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<a name="Higher-moments-_0028skewness-and-kurtosis_0029"></a>
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<p>
Next: <a href="Autocorrelation.html#Autocorrelation" accesskey="n" rel="next">Autocorrelation</a>, Previous: <a href="Absolute-deviation.html#Absolute-deviation" accesskey="p" rel="previous">Absolute deviation</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>
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<hr>
<a name="Higher-moments-_0028skewness-and-kurtosis_0029-1"></a>
<h3 class="section">21.3 Higher moments (skewness and kurtosis)</h3>
<a name="index-skewness"></a>
<a name="index-kurtosis"></a>

<dl>
<dt><a name="index-gsl_005fstats_005fskew"></a>Function: <em>double</em> <strong>gsl_stats_skew</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dd><p>This function computes the skewness of <var>data</var>, a dataset of length
<var>n</var> with stride <var>stride</var>.  The skewness is defined as,
</p>
<div class="example">
<pre class="example">skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3
</pre></div>

<p>where <em>x_i</em> are the elements of the dataset <var>data</var>.  The skewness
measures the asymmetry of the tails of a distribution.
</p>
<p>The function computes the mean and estimated standard deviation of
<var>data</var> via calls to <code>gsl_stats_mean</code> and <code>gsl_stats_sd</code>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fskew_005fm_005fsd"></a>Function: <em>double</em> <strong>gsl_stats_skew_m_sd</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>mean</var>, double <var>sd</var>)</em></dt>
<dd><p>This function computes the skewness of the dataset <var>data</var> using the
given values of the mean <var>mean</var> and standard deviation <var>sd</var>,
</p>
<div class="example">
<pre class="example">skew = (1/N) \sum ((x_i - mean)/sd)^3
</pre></div>

<p>These functions are useful if you have already computed the mean and
standard deviation of <var>data</var> and want to avoid recomputing them.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fkurtosis"></a>Function: <em>double</em> <strong>gsl_stats_kurtosis</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dd><p>This function computes the kurtosis of <var>data</var>, a dataset of length
<var>n</var> with stride <var>stride</var>.  The kurtosis is defined as,
</p>
<div class="example">
<pre class="example">kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4)  - 3
</pre></div>

<p>The kurtosis measures how sharply peaked a distribution is, relative to
its width.  The kurtosis is normalized to zero for a Gaussian
distribution.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fkurtosis_005fm_005fsd"></a>Function: <em>double</em> <strong>gsl_stats_kurtosis_m_sd</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>mean</var>, double <var>sd</var>)</em></dt>
<dd><p>This function computes the kurtosis of the dataset <var>data</var> using the
given values of the mean <var>mean</var> and standard deviation <var>sd</var>,
</p>
<div class="example">
<pre class="example">kurtosis = ((1/N) \sum ((x_i - mean)/sd)^4) - 3
</pre></div>

<p>This function is useful if you have already computed the mean and
standard deviation of <var>data</var> and want to avoid recomputing them.
</p></dd></dl>

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Next: <a href="Autocorrelation.html#Autocorrelation" accesskey="n" rel="next">Autocorrelation</a>, Previous: <a href="Absolute-deviation.html#Absolute-deviation" accesskey="p" rel="previous">Absolute deviation</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>
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