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<HTML>
<HEAD>
<TITLE>STATISTICS command</TITLE>
</HEAD>
<BODY BGCOLOR="#FFFFFF" TEXT="#000000">
<p><font size="+3" color="green"><B>STATISTICS command</B></font></P>
<TABLE border="1" cols="2" frame="box" rules="all" width="572">
<TR>
<TD width="15%" valign="top"><B>Syntax</B>:</TD>
<TD width="85%" valign="top"><CODE>
STATISTICS x { s1\keyword { s2\keyword ... }}<br />
STATISTICS\PEARSON x y { rcof prob }<br />
STATISTICS\MOMENTS w x n { sout }</CODE>
</TD></TR>
<TR>
<TD valign="top"><B>Qualifiers</B>:</TD>
<TD valign="top"><CODE>\MESSAGES, \WEIGHTS, \MOMENTS, \PEARSON</CODE></TD></TR>
<TR>
<TD valign="top"><B>Defaults</B>:</TD>
<TD valign="top"><CODE>\MESSAGES, \-WEIGHTS</CODE></TD></TR>
<TR>
<TD valign="top"><B>Examples</B>:</TD>
<TD valign="top"><CODE>
STATISTICS X<br />
STATISTICS\-MESS X XMED\MEDIAN XMEAN\XMEAN<BR />
STATISTICS\WEIGHTS W X XVAR\VARIANCE XSUM\SUM<BR />
STATISTICS\MOMENTS Y X 3 M3</CODE>
</TD></TR>
</TABLE>
<P>
The <CODE>STATISTICS</CODE> command calculates various statistics
for the input variable <CODE>x</CODE>, which can be
a vector or a matrix. Specific statistics are chosen with qualifier keywords
which are appended to the output parameters with the backslash, \. All
vectors must be the same size.</P>
<P>
Table 1 below shows the parameter qualifier keywords and corresponding output values for extrema.
Table 2 shows the parameter qualifier keywords and corresponding output values for central measures.
Table 3 shows the parameter qualifier keywords and corresponding output values for dispersion and
skewness.</p>
<p>
<center><table border="1" width="400">
<tr>
<td><i>Keyword</i></td>
<td><i>Output Value</i></td>
</tr><tr>
<td><CODE>\MAX</CODE></td>
<td>maximum value of <CODE>x</CODE></td>
</tr><tr>
<td><CODE>\IMAX</CODE></td>
<td>index of the maximum if <CODE>x</CODE> is a vector<br />
row index of the maximum if <CODE>x</CODE> is a matrix</td>
</tr><tr>
<td><CODE>\JMAX</CODE></td>
<td>column index of the maximum if <CODE>x is a matrix</CODE></td>
</tr><tr>
<td><CODE>\MIN</CODE></td>
<td>minimum value of <CODE>x</CODE></td>
</tr><tr>
<td><CODE>\IMIN</CODE></td>
<td>index of the minimum if <CODE>x</CODE> is a vector<br />
row index of the minimum if <CODE>x</CODE> is a matrix</td>
</tr><tr>
<td><CODE>\JMIN</CODE></td>
<td>column index of the minimum value if <CODE>x</CODE> is a matrix</td>
</tr></table>
<table width="400" border="0">
<tr><td align="center"><b>Table 1:</b> Extrema keywords</td>
</tr></table></center></p>
<p>
<center><table border="1" width="400">
<tr>
<td><i>Keyword</i></td><td><i>Output Value</i></td>
</tr><tr>
<td><CODE>\SUM</CODE></td><td>arithmetic sum (unweighted)</td>
</tr><tr>
<td><CODE>\MEAN</CODE></td><td>arithmetic mean</td>
</tr><tr>
<td><CODE>\GMEAN</CODE></td><td>geometric mean</td>
</tr><tr>
<td><CODE>\MEDIAN</CODE></td><td>median value</td>
</tr><tr>
<td><CODE>\RMS</CODE></td><td>root-mean-square</td>
</tr></table>
<table width="400" border="0">
<tr><td align="center"><b>Table 2:</b> Central measure keywords</td>
</td></table></center></p>
<p>
<center><table border="1" width="400">
<tr>
<td><i>Keyword</i></td><td><i>Output Value</i></td>
</tr><tr>
<td><CODE>\VARIANCE</CODE></td><td>variance</td>
</tr><tr>
<td><CODE>\SDEV</CODE></td><td>standard deviation</td>
</tr><tr>
<td><CODE>\ADEV</CODE></td><td>average deviation</td>
</tr><tr>
<td><CODE>\KURTOSIS</CODE></td><td>kurtosis</td>
</tr><tr>
<td><CODE>\SKEWNESS</CODE></td><td>skewness</td>
</tr></table>
<table width="400" border="0">
<tr><td align="center"><b>Table 3:</b> Dispersion and skewness keywords</td>
</tr></table></center></p>
<p>
<font size="+2" color="green">Informational messages</font></p>
<p>
The default is to display all the calculated statistics. If the
<CODE>\-MESSAGES</CODE> command qualifier is used, and if at least one output scalar is entered,
then the statistics values will not be displayed.</p>
<p>
<font size="+2" color="green">Weights</font></p>
<p>
<TABLE border="1" cols="2" frame="box" rules="all" width="572">
<TR>
<TD width="15%" valign="top"><B>Syntax</B>:</TD>
<TD width="85%" valign="top"><CODE>
STATISTICS\WEIGHTS w x { s1\keyword { s2\keyword ... }}</CODE>
</TD></TR></TABLE></p>
<p>
You <EM>must</EM> use the <CODE>\WEIGHTS</CODE>
qualifier to indicate that a weight vector is present. Weights cannot be
applied to matrix data.</p>
<P>
A weighting factor, <CODE>w[i] ≥ 0</CODE>,
could be the frequency, the probability, the mass, the reliability, or some
other multiplier. The lengths of <CODE>w</CODE> and <CODE>x</CODE> must be equal.</p>
<p>
<font size="+2" color="green">Definitions</font></p>
<p>
Suppose that <code>x</code> is a vector with <code>N</code> elements.</P>
<P>
If a weight vector, <code>w</code>, is entered, remember to use the
<CODE>\WEIGHTS</CODE> command qualifier. The
length of <code>w</code> is assumed to also be <code>N</code>. If no weights are entered,
let <code>w<sub>i</sub></code> default to <CODE>1</CODE>, for <code>i = 1,2,...,N</code>.
Define the total weight: <code>W = w<sub>1</sub> + w<sub>2</sub> + ... + w<sub>N</sub></code></p>
<P>
<font size="+1" color="green">Sum</font></p>
<P>
The sum is defined by <code>x<sub>1</sub> + x<sub>2</sub> + ... + x<sub>N</sub></code></p>
<P>
<font size="+1" color="green">Mean value</font></p>
<P>
The mean value, <code>M</code>, is defined by</p>
<p>
<center><code>M = (1/W)*[w<sub>1</sub>x<sub>1</sub> +
w<sub>2</sub>x<sub>2</sub> + ... + w<sub>N</sub>x<sub>N</sub>]</code></center></p>
<P>
<font size="+1" color="green">Geometric mean</font></p>
<P>
The geometric mean, <code>G<sub>x</sub></code>, is defined if each <code>x<sub>i</sub> ≥ 0</code>
by:</p>
<p>
<center><code>G<sub>x</sub> = exp(1/W)*[w<sub>1</sub>log(x<sub>1</sub>) +
w<sub>2</sub>log(x<sub>2</sub>) + ... +
w<sub>N</sub>log(x<sub>N</sub>)]</code></center></p>
<P>
<font size="+1" color="green">Median</font></p>
<P>
The median is the element of <code>x</code> which has equal numbers of values above
it and below it. If <code>N</code> is even, the median is the average of the unique
two central values.</p>
<P>
<font size="+1" color="green">Root-mean-square</font></p>
<P>
The root-mean-square, <code>RMS</code>, is defined by</p>
<p>
<center><code>RMS = sqrt([1/W]*[w<sub>1</sub>x<sub>1</sub><sup>2</sup> +
w<sub>2</sub>x<sub>2</sub><sup>2</sup>
+ ... + w<sub>N</sub>x<sub>N</sub><sup>2</sup>])</code></center></p>
<P>
<font size="+1" color="green">Variance</font></p>
<P>
The variance, <code>μ</code>, is defined by</p>
<p>
<center><code>μ = [N/W(N-1)]*[w<sub>1</sub>(x<sub>1</sub>-M)<sup>2</sup> +
w<sub>2</sub>(x<sub>2</sub>-M)<sup>2</sup> + ... +
w<sub>N</sub>(x<sub>N</sub>-M)<sup>2</sup>]</code></center></p>
<P>
<font size="+1" color="green">Standard deviation</font></p>
<P>
The standard deviation, <code>σ</code>, is defined by <code>σ = sqrt(μ)</code></p>
<P>
<font size="+1" color="green">Average deviation</font></p>
<P>
The average deviation, or mean deviation, <code>δ</code>, is defined by</p>
<p>
<center><code>δ = (1/W)*[w<sub>1</sub>|x<sub>1</sub>-M| + w<sub>2</sub>|x<sub>2</sub>-M| + ... +
w<sub>N</sub>|x<sub>N</sub>-M|]</code></center></p>
<P>
<font size="+1" color="green">Skewness</font></p>
<P>
The skewness, or third moment, <code>skew</code>, is a nondimensional quantity that
characterizes the degree of asymmetry of a distribution around its mean. The
skewness is a pure number that characterizes only the shape of the
distribution, and is defined by</p>
<p>
<center><code>skew = (1/W)*{w<sub>1</sub>[(x<sub>1</sub>-M)/σ]<sup>3</sup> +
w<sub>2</sub>[(x<sub>2</sub>-M)/σ]<sup>3</sup> + ... +
w<sub>N</sub>[(x<sub>N</sub>-M)/σ]<sup>3</sup>}</code></center></p>
<P>
A positive value of skewness signifies a distribution with an asymmetric tail
extending out towards more positive <i>x</i>; a negative value signifies a
distribution whose tail extends out towards more negative <i>x</i>.</p>
<P>
<font size="+1" color="green">Kurtosis</font></p>
<P>
The kurtosis, <code>kurt</code>, is a nondimensional quantity which measures the
relative peakedness or flatness of a distribution, relative to a normal
distribution. A distribution with positive kurtosis is termed leptokurtic;
a distribution with negative kurtosis is termed platykurtic. An in-between
distribution is termed mesokurtic. The kurtosis is defined by</p>
<p>
<center><code>kurt =
w<sub>1</sub>[(x<sub>1</sub>-M)/σ]<sup>4</sup> +
w<sub>2</sub>[(x<sub>2</sub>-M)/σ]<sup>4</sup> + ... +
w<sub>N</sub>[(x<sub>N</sub>-M)/σ]<sup>4</sup> - 3</code></center></P>
<P>
where the <i>-3</i> term makes the value zero for a normal distribution.</p>
<p>
<font size="+2" color="green">Moments</font></p>
<TABLE border="1" cols="2" frame="box" rules="all" width="572">
<TR>
<TD width="15%" valign="top"><B>Syntax</B>:</TD>
<TD width="85%" valign="top"><CODE>
STATISTICS\MOMENTS w x n { s }</CODE>
</TD></TR></TABLE>
<p>
If the <CODE>\MOMENTS</CODE> command qualifier is used, the <CODE>n</CODE><sup>th</sup>
moment of vector <CODE>x</CODE>, with weight <CODE>w</CODE>, is calculated and optionally
stored in output scalar <CODE>s</CODE>. The moment number, <CODE>n</CODE>, can be any integer
<code>> 0</code>.</p>
<P>
<center><code>s = (1/W)*[w<sub>1</sub>x<sub>1</sub><sup>n</sup> +
w<sub>2</sub>x<sub>2</sub><sup>n</sup> + ... +
w<sub>N</sub>x<sub>N</sub><sup>n</sup>]</code></center></p>
<P>
<font size="+2" color="green">Linear correlation coefficient</font></p>
<TABLE border="1" cols="2" frame="box" rules="all" width="572">
<TR>
<TD width="15%" valign="top"><B>Syntax</B>:</TD>
<TD width="85%" valign="top"><CODE>
STATISTICS\PEARSON x y { r p }</CODE>
</TD></TR></TABLE>
<p>
Pearson's <code>r</code>, or the linear correlation coefficient, is widely used as
a measure of association between variables that are continuous. For pairs
of quantities <code>(x<sub>i</sub>,y<sub>i</sub>)</code>, for <code>i = 1,2,...,N</code>, the
linear correlation coefficient <code>r</code> is given by the formula:</p>
<P>
<IMG SRC="img33.gif"></P>
<P>
where <IMG SRC="img12.gif"> is the mean of <code>x</code>, and
<IMG SRC="img35.gif"> is the mean of <code>y</code>.</p>
<P>
The value of <i>r</i> lies between <i>-1</i> and <i>+1</i>, inclusive. It
takes on a value of <i>+1</i> when the data points lie on a straight line
with positive slope, <code>x</code> and <code>y</code> increase together. The value
<i>+1</i> holds independent of the magnitude of this slope. If the data
points lie on a straight line with negative slope, <code>y</code> decreases as
<code>x</code> increases, then <code>r</code> has the value <i>-1</i>. A value of
<code>r</code> near zero indicates that the variables <code>x</code> and <code>y</code> are
uncorrelated.</p>
<P>
<code>r</code> is a way of summarizing the strength of a correlation which is
known to be significant, but it is a poor statistic for deciding whether an
observed correlation is statistically significant, and/or whether one observed
correlation is significantly stronger than another. The reason is that
<code>r</code> is ignorant of the individual distributions of <code>x</code> and
<code>y</code>, so there is no universal way to compute its distribution in the
case of the null hypothesis.</p>
<P>
The <CODE>STATISTICS\PEARSON</CODE> command returns Pearson's <code>r</code> in the scalar variable
<CODE>r</CODE>. It also returns scalar <CODE>p</CODE>, the significance
level at which the null hypothesis of zero correlation is disproved.
A small value of <CODE>p</CODE> indicates a significant correlation.</p>
<P>
<IMG SRC="img37.gif"></P>
<P>
where <code>I</code> is the incomplete Beta function and <code>t</code> is defined by:</p>
<p>
<center><IMG SRC="img39.gif"></center></P>
<P>
<font size="+1" color="green">Examples</font></p>
<p>
Suppose you have a vector <code>X=[1.2;2.1;3.2;4.5;5;6;7]</code>. Entering
<code><font color="blue">STATISTICS X</font></code> produces the following display:</P>
<p>
<IMG SRC="ex1.png"></p>
<p>
If you want to use the values for the maximum, minimum and mean of <TT>X</TT>,
enter:</p>
<P>
<code><font color="blue">STATISTICS X XMEAN\MEAN XMIN\MIN XMAX\MAX</font></code></p>
<P>
and you will have the scalars: <code>XMAX=7</code>, <code>XMIN=1.2</code>, and
<code>XMEAN=4.142857</code></p>
<P>
If you also want the index values for the maximum and the minimum of
<TT>X</TT>, enter:</p>
<P>
<code><font color="blue">STATISTICS X XMEAN\MEAN XMIN\MIN XMAX\MAX IMX\IMAX IMN\IMIN</font></code></p>
<P>
and you will also have scalars: <code>IMX=7</code> and <code>IMN=1</code>.</p>
</BODY>
</HTML>
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