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.
-
This function returns the weighted mean of the dataset data using the
set of weights w. The weighted mean is defined as
 |
(11.16) |
-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
Release 0.9, documentation updated on October, 2008.