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<title>GNU Scientific Library &ndash; Reference Manual: Mean and standard deviation and variance</title>

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<a name="Mean-and-standard-deviation-and-variance"></a>
<div class="header">
<p>
Next: <a href="Absolute-deviation.html#Absolute-deviation" accesskey="n" rel="next">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>
</div>
<hr>
<a name="Mean_002c-Standard-Deviation-and-Variance"></a>
<h3 class="section">21.1 Mean, Standard Deviation and Variance</h3>

<dl>
<dt><a name="index-gsl_005fstats_005fmean"></a>Function: <em>double</em> <strong>gsl_stats_mean</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dd><p>This function returns the arithmetic mean of <var>data</var>, a dataset of
length <var>n</var> with stride <var>stride</var>.  The arithmetic mean, or
<em>sample mean</em>, is denoted by <em>\Hat\mu</em> and defined as,
</p>
<div class="example">
<pre class="example">\Hat\mu = (1/N) \sum x_i
</pre></div>

<p>where <em>x_i</em> are the elements of the dataset <var>data</var>.  For
samples drawn from a gaussian distribution the variance of
<em>\Hat\mu</em> is <em>\sigma^2 / N</em>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fvariance"></a>Function: <em>double</em> <strong>gsl_stats_variance</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dd><p>This function returns the estimated, or <em>sample</em>, variance of
<var>data</var>, a dataset of length <var>n</var> with stride <var>stride</var>.  The
estimated variance is denoted by <em>\Hat\sigma^2</em> and is defined by,
</p>
<div class="example">
<pre class="example">\Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2
</pre></div>

<p>where <em>x_i</em> are the elements of the dataset <var>data</var>.  Note that
the normalization factor of <em>1/(N-1)</em> results from the derivation
of <em>\Hat\sigma^2</em> as an unbiased estimator of the population
variance <em>\sigma^2</em>.  For samples drawn from a Gaussian distribution
the variance of <em>\Hat\sigma^2</em> itself is <em>2 \sigma^4 / N</em>.
</p>
<p>This function computes the mean via a call to <code>gsl_stats_mean</code>.  If
you have already computed the mean then you can pass it directly to
<code>gsl_stats_variance_m</code>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fvariance_005fm"></a>Function: <em>double</em> <strong>gsl_stats_variance_m</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>mean</var>)</em></dt>
<dd><p>This function returns the sample variance of <var>data</var> relative to the
given value of <var>mean</var>.  The function is computed with <em>\Hat\mu</em>
replaced by the value of <var>mean</var> that you supply,
</p>
<div class="example">
<pre class="example">\Hat\sigma^2 = (1/(N-1)) \sum (x_i - mean)^2
</pre></div>
</dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fsd"></a>Function: <em>double</em> <strong>gsl_stats_sd</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dt><a name="index-gsl_005fstats_005fsd_005fm"></a>Function: <em>double</em> <strong>gsl_stats_sd_m</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>mean</var>)</em></dt>
<dd><p>The standard deviation is defined as the square root of the variance.
These functions return the square root of the corresponding variance
functions above.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005ftss"></a>Function: <em>double</em> <strong>gsl_stats_tss</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>)</em></dt>
<dt><a name="index-gsl_005fstats_005ftss_005fm"></a>Function: <em>double</em> <strong>gsl_stats_tss_m</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>mean</var>)</em></dt>
<dd><p>These functions return the total sum of squares (TSS) of <var>data</var> about
the mean.  For <code>gsl_stats_tss_m</code> the user-supplied value of
<var>mean</var> is used, and for <code>gsl_stats_tss</code> it is computed using
<code>gsl_stats_mean</code>.
</p>
<div class="example">
<pre class="example">TSS =  \sum (x_i - mean)^2
</pre></div>
</dd></dl>

<dl>
<dt><a name="index-gsl_005fstats_005fvariance_005fwith_005ffixed_005fmean"></a>Function: <em>double</em> <strong>gsl_stats_variance_with_fixed_mean</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>mean</var>)</em></dt>
<dd><p>This function computes an unbiased estimate of the variance of
<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 uses the factor <em>1/N</em> and the sample mean
<em>\Hat\mu</em> is replaced by the known population mean <em>\mu</em>,
</p>
<div class="example">
<pre class="example">\Hat\sigma^2 = (1/N) \sum (x_i - \mu)^2
</pre></div>
</dd></dl>


<dl>
<dt><a name="index-gsl_005fstats_005fsd_005fwith_005ffixed_005fmean"></a>Function: <em>double</em> <strong>gsl_stats_sd_with_fixed_mean</strong> <em>(const double <var>data</var>[], size_t <var>stride</var>, size_t <var>n</var>, double <var>mean</var>)</em></dt>
<dd><p>This function calculates the standard deviation of <var>data</var> for a
fixed population mean <var>mean</var>.  The result is the square root of the
corresponding variance function.
</p></dd></dl>

<hr>
<div class="header">
<p>
Next: <a href="Absolute-deviation.html#Absolute-deviation" accesskey="n" rel="next">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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