11.2.8 Weighted Samples

The functions described in this section allow the computation of statistics for weighted samples. The functions accept an array of samples, , with associated weights, . Each sample is considered as having been drawn from a Gaussian distribution with variance . The sample weight is defined as the reciprocal of this variance, . Setting a weight to zero corresponds to removing a sample from a dataset.

(`wmean``w, data`)-
This function returns the weighted mean of the dataset
`data`using the set of weights`w`. The weighted mean is defined as(11.16)

`wvariance``(``w, data`)-
This function returns the estimated variance of the dataset
`data`, using the set of weights`w`. The estimated variance of a weighted dataset is defined as(11.17)

Note that this expression reduces to an unweighted variance with the familiar factor when there are equal non-zero weights.

(`wvariance_m``w, data, wmean`)-
This function returns the estimated variance of the weighted dataset
`data`using the given weighted mean`wmean`.

(`wsd``w, data`)-
The standard deviation is defined as the square root of the variance. This
function returns the square root of the corresponding variance function
`wvariance`above.

(`wsd_m``w, data, wmean`)-
This function returns the square root of the corresponding variance function
`wvariance_m`above.

(`wvariance_with_fixed_mean``w, data, mean`)-
This function computes an unbiased estimate of the variance of weighted
dataset
`data`when the population mean`mean`of the underlying distribution is known _a priori_. In this case the estimator for the variance replaces the sample mean by the known population mean ,(11.18)

(`wsd_with_fixed_mean``w, data, mean`)- The standard deviation is defined as the square root of the variance. This function returns the square root of the corresponding variance function above.

(`wabsdev``w, data`)-
This function computes the weighted absolute deviation from the weighted
mean of
`data`. The absolute deviation from the mean is defined as(11.19)

(`wabsdev_m``w, data, wmean`)- This function computes the absolute deviation of the weighted dataset DATA about the given weighted mean WMEAN.

(`wskew``w, data`)-
This function computes the weighted skewness of the dataset DATA.
(11.20)

(`wskew_m_sd``w, data, mean, wsd`)-
This function computes the weighted skewness of the dataset
`data`using the given values of the weighted mean and weighted standard deviation,`wmean`and`wsd`.

(`wkurtosis``w, data`)-
This function computes the weighted kurtosis of the dataset
`data`. The kurtosis is defined as(11.21)

(`wkurtosis_m_sd``w, data, mean, wsd`)-
This function computes the weighted kurtosis of the dataset
`data`using the given values of the weighted mean and weighted standard deviation,`wmean`and`wsd`.