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<p>
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Next: <a rel="next" accesskey="n" href="Absolute-deviation.html">Absolute deviation</a>,
Up: <a rel="up" accesskey="u" href="Statistics.html">Statistics</a>
<hr>
</div>
<h3 class="section">20.1 Mean, Standard Deviation and Variance</h3>
<div class="defun">
— Function: double <b>gsl_stats_mean</b> (<var>const double data</var>[]<var>, size_t stride, size_t n</var>)<var><a name="index-gsl_005fstats_005fmean-1818"></a></var><br>
<blockquote><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
<dfn>sample mean</dfn>, is denoted by \Hat\mu and defined as,
<pre class="example"> \Hat\mu = (1/N) \sum x_i
</pre>
<p class="noindent">where x_i are the elements of the dataset <var>data</var>. For
samples drawn from a gaussian distribution the variance of
\Hat\mu is \sigma^2 / N.
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_variance</b> (<var>const double data</var>[]<var>, size_t stride, size_t n</var>)<var><a name="index-gsl_005fstats_005fvariance-1819"></a></var><br>
<blockquote><p>This function returns the estimated, or <dfn>sample</dfn>, variance of
<var>data</var>, a dataset of length <var>n</var> with stride <var>stride</var>. The
estimated variance is denoted by \Hat\sigma^2 and is defined by,
<pre class="example"> \Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2
</pre>
<p class="noindent">where x_i are the elements of the dataset <var>data</var>. Note that
the normalization factor of 1/(N-1) results from the derivation
of \Hat\sigma^2 as an unbiased estimator of the population
variance \sigma^2. For samples drawn from a gaussian distribution
the variance of \Hat\sigma^2 itself is 2 \sigma^4 / N.
<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></blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_variance_m</b> (<var>const double data</var>[]<var>, size_t stride, size_t n, double mean</var>)<var><a name="index-gsl_005fstats_005fvariance_005fm-1820"></a></var><br>
<blockquote><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 \Hat\mu
replaced by the value of <var>mean</var> that you supply,
<pre class="example"> \Hat\sigma^2 = (1/(N-1)) \sum (x_i - mean)^2
</pre>
</blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_sd</b> (<var>const double data</var>[]<var>, size_t stride, size_t n</var>)<var><a name="index-gsl_005fstats_005fsd-1821"></a></var><br>
— Function: double <b>gsl_stats_sd_m</b> (<var>const double data</var>[]<var>, size_t stride, size_t n, double mean</var>)<var><a name="index-gsl_005fstats_005fsd_005fm-1822"></a></var><br>
<blockquote><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></blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_variance_with_fixed_mean</b> (<var>const double data</var>[]<var>, size_t stride, size_t n, double mean</var>)<var><a name="index-gsl_005fstats_005fvariance_005fwith_005ffixed_005fmean-1823"></a></var><br>
<blockquote><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 1/N and the sample mean
\Hat\mu is replaced by the known population mean \mu,
<pre class="example"> \Hat\sigma^2 = (1/N) \sum (x_i - \mu)^2
</pre>
</blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_sd_with_fixed_mean</b> (<var>const double data</var>[]<var>, size_t stride, size_t n, double mean</var>)<var><a name="index-gsl_005fstats_005fsd_005fwith_005ffixed_005fmean-1824"></a></var><br>
<blockquote><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></blockquote></div>
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