Thirdly, Bessel's correction is only necessary when the population mean is unknown, and one is estimating " both " population mean " and " population variance from a given sample set, using the sample mean to estimate the population mean.
42.
When estimating a scale parameter, using a trimmed estimator as a robust measures of scale, such as to estimate the population variance or population standard deviation, one generally must multiply by a scale factor to make it an unbiased consistent estimator; see scale parameter : estimation.
43.
When estimating a scale parameter, such as when using an L-estimator as a robust measures of scale, such as to estimate the population variance or population standard deviation, one generally must multiply by a scale factor to make it an unbiased consistent estimator; see scale parameter : estimation.
44.
For example, dividing the IQR by 2 \ sqrt { 2 } \ operatorname { erf } ^ {-1 } ( 1 / 2 ) \ approx 1.349 ( using the error function ) makes it an unbiased, consistent estimator for the population variance if the data follow a normal distribution.
45.
Further, mean-unbiasedness is not preserved under non-linear transformations, though median-unbiasedness is ( see effect of transformations ); for example, the sample variance is an unbiased estimator for the population variance, but its square root, the sample standard deviation, is a biased estimator for the population standard deviation.
46.
The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the "'sample mean "'and "'( uncorrected ) sample variance "' these are consistent estimators ( they converge to the correct value as the number of samples increases ), but can be improved.
47.
When estimating the population variance and standard deviation of a sample where the population mean is unknown, the sample variance estimated as the " mean " of the squared deviations of sample values from their mean ( that is, using a multiplicative factor ) is a biased estimator of the population variance, and for the average sample underestimates it.
48.
When estimating the population variance and standard deviation of a sample where the population mean is unknown, the sample variance estimated as the " mean " of the squared deviations of sample values from their mean ( that is, using a multiplicative factor ) is a biased estimator of the population variance, and for the average sample underestimates it.
49.
Firstly, while the sample variance ( using Bessel's correction ) is an unbiased estimate of the population variance, its square root, the sample standard deviation, is a " biased " estimate of the population standard deviation; because the square root is a concave function, the bias is downward, by Jensen's inequality.
50.
Where df " t " is the degrees of freedom " n " 1 of the estimate of the population variance of the dependent variable, and df " e " is the degrees of freedom " n " " p " 1 of the estimate of the underlying population error variance.
