mean, " m " 2 is the second sample moment about the mean, " x " " i " is the " i " th value, and \ bar { x } is the sample mean.
46.
Where " k " 4 is the unique symmetric unbiased estimator of the fourth cumulant, " k " 2 is the unbiased estimate of the second cumulant ( identical to the unbiased estimate of the sample variance ), " m " 4 is the fourth sample moment about the mean, " m " 2 is the second sample moment about the mean, " x " " i " is the " i " th value, and \ bar { x } is the sample mean.
