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<div class="subTitle">edu.rit.numeric</div>
<h2 title="Class Statistics" class="title">Class Statistics</h2>
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<pre>public class <span class="strong">Statistics</span>
extends <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></pre>
<div class="block">Class Statistics provides static methods for doing statistical tests.
<P>
For each statistical test, there is a method that returns the "p-value" of
the test statistic. This is the probability that the test statistic would
have a value greater than or equal to the observed value if the null
hypothesis is true.</div>
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<h3>Method Summary</h3>
<table class="overviewSummary" border="0" cellpadding="3" cellspacing="0" summary="Method Summary table, listing methods, and an explanation">
<caption><span>Methods</span><span class="tabEnd">&nbsp;</span></caption>
<tr>
<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Method and Description</th>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../edu/rit/numeric/Statistics.html#bernoulliChiSquarePvalue(double)">bernoulliChiSquarePvalue</a></strong>(double&nbsp;chisqr)</code>
<div class="block">Returns the p-value of a Bernoulli chi-square statistic.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../edu/rit/numeric/Statistics.html#bernoulliChiSquareTest(long, long)">bernoulliChiSquareTest</a></strong>(long&nbsp;total,
long&nbsp;measured)</code>
<div class="block">Do a Bernoulli chi-square test on the given data.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../edu/rit/numeric/Statistics.html#chiSquarePvalue(double, double)">chiSquarePvalue</a></strong>(double&nbsp;N,
double&nbsp;chisqr)</code>
<div class="block">Returns the p-value of a chi-square statistic.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../edu/rit/numeric/Statistics.html#chiSquareTest(double[], double[])">chiSquareTest</a></strong>(double[]&nbsp;measured,
double[]&nbsp;expected)</code>
<div class="block">Do a chi-square test on the given data.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../edu/rit/numeric/Statistics.html#ksPvalue(double, double)">ksPvalue</a></strong>(double&nbsp;N,
double&nbsp;D)</code>
<div class="block">Returns the p-value of a K-S statistic.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../edu/rit/numeric/Statistics.html#ksTest(double[])">ksTest</a></strong>(double[]&nbsp;data)</code>
<div class="block">Do a Kolmogorov-Smirnov (K-S) test on the given data.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../edu/rit/numeric/Statistics.html#ksTest(double[], edu.rit.numeric.Function)">ksTest</a></strong>(double[]&nbsp;data,
<a href="../../../edu/rit/numeric/Function.html" title="interface in edu.rit.numeric">Function</a>&nbsp;cdf)</code>
<div class="block">Do a Kolmogorov-Smirnov (K-S) test on the given data.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../edu/rit/numeric/Statistics.html#normalPvalue(double, double, double)">normalPvalue</a></strong>(double&nbsp;x,
double&nbsp;mean,
double&nbsp;stddev)</code>
<div class="block">Returns the p-value of a statistic drawn from a normal distribution.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static double[]</code></td>
<td class="colLast"><code><strong><a href="../../../edu/rit/numeric/Statistics.html#tTestUnequalVariance(edu.rit.numeric.Series, edu.rit.numeric.Series)">tTestUnequalVariance</a></strong>(<a href="../../../edu/rit/numeric/Series.html" title="class in edu.rit.numeric">Series</a>&nbsp;data1,
<a href="../../../edu/rit/numeric/Series.html" title="class in edu.rit.numeric">Series</a>&nbsp;data2)</code>
<div class="block">Do an unequal-variance <I>t</I>-test on the two given data series.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../edu/rit/numeric/Statistics.html#ySquarePvalue(double, double)">ySquarePvalue</a></strong>(double&nbsp;N,
double&nbsp;ysqr)</code>
<div class="block">Returns the p-value of a Y-square statistic.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../edu/rit/numeric/Statistics.html#ySquareTest(int, double[], double[])">ySquareTest</a></strong>(int&nbsp;N,
double[]&nbsp;measured,
double[]&nbsp;expected)</code>
<div class="block">Do a Y-square test on the given data.</div>
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<h3>Methods inherited from class&nbsp;java.lang.<a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></h3>
<code><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#clone()" title="class or interface in java.lang">clone</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#equals(java.lang.Object)" title="class or interface in java.lang">equals</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#finalize()" title="class or interface in java.lang">finalize</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#getClass()" title="class or interface in java.lang">getClass</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#hashCode()" title="class or interface in java.lang">hashCode</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notify()" title="class or interface in java.lang">notify</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notifyAll()" title="class or interface in java.lang">notifyAll</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#toString()" title="class or interface in java.lang">toString</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait()" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long)" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long, int)" title="class or interface in java.lang">wait</a></code></li>
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<h3>Method Detail</h3>
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<h4>chiSquareTest</h4>
<pre>public static&nbsp;double&nbsp;chiSquareTest(double[]&nbsp;measured,
double[]&nbsp;expected)</pre>
<div class="block">Do a chi-square test on the given data. The null hypothesis is that the
data was drawn from the distribution given by <TT>expected</TT>. The
<TT>measured</TT> and <TT>expected</TT> arrays must be the same length.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>measured</code> - Measured count in each bin.</dd><dd><code>expected</code> - Expected count in each bin.</dd>
<dt><span class="strong">Returns:</span></dt><dd>Chi-square statistic.</dd></dl>
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<h4>chiSquarePvalue</h4>
<pre>public static&nbsp;double&nbsp;chiSquarePvalue(double&nbsp;N,
double&nbsp;chisqr)</pre>
<div class="block">Returns the p-value of a chi-square statistic.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>N</code> - Degrees of freedom.</dd><dd><code>chisqr</code> - Chi-square statistic.</dd>
<dt><span class="strong">Returns:</span></dt><dd>P-value.</dd></dl>
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<h4>bernoulliChiSquareTest</h4>
<pre>public static&nbsp;double&nbsp;bernoulliChiSquareTest(long&nbsp;total,
long&nbsp;measured)</pre>
<div class="block">Do a Bernoulli chi-square test on the given data. The null hypothesis is
that the data was drawn from a Bernoulli distribution with both outcomes
equally likely (e.g., a fair coin). <TT>total</TT> is the total number of
trials. <TT>measured</TT> is the number of trials yielding one of the
outcomes. (<TT>total</TT>&nbsp;&minus;&nbsp;<TT>measured</TT>) is the
number of trials yielding the other outcome.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>total</code> - Total number of trials.</dd><dd><code>measured</code> - Number of trials yielding one of the outcomes.</dd>
<dt><span class="strong">Returns:</span></dt><dd>Chi-square statistic.</dd></dl>
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<pre>public static&nbsp;double&nbsp;bernoulliChiSquarePvalue(double&nbsp;chisqr)</pre>
<div class="block">Returns the p-value of a Bernoulli chi-square statistic.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>chisqr</code> - Chi-square statistic.</dd>
<dt><span class="strong">Returns:</span></dt><dd>P-value.</dd></dl>
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<h4>ySquareTest</h4>
<pre>public static&nbsp;double&nbsp;ySquareTest(int&nbsp;N,
double[]&nbsp;measured,
double[]&nbsp;expected)</pre>
<div class="block">Do a Y-square test on the given data. The null hypothesis is that the
data was drawn from the distribution given by <TT>expected</TT>. The
<TT>measured</TT> and <TT>expected</TT> arrays must be the same length.
<P>
The Y-square test is similar to the chi-square test, except the Y-square
statistic is valid even if the expected counts in some of the bins are
small, which is not true of the chi-square statistic. For further
information, see:
<P>
L. Lucy. Hypothesis testing for meagre data sets. <I>Monthly Notices of
the Royal Astronomical Society,</I> 318(1):92-100, October 2000.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>N</code> - Degrees of freedom.</dd><dd><code>measured</code> - Measured count in each bin.</dd><dd><code>expected</code> - Expected count in each bin.</dd>
<dt><span class="strong">Returns:</span></dt><dd>Y-square statistic.</dd></dl>
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<pre>public static&nbsp;double&nbsp;ySquarePvalue(double&nbsp;N,
double&nbsp;ysqr)</pre>
<div class="block">Returns the p-value of a Y-square statistic.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>N</code> - Degrees of freedom.</dd><dd><code>ysqr</code> - Y-square statistic.</dd>
<dt><span class="strong">Returns:</span></dt><dd>P-value.</dd></dl>
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<h4>ksTest</h4>
<pre>public static&nbsp;double&nbsp;ksTest(double[]&nbsp;data)</pre>
<div class="block">Do a Kolmogorov-Smirnov (K-S) test on the given data. The null hypothesis
is that the data was drawn from a uniform distribution between 0.0 and
1.0.
<P>
The values in the <TT>data</TT> array must all be in the range 0.0
through 1.0 and must be in ascending numerical order. The
<TT>ksTest()</TT> method does not sort the data itself because the
process that produced the data might already have sorted the data. If
necessary, call <a href="../../../edu/rit/util/Sorting.html#sort(double[])"><code>Sorting.sort</code></a><TT>(data)</TT> before calling <TT>ksTest(data)</TT>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>data</code> - Data array.</dd>
<dt><span class="strong">Returns:</span></dt><dd>K-S statistic.</dd></dl>
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<h4>ksTest</h4>
<pre>public static&nbsp;double&nbsp;ksTest(double[]&nbsp;data,
<a href="../../../edu/rit/numeric/Function.html" title="interface in edu.rit.numeric">Function</a>&nbsp;cdf)</pre>
<div class="block">Do a Kolmogorov-Smirnov (K-S) test on the given data. The null hypothesis
is that the data was drawn from the distribution specified by the given
<a href="../../../edu/rit/numeric/Function.html" title="interface in edu.rit.numeric">Function</a>. <TT>cdf.f(x)</TT> must return the value of the
cumulative distribution function at <I>x</I>, in the range 0.0 through
1.0.
<P>
The values in the <TT>data</TT> array must all be in the domain of
<TT>cdf</TT> and must be in ascending numerical order. The
<TT>ksTest()</TT> method does not sort the data itself because the
process that produced the data might already have sorted the data. If
necessary, call <a href="../../../edu/rit/util/Sorting.html#sort(double[])"><code>Sorting.sort</code></a><TT>(data)</TT> before calling <TT>ksTest(data)</TT>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>data</code> - Data array.</dd><dd><code>cdf</code> - Cumulative distribution function.</dd>
<dt><span class="strong">Returns:</span></dt><dd>K-S statistic.</dd></dl>
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<h4>ksPvalue</h4>
<pre>public static&nbsp;double&nbsp;ksPvalue(double&nbsp;N,
double&nbsp;D)</pre>
<div class="block">Returns the p-value of a K-S statistic.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>N</code> - Number of data points.</dd><dd><code>D</code> - K-S statistic.</dd>
<dt><span class="strong">Returns:</span></dt><dd>P-value.</dd></dl>
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<h4>normalPvalue</h4>
<pre>public static&nbsp;double&nbsp;normalPvalue(double&nbsp;x,
double&nbsp;mean,
double&nbsp;stddev)</pre>
<div class="block">Returns the p-value of a statistic drawn from a normal distribution.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - Statistic.</dd><dd><code>mean</code> - Mean of the normal distribution.</dd><dd><code>stddev</code> - Standard deviation of the normal distribution.</dd>
<dt><span class="strong">Returns:</span></dt><dd>P-value.</dd></dl>
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<h4>tTestUnequalVariance</h4>
<pre>public static&nbsp;double[]&nbsp;tTestUnequalVariance(<a href="../../../edu/rit/numeric/Series.html" title="class in edu.rit.numeric">Series</a>&nbsp;data1,
<a href="../../../edu/rit/numeric/Series.html" title="class in edu.rit.numeric">Series</a>&nbsp;data2)</pre>
<div class="block">Do an unequal-variance <I>t</I>-test on the two given data series. The
null hypothesis is that the two data series have the same mean; however,
the two series are assumed to have different variances. An array of two
doubles is returned; element 0 gives the <I>t</I> statistic; element 1
gives the <I>p</I>-value (significance) of the <I>t</I> statistic.
Roughly speaking, the <I>p</I>-value is the probability that the
hypothesis is true. If the <I>p</I>-value falls below a significance
threshold, the hypothesis is not true, and the two data series have
different means.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>data1</code> - First data series.</dd><dd><code>data2</code> - Second data series.</dd>
<dt><span class="strong">Returns:</span></dt><dd><I>t</I> statistic and its <I>p</I>-value.</dd></dl>
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