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<a name="Descriptive-Statistics"></a>
<p>
Next: <a rel="next" accesskey="n" href="Basic-Statistical-Functions.html#Basic-Statistical-Functions">Basic Statistical Functions</a>,
Up: <a rel="up" accesskey="u" href="Statistics.html#Statistics">Statistics</a>
<hr>
</div>
<h3 class="section">26.1 Descriptive Statistics</h3>
<p>One principal goal of descriptive statistics is to represent the essence of a
large data set concisely. Octave provides the mean, median, and mode functions
which all summarize a data set with just a single number corresponding to
the central tendency of the data.
<!-- mean scripts/statistics/base/mean.m -->
<p><a name="doc_002dmean"></a>
<div class="defun">
— Function File: <b>mean</b> (<var>x</var>)<var><a name="index-mean-2407"></a></var><br>
— Function File: <b>mean</b> (<var>x, dim</var>)<var><a name="index-mean-2408"></a></var><br>
— Function File: <b>mean</b> (<var>x, opt</var>)<var><a name="index-mean-2409"></a></var><br>
— Function File: <b>mean</b> (<var>x, dim, opt</var>)<var><a name="index-mean-2410"></a></var><br>
<blockquote><p>Compute the mean of the elements of the vector <var>x</var>.
<pre class="example"> mean (x) = SUM_i x(i) / N
</pre>
<p>If <var>x</var> is a matrix, compute the mean for each column and return them
in a row vector.
<p>The optional argument <var>opt</var> selects the type of mean to compute.
The following options are recognized:
<dl>
<dt>"a"<dd>Compute the (ordinary) arithmetic mean. [default]
<br><dt>"g"<dd>Compute the geometric mean.
<br><dt>"h"<dd>Compute the harmonic mean.
</dl>
<p>If the optional argument <var>dim</var> is given, operate along this dimension.
<p>Both <var>dim</var> and <var>opt</var> are optional. If both are supplied,
either may appear first.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmedian.html#doc_002dmedian">median</a>, <a href="doc_002dmode.html#doc_002dmode">mode</a>.
</p></blockquote></div>
<!-- median scripts/statistics/base/median.m -->
<p><a name="doc_002dmedian"></a>
<div class="defun">
— Function File: <b>median</b> (<var>x</var>)<var><a name="index-median-2411"></a></var><br>
— Function File: <b>median</b> (<var>x, dim</var>)<var><a name="index-median-2412"></a></var><br>
<blockquote><p>Compute the median value of the elements of the vector <var>x</var>.
If the elements of <var>x</var> are sorted, the median is defined
as
<pre class="example"> x(ceil(N/2)) N odd
median (x) =
(x(N/2) + x((N/2)+1))/2 N even
</pre>
<p>If <var>x</var> is a matrix, compute the median value for each
column and return them in a row vector. If the optional <var>dim</var>
argument is given, operate along this dimension.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmean.html#doc_002dmean">mean</a>, <a href="doc_002dmode.html#doc_002dmode">mode</a>.
</p></blockquote></div>
<!-- mode scripts/statistics/base/mode.m -->
<p><a name="doc_002dmode"></a>
<div class="defun">
— Function File: <b>mode</b> (<var>x</var>)<var><a name="index-mode-2413"></a></var><br>
— Function File: <b>mode</b> (<var>x, dim</var>)<var><a name="index-mode-2414"></a></var><br>
— Function File: [<var>m</var>, <var>f</var>, <var>c</var>] = <b>mode</b> (<var><small class="dots">...</small></var>)<var><a name="index-mode-2415"></a></var><br>
<blockquote><p>Compute the most frequently occurring value in a dataset (mode).
<code>mode</code> determines the frequency of values along the first non-singleton
dimension and returns the value with the highest frequency. If two, or
more, values have the same frequency <code>mode</code> returns the smallest.
<p>If the optional argument <var>dim</var> is given, operate along this dimension.
<p>The return variable <var>f</var> is the number of occurrences of the mode in
in the dataset. The cell array <var>c</var> contains all of the elements
with the maximum frequency.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmean.html#doc_002dmean">mean</a>, <a href="doc_002dmedian.html#doc_002dmedian">median</a>.
</p></blockquote></div>
<p>Using just one number, such as the mean, to represent an entire data set may
not give an accurate picture of the data. One way to characterize the fit is
to measure the dispersion of the data. Octave provides several functions for
measuring dispersion.
<!-- range scripts/statistics/base/range.m -->
<p><a name="doc_002drange"></a>
<div class="defun">
— Function File: <b>range</b> (<var>x</var>)<var><a name="index-range-2416"></a></var><br>
— Function File: <b>range</b> (<var>x, dim</var>)<var><a name="index-range-2417"></a></var><br>
<blockquote><p>Return the range, i.e., the difference between the maximum and the minimum
of the input data. If <var>x</var> is a vector, the range is calculated over
the elements of <var>x</var>. If <var>x</var> is a matrix, the range is calculated
over each column of <var>x</var>.
<p>If the optional argument <var>dim</var> is given, operate along this dimension.
<p>The range is a quickly computed measure of the dispersion of a data set, but
is less accurate than <code>iqr</code> if there are outlying data points.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002diqr.html#doc_002diqr">iqr</a>, <a href="doc_002dstd.html#doc_002dstd">std</a>.
</p></blockquote></div>
<!-- iqr scripts/statistics/base/iqr.m -->
<p><a name="doc_002diqr"></a>
<div class="defun">
— Function File: <b>iqr</b> (<var>x</var>)<var><a name="index-iqr-2418"></a></var><br>
— Function File: <b>iqr</b> (<var>x, dim</var>)<var><a name="index-iqr-2419"></a></var><br>
<blockquote><p>Return the interquartile range, i.e., the difference between the upper
and lower quartile of the input data. If <var>x</var> is a matrix, do the
above for first non-singleton dimension of <var>x</var>.
<p>If the optional argument <var>dim</var> is given, operate along this dimension.
<p>As a measure of dispersion, the interquartile range is less affected by
outliers than either <code>range</code> or <code>std</code>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002drange.html#doc_002drange">range</a>, <a href="doc_002dstd.html#doc_002dstd">std</a>.
</p></blockquote></div>
<!-- meansq scripts/statistics/base/meansq.m -->
<p><a name="doc_002dmeansq"></a>
<div class="defun">
— Function File: <b>meansq</b> (<var>x</var>)<var><a name="index-meansq-2420"></a></var><br>
— Function File: <b>meansq</b> (<var>x, dim</var>)<var><a name="index-meansq-2421"></a></var><br>
<blockquote><p>Compute the mean square of the elements of the vector <var>x</var>.
<pre class="example"> std (x) = 1/N SUM_i x(i)^2
</pre>
<p>For matrix arguments, return a row vector containing the mean square
of each column.
<p>If the optional argument <var>dim</var> is given, operate along this dimension.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dvar.html#doc_002dvar">var</a>, <a href="doc_002dstd.html#doc_002dstd">std</a>, <a href="doc_002dmoment.html#doc_002dmoment">moment</a>.
</p></blockquote></div>
<!-- std scripts/statistics/base/std.m -->
<p><a name="doc_002dstd"></a>
<div class="defun">
— Function File: <b>std</b> (<var>x</var>)<var><a name="index-std-2422"></a></var><br>
— Function File: <b>std</b> (<var>x, opt</var>)<var><a name="index-std-2423"></a></var><br>
— Function File: <b>std</b> (<var>x, opt, dim</var>)<var><a name="index-std-2424"></a></var><br>
<blockquote><p>Compute the standard deviation of the elements of the vector <var>x</var>.
<pre class="example"> std (x) = sqrt ( 1/(N-1) SUM_i (x(i) - mean(x))^2 )
</pre>
<p class="noindent">where N is the number of elements.
<p>If <var>x</var> is a matrix, compute the standard deviation for
each column and return them in a row vector.
<p>The argument <var>opt</var> determines the type of normalization to use.
Valid values are
<dl>
<dt>0:<dd> normalize with N-1, provides the square root of the best unbiased
estimator of the variance [default]
<br><dt>1:<dd> normalize with N, this provides the square root of the second
moment around the mean
</dl>
<p>If the optional argument <var>dim</var> is given, operate along this dimension.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dvar.html#doc_002dvar">var</a>, <a href="doc_002drange.html#doc_002drange">range</a>, <a href="doc_002diqr.html#doc_002diqr">iqr</a>, <a href="doc_002dmean.html#doc_002dmean">mean</a>, <a href="doc_002dmedian.html#doc_002dmedian">median</a>.
</p></blockquote></div>
<p>In addition to knowing the size of a dispersion it is useful to know the shape
of the data set. For example, are data points massed to the left or right
of the mean? Octave provides several common measures to describe the shape
of the data set. Octave can also calculate moments allowing arbitrary shape
measures to be developed.
<!-- var scripts/statistics/base/var.m -->
<p><a name="doc_002dvar"></a>
<div class="defun">
— Function File: <b>var</b> (<var>x</var>)<var><a name="index-var-2425"></a></var><br>
— Function File: <b>var</b> (<var>x, opt</var>)<var><a name="index-var-2426"></a></var><br>
— Function File: <b>var</b> (<var>x, opt, dim</var>)<var><a name="index-var-2427"></a></var><br>
<blockquote><p>Compute the variance of the elements of the vector <var>x</var>.
<pre class="example"> var (x) = 1/(N-1) SUM_i (x(i) - mean(x))^2
</pre>
<p>If <var>x</var> is a matrix, compute the variance for each column
and return them in a row vector.
<p>The argument <var>opt</var> determines the type of normalization to use.
Valid values are
<dl>
<dt>0:<dd> normalize with N-1, provides the best unbiased estimator of the
variance [default]
<br><dt>1:<dd> normalizes with N, this provides the second moment around the mean
</dl>
<p>If the optional argument <var>dim</var> is given, operate along this dimension.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dcov.html#doc_002dcov">cov</a>, <a href="doc_002dstd.html#doc_002dstd">std</a>, <a href="doc_002dskewness.html#doc_002dskewness">skewness</a>, <a href="doc_002dkurtosis.html#doc_002dkurtosis">kurtosis</a>, <a href="doc_002dmoment.html#doc_002dmoment">moment</a>.
</p></blockquote></div>
<!-- skewness scripts/statistics/base/skewness.m -->
<p><a name="doc_002dskewness"></a>
<div class="defun">
— Function File: <b>skewness</b> (<var>x</var>)<var><a name="index-skewness-2428"></a></var><br>
— Function File: <b>skewness</b> (<var>x, dim</var>)<var><a name="index-skewness-2429"></a></var><br>
<blockquote><p>Compute the skewness of the elements of the vector <var>x</var>.
<pre class="example"> skewness (x) = 1/N std(x)^(-3) sum ((x - mean(x)).^3)
</pre>
<p class="noindent">If <var>x</var> is a matrix, return the skewness along the
first non-singleton dimension of the matrix. If the optional
<var>dim</var> argument is given, operate along this dimension.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dvar.html#doc_002dvar">var</a>, <a href="doc_002dkurtosis.html#doc_002dkurtosis">kurtosis</a>, <a href="doc_002dmoment.html#doc_002dmoment">moment</a>.
</p></blockquote></div>
<!-- kurtosis scripts/statistics/base/kurtosis.m -->
<p><a name="doc_002dkurtosis"></a>
<div class="defun">
— Function File: <b>kurtosis</b> (<var>x</var>)<var><a name="index-kurtosis-2430"></a></var><br>
— Function File: <b>kurtosis</b> (<var>x, dim</var>)<var><a name="index-kurtosis-2431"></a></var><br>
<blockquote><p>Compute the kurtosis of the elements of the vector <var>x</var>.
<pre class="example"> kurtosis (x) = 1/N std(x)^(-4) sum ((x - mean(x)).^4) - 3
</pre>
<p>If <var>x</var> is a matrix, return the kurtosis over the
first non-singleton dimension of the matrix. If the optional
<var>dim</var> argument is given, operate along this dimension.
<p>Note: The definition of kurtosis above yields a kurtosis of zero for the
stdnormal distribution and is sometimes referred to as "excess kurtosis".
To calculate kurtosis without the normalization factor of -3 use
<code>moment (</code><var>x</var><code>, 4, 'c') / std (</code><var>x</var><code>)^4</code>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dvar.html#doc_002dvar">var</a>, <a href="doc_002dskewness.html#doc_002dskewness">skewness</a>, <a href="doc_002dmoment.html#doc_002dmoment">moment</a>.
</p></blockquote></div>
<!-- moment scripts/statistics/base/moment.m -->
<p><a name="doc_002dmoment"></a>
<div class="defun">
— Function File: <b>moment</b> (<var>x, p</var>)<var><a name="index-moment-2432"></a></var><br>
— Function File: <b>moment</b> (<var>x, p, type</var>)<var><a name="index-moment-2433"></a></var><br>
— Function File: <b>moment</b> (<var>x, p, dim</var>)<var><a name="index-moment-2434"></a></var><br>
— Function File: <b>moment</b> (<var>x, p, type, dim</var>)<var><a name="index-moment-2435"></a></var><br>
— Function File: <b>moment</b> (<var>x, p, dim, type</var>)<var><a name="index-moment-2436"></a></var><br>
<blockquote><p>Compute the <var>p</var>-th moment of the vector <var>x</var> about zero.
<pre class="example"> moment (x) = 1/N SUM_i x(i)^p
</pre>
<p>If <var>x</var> is a matrix, return the row vector containing the
<var>p</var>-th moment of each column.
<p>The optional string <var>type</var> specifies the type of moment to be computed.
Valid options are:
<dl>
<dt>"c"<dd> Central Moment. The moment about the mean defined as
<pre class="example"> 1/N SUM_i (x(i) - mean(x))^p
</pre>
<br><dt>"a"<dd> Absolute Moment. The moment about zero ignoring sign defined as
<pre class="example"> 1/N SUM_i ( abs (x(i)) )^p
</pre>
<br><dt>"ac"<dd> Absolute Central Moment. Defined as
<pre class="example"> 1/N SUM_i ( abs (x(i) - mean(x)) )^p
</pre>
</dl>
<p>If the optional argument <var>dim</var> is given, operate along this dimension.
<p>If both <var>type</var> and <var>dim</var> are given they may appear in any order.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dvar.html#doc_002dvar">var</a>, <a href="doc_002dskewness.html#doc_002dskewness">skewness</a>, <a href="doc_002dkurtosis.html#doc_002dkurtosis">kurtosis</a>.
</p></blockquote></div>
<!-- quantile scripts/statistics/base/quantile.m -->
<p><a name="doc_002dquantile"></a>
<div class="defun">
— Function File: <var>q</var> = <b>quantile</b> (<var>x, p</var>)<var><a name="index-quantile-2437"></a></var><br>
— Function File: <var>q</var> = <b>quantile</b> (<var>x, p, dim</var>)<var><a name="index-quantile-2438"></a></var><br>
— Function File: <var>q</var> = <b>quantile</b> (<var>x, p, dim, method</var>)<var><a name="index-quantile-2439"></a></var><br>
<blockquote><p>For a sample, <var>x</var>, calculate the quantiles, <var>q</var>, corresponding to
the cumulative probability values in <var>p</var>. All non-numeric values (NaNs)
of <var>x</var> are ignored.
<p>If <var>x</var> is a matrix, compute the quantiles for each column and
return them in a matrix, such that the i-th row of <var>q</var> contains
the <var>p</var>(i)th quantiles of each column of <var>x</var>.
<p>The optional argument <var>dim</var> determines the dimension along which
the quantiles are calculated. If <var>dim</var> is omitted, and <var>x</var> is
a vector or matrix, it defaults to 1 (column-wise quantiles). If
<var>x</var> is an N-D array, <var>dim</var> defaults to the first non-singleton
dimension.
<p>The methods available to calculate sample quantiles are the nine methods
used by R (http://www.r-project.org/). The default value is METHOD = 5.
<p>Discontinuous sample quantile methods 1, 2, and 3
<ol type=1 start=1>
<li>Method 1: Inverse of empirical distribution function.
<li>Method 2: Similar to method 1 but with averaging at discontinuities.
<li>Method 3: SAS definition: nearest even order statistic.
</ol>
<p>Continuous sample quantile methods 4 through 9, where p(k) is the linear
interpolation function respecting each methods' representative cdf.
<ol type=1 start=4>
<li>Method 4: p(k) = k / n. That is, linear interpolation of the
empirical cdf.
<li>Method 5: p(k) = (k - 0.5) / n. That is a piecewise linear function
where the knots are the values midway through the steps of the empirical
cdf.
<li>Method 6: p(k) = k / (n + 1).
<li>Method 7: p(k) = (k - 1) / (n - 1).
<li>Method 8: p(k) = (k - 1/3) / (n + 1/3). The resulting quantile
estimates are approximately median-unbiased regardless of the distribution
of <var>x</var>.
<li>Method 9: p(k) = (k - 3/8) / (n + 1/4). The resulting quantile
estimates are approximately unbiased for the expected order statistics if
<var>x</var> is normally distributed.
</ol>
<p>Hyndman and Fan (1996) recommend method 8. Maxima, S, and R
(versions prior to 2.0.0) use 7 as their default. Minitab and SPSS
use method 6. <span class="sc">matlab</span> uses method 5.
<p>References:
<ul>
<li>Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New
S Language. Wadsworth & Brooks/Cole.
<li>Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in
statistical packages, American Statistician, 50, 361–365.
<li>R: A Language and Environment for Statistical Computing;
<a href="http://cran.r-project.org/doc/manuals/fullrefman.pdf">http://cran.r-project.org/doc/manuals/fullrefman.pdf</a>.
</ul>
<p>Examples:
<!-- Set example in small font to prevent overfull line -->
<pre class="smallexample"> x = randi (1000, [10, 1]); # Create empirical data in range 1-1000
q = quantile (x, [0, 1]); # Return minimum, maximum of distribution
q = quantile (x, [0.25 0.5 0.75]); # Return quartiles of distribution
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dprctile.html#doc_002dprctile">prctile</a>.
</p></blockquote></div>
<!-- prctile scripts/statistics/base/prctile.m -->
<p><a name="doc_002dprctile"></a>
<div class="defun">
— Function File: <var>q</var> = <b>prctile</b> (<var>x</var>)<var><a name="index-prctile-2440"></a></var><br>
— Function File: <var>q</var> = <b>prctile</b> (<var>x, p</var>)<var><a name="index-prctile-2441"></a></var><br>
— Function File: <var>q</var> = <b>prctile</b> (<var>x, p, dim</var>)<var><a name="index-prctile-2442"></a></var><br>
<blockquote><p>For a sample <var>x</var>, compute the quantiles, <var>q</var>, corresponding
to the cumulative probability values, <var>p</var>, in percent. All non-numeric
values (NaNs) of <var>x</var> are ignored.
<p>If <var>x</var> is a matrix, compute the percentiles for each column and
return them in a matrix, such that the i-th row of <var>y</var> contains the
<var>p</var>(i)th percentiles of each column of <var>x</var>.
<p>If <var>p</var> is unspecified, return the quantiles for <code>[0 25 50 75 100]</code>.
The optional argument <var>dim</var> determines the dimension along which
the percentiles are calculated. If <var>dim</var> is omitted, and <var>x</var> is
a vector or matrix, it defaults to 1 (column-wise quantiles). When
<var>x</var> is an N-D array, <var>dim</var> defaults to the first non-singleton
dimension.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dquantile.html#doc_002dquantile">quantile</a>.
</p></blockquote></div>
<p>A summary view of a data set can be generated quickly with the
<code>statistics</code> function.
<!-- statistics scripts/statistics/base/statistics.m -->
<p><a name="doc_002dstatistics"></a>
<div class="defun">
— Function File: <b>statistics</b> (<var>x</var>)<var><a name="index-statistics-2443"></a></var><br>
— Function File: <b>statistics</b> (<var>x, dim</var>)<var><a name="index-statistics-2444"></a></var><br>
<blockquote><p>Return a vector with the minimum, first quartile, median, third quartile,
maximum, mean, standard deviation, skewness, and kurtosis of the elements of
the vector <var>x</var>.
<p>If <var>x</var> is a matrix, calculate statistics over the first
non-singleton dimension.
If the optional argument <var>dim</var> is given, operate along this dimension.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmin.html#doc_002dmin">min</a>, <a href="doc_002dmax.html#doc_002dmax">max</a>, <a href="doc_002dmedian.html#doc_002dmedian">median</a>, <a href="doc_002dmean.html#doc_002dmean">mean</a>, <a href="doc_002dstd.html#doc_002dstd">std</a>, <a href="doc_002dskewness.html#doc_002dskewness">skewness</a>, <a href="doc_002dkurtosis.html#doc_002dkurtosis">kurtosis</a>.
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