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<a name="Basic-Statistical-Functions"></a>
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Next: <a href="Correlation-and-Regression-Analysis.html#Correlation-and-Regression-Analysis" accesskey="n" rel="next">Correlation and Regression Analysis</a>, Previous: <a href="Statistics-on-Sliding-Windows-of-Data.html#Statistics-on-Sliding-Windows-of-Data" accesskey="p" rel="prev">Statistics on Sliding Windows of Data</a>, Up: <a href="Statistics.html#Statistics" accesskey="u" rel="up">Statistics</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Basic-Statistical-Functions-1"></a>
<h3 class="section">26.3 Basic Statistical Functions</h3>
<p>Octave supports various helpful statistical functions. Many are useful as
initial steps to prepare a data set for further analysis. Others provide
different measures from those of the basic descriptive statistics.
</p>
<a name="XREFcenter"></a><dl>
<dt><a name="index-center"></a><em></em> <strong>center</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-center-1"></a><em></em> <strong>center</strong> <em>(<var>x</var>, <var>dim</var>)</em></dt>
<dd><p>Center data by subtracting its mean.
</p>
<p>If <var>x</var> is a vector, subtract its mean.
</p>
<p>If <var>x</var> is a matrix, do the above for each column.
</p>
<p>If the optional argument <var>dim</var> is given, operate along this dimension.
</p>
<p>Programming Note: <code>center</code> has obvious application for normalizing
statistical data. It is also useful for improving the precision of general
numerical calculations. Whenever there is a large value that is common
to a batch of data, the mean can be subtracted off, the calculation
performed, and then the mean added back to obtain the final answer.
</p>
<p><strong>See also:</strong> <a href="#XREFzscore">zscore</a>.
</p></dd></dl>
<a name="XREFzscore"></a><dl>
<dt><a name="index-zscore"></a><em><var>z</var> =</em> <strong>zscore</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-zscore-1"></a><em><var>z</var> =</em> <strong>zscore</strong> <em>(<var>x</var>, <var>opt</var>)</em></dt>
<dt><a name="index-zscore-2"></a><em><var>z</var> =</em> <strong>zscore</strong> <em>(<var>x</var>, <var>opt</var>, <var>dim</var>)</em></dt>
<dt><a name="index-zscore-3"></a><em>[<var>z</var>, <var>mu</var>, <var>sigma</var>] =</em> <strong>zscore</strong> <em>(…)</em></dt>
<dd><p>Compute the Z score of <var>x</var>.
</p>
<p>If <var>x</var> is a vector, subtract its mean and divide by its standard
deviation. If the standard deviation is zero, divide by 1 instead.
</p>
<p>The optional parameter <var>opt</var> determines the normalization to use when
computing the standard deviation and has the same definition as the
corresponding parameter for <code>std</code>.
</p>
<p>If <var>x</var> is a matrix, calculate along the first non-singleton dimension.
If the third optional argument <var>dim</var> is given, operate along this
dimension.
</p>
<p>The optional outputs <var>mu</var> and <var>sigma</var> contain the mean and standard
deviation.
</p>
<p><strong>See also:</strong> <a href="Descriptive-Statistics.html#XREFmean">mean</a>, <a href="Descriptive-Statistics.html#XREFstd">std</a>, <a href="#XREFcenter">center</a>.
</p></dd></dl>
<a name="XREFhistc"></a><dl>
<dt><a name="index-histc"></a><em><var>n</var> =</em> <strong>histc</strong> <em>(<var>x</var>, <var>edges</var>)</em></dt>
<dt><a name="index-histc-1"></a><em><var>n</var> =</em> <strong>histc</strong> <em>(<var>x</var>, <var>edges</var>, <var>dim</var>)</em></dt>
<dt><a name="index-histc-2"></a><em>[<var>n</var>, <var>idx</var>] =</em> <strong>histc</strong> <em>(…)</em></dt>
<dd><p>Compute histogram counts.
</p>
<p>When <var>x</var> is a vector, the function counts the number of elements of
<var>x</var> that fall in the histogram bins defined by <var>edges</var>. This
must be a vector of monotonically increasing values that define the edges
of the histogram bins.
<code><var>n</var>(k)</code>
contains the number of elements in <var>x</var> for which
<code><var>edges</var>(k) <= <var>x</var> < <var>edges</var>(k+1)</code>.
The final element of <var>n</var> contains the number of elements of <var>x</var>
exactly equal to the last element of <var>edges</var>.
</p>
<p>When <var>x</var> is an <em>N</em>-dimensional array, the computation is carried
out along dimension <var>dim</var>. If not specified <var>dim</var> defaults to the
first non-singleton dimension.
</p>
<p>When a second output argument is requested an index matrix is also returned.
The <var>idx</var> matrix has the same size as <var>x</var>. Each element of
<var>idx</var> contains the index of the histogram bin in which the
corresponding element of <var>x</var> was counted.
</p>
<p><strong>See also:</strong> <a href="Two_002dDimensional-Plots.html#XREFhist">hist</a>.
</p></dd></dl>
<p><code>unique</code> function documented at <a href="Sets.html#XREFunique">unique</a> is often
useful for statistics.
</p>
<a name="XREFnchoosek"></a><dl>
<dt><a name="index-nchoosek"></a><em><var>c</var> =</em> <strong>nchoosek</strong> <em>(<var>n</var>, <var>k</var>)</em></dt>
<dt><a name="index-nchoosek-1"></a><em><var>c</var> =</em> <strong>nchoosek</strong> <em>(<var>set</var>, <var>k</var>)</em></dt>
<dd>
<p>Compute the binomial coefficient of <var>n</var> or list all possible
combinations of a <var>set</var> of items.
</p>
<p>If <var>n</var> is a scalar then calculate the binomial coefficient
of <var>n</var> and <var>k</var> which is defined as
</p>
<div class="example">
<pre class="example"> / \
| n | n (n-1) (n-2) … (n-k+1) n!
| | = ------------------------- = ---------
| k | k! k! (n-k)!
\ /
</pre></div>
<p>This is the number of combinations of <var>n</var> items taken in groups of
size <var>k</var>.
</p>
<p>If the first argument is a vector, <var>set</var>, then generate all
combinations of the elements of <var>set</var>, taken <var>k</var> at a time, with
one row per combination. The result <var>c</var> has <var>k</var> columns and
<code>nchoosek (length (<var>set</var>), <var>k</var>)</code><!-- /@w --> rows.
</p>
<p>For example:
</p>
<p>How many ways can three items be grouped into pairs?
</p>
<div class="example">
<pre class="example">nchoosek (3, 2)
⇒ 3
</pre></div>
<p>What are the possible pairs?
</p>
<div class="example">
<pre class="example">nchoosek (1:3, 2)
⇒ 1 2
1 3
2 3
</pre></div>
<p>Programming Note: When calculating the binomial coefficient <code>nchoosek</code>
works only for non-negative, integer arguments. Use <code>bincoeff</code> for
non-integer and negative scalar arguments, or for computing many binomial
coefficients at once with vector inputs for <var>n</var> or <var>k</var>.
</p>
<p><strong>See also:</strong> <a href="Special-Functions.html#XREFbincoeff">bincoeff</a>, <a href="#XREFperms">perms</a>.
</p></dd></dl>
<a name="XREFperms"></a><dl>
<dt><a name="index-perms"></a><em></em> <strong>perms</strong> <em>(<var>v</var>)</em></dt>
<dd><p>Generate all permutations of vector <var>v</var> with one row per permutation.
</p>
<p>Results are returned in inverse lexicographic order. The result has size
<code>factorial (<var>n</var>) * <var>n</var></code>, where <var>n</var> is the length of
<var>v</var>. Any repetitions are included in the output. To generate just the
unique permutations use <code>unique (perms (<var>v</var>), "rows")(end:-1:1,:)</code>.
</p>
<p>Example
</p>
<div class="example">
<pre class="example">perms ([1, 2, 3])
⇒
3 2 1
3 1 2
2 3 1
2 1 3
1 3 2
1 2 3
</pre></div>
<p>Programming Note: The maximum length of <var>v</var> should be less than or
equal to 10 to limit memory consumption.
</p>
<p><strong>See also:</strong> <a href="Rearranging-Matrices.html#XREFpermute">permute</a>, <a href="Special-Utility-Matrices.html#XREFrandperm">randperm</a>, <a href="#XREFnchoosek">nchoosek</a>.
</p></dd></dl>
<a name="XREFranks"></a><dl>
<dt><a name="index-ranks"></a><em></em> <strong>ranks</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-ranks-1"></a><em></em> <strong>ranks</strong> <em>(<var>x</var>, <var>dim</var>)</em></dt>
<dt><a name="index-ranks-2"></a><em></em> <strong>ranks</strong> <em>(<var>x</var>, <var>dim</var>, <var>rtype</var>)</em></dt>
<dd><p>Return the ranks (in the sense of order statistics) of <var>x</var> along the
first non-singleton dimension adjusted for ties.
</p>
<p>If the optional <var>dim</var> argument is given, operate along this dimension.
</p>
<p>The optional parameter <var>rtype</var> determines how ties are handled. All
examples below assume an input of <code>[ 1, 2, 2, 4 ]</code>.
</p>
<dl compact="compact">
<dt>0 or <code>"fractional"</code> (default) for fractional ranking (1, 2.5,</dt>
<dd><p>2.5, 4);
</p>
</dd>
<dt>1 or <code>"competition"</code> for competition ranking (1, 2, 2, 4);</dt>
<dt>2 or <code>"modified"</code> for modified competition ranking (1, 3, 3, 4);</dt>
<dt>3 or <code>"ordinal"</code> for ordinal ranking (1, 2, 3, 4);</dt>
<dt>4 or <code>"dense"</code> for dense ranking (1, 2, 2, 3).</dt>
</dl>
<p><strong>See also:</strong> <a href="Correlation-and-Regression-Analysis.html#XREFspearman">spearman</a>, <a href="Correlation-and-Regression-Analysis.html#XREFkendall">kendall</a>.
</p></dd></dl>
<a name="XREFrun_005fcount"></a><dl>
<dt><a name="index-run_005fcount"></a><em></em> <strong>run_count</strong> <em>(<var>x</var>, <var>n</var>)</em></dt>
<dt><a name="index-run_005fcount-1"></a><em></em> <strong>run_count</strong> <em>(<var>x</var>, <var>n</var>, <var>dim</var>)</em></dt>
<dd><p>Count the upward runs along the first non-singleton dimension of <var>x</var>
of length 1, 2, …, <var>n</var>-1 and greater than or equal to <var>n</var>.
</p>
<p>If the optional argument <var>dim</var> is given then operate along this
dimension.
</p>
<p><strong>See also:</strong> <a href="#XREFrunlength">runlength</a>.
</p></dd></dl>
<a name="XREFrunlength"></a><dl>
<dt><a name="index-runlength"></a><em>count =</em> <strong>runlength</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-runlength-1"></a><em>[count, value] =</em> <strong>runlength</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Find the lengths of all sequences of common values.
</p>
<p><var>count</var> is a vector with the lengths of each repeated value.
</p>
<p>The optional output <var>value</var> contains the value that was repeated in
the sequence.
</p>
<div class="example">
<pre class="example">runlength ([2, 2, 0, 4, 4, 4, 0, 1, 1, 1, 1])
⇒ 2 1 3 1 4
</pre></div>
<p><strong>See also:</strong> <a href="#XREFrun_005fcount">run_count</a>.
</p></dd></dl>
<hr>
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Next: <a href="Correlation-and-Regression-Analysis.html#Correlation-and-Regression-Analysis" accesskey="n" rel="next">Correlation and Regression Analysis</a>, Previous: <a href="Statistics-on-Sliding-Windows-of-Data.html#Statistics-on-Sliding-Windows-of-Data" accesskey="p" rel="prev">Statistics on Sliding Windows of Data</a>, Up: <a href="Statistics.html#Statistics" accesskey="u" rel="up">Statistics</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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