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<title>GNU Scientific Library &ndash; Reference Manual: Weighted Samples</title>

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<a name="Weighted-Samples"></a>
<div class="header">
<p>
Next: <a href="Maximum-and-Minimum-values.html#Maximum-and-Minimum-values" accesskey="n" rel="next">Maximum and Minimum values</a>, Previous: <a href="Correlation.html#Correlation" accesskey="p" rel="previous">Correlation</a>, Up: <a href="Statistics.html#Statistics" accesskey="u" rel="up">Statistics</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Weighted-Samples-1"></a>
<h3 class="section">21.7 Weighted Samples</h3>

<p>The functions described in this section allow the computation of
statistics for weighted samples.  The functions accept an array of
samples, <em>x_i</em>, with associated weights, <em>w_i</em>.  Each sample
<em>x_i</em> is considered as having been drawn from a Gaussian
distribution with variance <em>\sigma_i^2</em>.  The sample weight
<em>w_i</em> is defined as the reciprocal of this variance, <em>w_i =
1/\sigma_i^2</em>.  Setting a weight to zero corresponds to removing a
sample from a dataset.
</p>
<dl>
<dt><a name="index-gsl_005fstats_005fwmean"></a>Function: <em>double</em> <strong>gsl_stats_wmean</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dd><p>This function returns the weighted mean of the dataset <var>data</var> with
stride <var>stride</var> and length <var>n</var>, using the set of weights <var>w</var>
with stride <var>wstride</var> and length <var>n</var>.  The weighted mean is defined as,
</p>
<div class="example">
<pre class="example">\Hat\mu = (\sum w_i x_i) / (\sum w_i)
</pre></div>
</dd></dl>


<dl>
<dt><a name="index-gsl_005fstats_005fwvariance"></a>Function: <em>double</em> <strong>gsl_stats_wvariance</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dd><p>This function returns the estimated variance of the dataset <var>data</var>
with stride <var>stride</var> and length <var>n</var>, using the set of weights
<var>w</var> with stride <var>wstride</var> and length <var>n</var>.  The estimated
variance of a weighted dataset is calculated as,
</p>
<div class="example">
<pre class="example">\Hat\sigma^2 = ((\sum w_i)/((\sum w_i)^2 - \sum (w_i^2))) 
                \sum w_i (x_i - \Hat\mu)^2
</pre></div>

<p>Note that this expression reduces to an unweighted variance with the
familiar <em>1/(N-1)</em> factor when there are <em>N</em> equal non-zero
weights.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fwvariance_005fm"></a>Function: <em>double</em> <strong>gsl_stats_wvariance_m</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>wmean</var>)</em></dt>
<dd><p>This function returns the estimated variance of the weighted dataset
<var>data</var> using the given weighted mean <var>wmean</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fwsd"></a>Function: <em>double</em> <strong>gsl_stats_wsd</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dd><p>The standard deviation is defined as the square root of the variance.
This function returns the square root of the corresponding variance
function <code>gsl_stats_wvariance</code> above.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fwsd_005fm"></a>Function: <em>double</em> <strong>gsl_stats_wsd_m</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>wmean</var>)</em></dt>
<dd><p>This function returns the square root of the corresponding variance
function <code>gsl_stats_wvariance_m</code> above.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fwvariance_005fwith_005ffixed_005fmean"></a>Function: <em>double</em> <strong>gsl_stats_wvariance_with_fixed_mean</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, const double <var>mean</var>)</em></dt>
<dd><p>This function computes an unbiased estimate of the variance of the weighted
dataset <var>data</var> when the population mean <var>mean</var> of the underlying
distribution is known <em>a priori</em>.  In this case the estimator for
the variance replaces the sample mean <em>\Hat\mu</em> by the known
population mean <em>\mu</em>,
</p>
<div class="example">
<pre class="example">\Hat\sigma^2 = (\sum w_i (x_i - \mu)^2) / (\sum w_i)
</pre></div>
</dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fwsd_005fwith_005ffixed_005fmean"></a>Function: <em>double</em> <strong>gsl_stats_wsd_with_fixed_mean</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, const double <var>mean</var>)</em></dt>
<dd><p>The standard deviation is defined as the square root of the variance.
This function returns the square root of the corresponding variance
function above.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fwtss"></a>Function: <em>double</em> <strong>gsl_stats_wtss</strong> <em>(const double <var>w</var>[], const size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dt><a name="index-gsl_005fstats_005fwtss_005fm"></a>Function: <em>double</em> <strong>gsl_stats_wtss_m</strong> <em>(const double <var>w</var>[], const size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>wmean</var>)</em></dt>
<dd><p>These functions return the weighted total sum of squares (TSS) of
<var>data</var> about the weighted mean.  For <code>gsl_stats_wtss_m</code> the
user-supplied value of <var>wmean</var> is used, and for <code>gsl_stats_wtss</code>
it is computed using <code>gsl_stats_wmean</code>.
</p>
<div class="example">
<pre class="example">TSS =  \sum w_i (x_i - wmean)^2
</pre></div>
</dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fwabsdev"></a>Function: <em>double</em> <strong>gsl_stats_wabsdev</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dd><p>This function computes the weighted absolute deviation from the weighted
mean of <var>data</var>.  The absolute deviation from the mean is defined as,
</p>
<div class="example">
<pre class="example">absdev = (\sum w_i |x_i - \Hat\mu|) / (\sum w_i)
</pre></div>
</dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fwabsdev_005fm"></a>Function: <em>double</em> <strong>gsl_stats_wabsdev_m</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>wmean</var>)</em></dt>
<dd><p>This function computes the absolute deviation of the weighted dataset
<var>data</var> about the given weighted mean <var>wmean</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fwskew"></a>Function: <em>double</em> <strong>gsl_stats_wskew</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dd><p>This function computes the weighted skewness of the dataset <var>data</var>.
</p>
<div class="example">
<pre class="example">skew = (\sum w_i ((x_i - \Hat x)/\Hat \sigma)^3) / (\sum w_i)
</pre></div>
</dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fwskew_005fm_005fsd"></a>Function: <em>double</em> <strong>gsl_stats_wskew_m_sd</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>wmean</var>, double <var>wsd</var>)</em></dt>
<dd><p>This function computes the weighted skewness of the dataset <var>data</var>
using the given values of the weighted mean and weighted standard
deviation, <var>wmean</var> and <var>wsd</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fwkurtosis"></a>Function: <em>double</em> <strong>gsl_stats_wkurtosis</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dd><p>This function computes the weighted kurtosis of the dataset <var>data</var>.
</p>
<div class="example">
<pre class="example">kurtosis = ((\sum w_i ((x_i - \Hat x)/\Hat \sigma)^4) / (\sum w_i)) - 3
</pre></div>
</dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fwkurtosis_005fm_005fsd"></a>Function: <em>double</em> <strong>gsl_stats_wkurtosis_m_sd</strong> <em>(const double <var>w</var>[], size_t <var>wstride</var>, const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>wmean</var>, double <var>wsd</var>)</em></dt>
<dd><p>This function computes the weighted kurtosis of the dataset <var>data</var>
using the given values of the weighted mean and weighted standard
deviation, <var>wmean</var> and <var>wsd</var>.
</p></dd></dl>

<hr>
<div class="header">
<p>
Next: <a href="Maximum-and-Minimum-values.html#Maximum-and-Minimum-values" accesskey="n" rel="next">Maximum and Minimum values</a>, Previous: <a href="Correlation.html#Correlation" accesskey="p" rel="previous">Correlation</a>, Up: <a href="Statistics.html#Statistics" accesskey="u" rel="up">Statistics</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>



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