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  <properties>
    <title>The Commons Math User Guide - Statistics</title>
  </properties>
  <body>
    <section name="1 Statistics">
      <subsection name="1.1 Overview">
        <p>
          The statistics package provides frameworks and implementations for
          basic Descriptive statistics, frequency distributions, bivariate regression,
          and t-, chi-square and ANOVA test statistics.
        </p>
        <p>
         <a href="#a1.2_Descriptive_statistics">Descriptive statistics</a><br></br>
         <a href="#a1.3_Frequency_distributions">Frequency distributions</a><br></br>
         <a href="#a1.4_Simple_regression">Simple Regression</a><br></br>
         <a href="#a1.5_Multiple_linear_regression">Multiple Regression</a><br></br>
         <a href="#a1.6_Rank_transformations">Rank transformations</a><br></br>
         <a href="#a1.7_Covariance_and_correlation">Covariance and correlation</a><br></br>
         <a href="#a1.8_Statistical_tests">Statistical Tests</a><br></br>
        </p>
      </subsection>
      <subsection name="1.2 Descriptive statistics">
        <p>
          The stat package includes a framework and default implementations for
           the following Descriptive statistics:
          <ul>
            <li>arithmetic and geometric means</li>
            <li>variance and standard deviation</li>
            <li>sum, product, log sum, sum of squared values</li>
            <li>minimum, maximum, median, and percentiles</li>
            <li>skewness and kurtosis</li>
            <li>first, second, third and fourth moments</li>
          </ul>
        </p>
        <p>
          With the exception of percentiles and the median, all of these
          statistics can be computed without maintaining the full list of input
          data values in memory.  The stat package provides interfaces and
          implementations that do not require value storage as well as
          implementations that operate on arrays of stored values.
        </p>
        <p>
          The top level interface is
          <a href="../apidocs/org/apache/commons/math3/stat/descriptive/UnivariateStatistic.html">
          UnivariateStatistic</a>.
          This interface, implemented by all statistics, consists of
          <code>evaluate()</code> methods that take double[] arrays as arguments
          and return the value of the statistic.   This interface is extended by
          <a href="../apidocs/org/apache/commons/math3/stat/descriptive/StorelessUnivariateStatistic.html">
          StorelessUnivariateStatistic</a>, which adds <code>increment(),</code>
          <code>getResult()</code> and associated methods to support
          "storageless" implementations that maintain counters, sums or other
          state information as values are added using the <code>increment()</code>
          method.
        </p>
        <p>
          Abstract implementations of the top level interfaces are provided in
          <a href="../apidocs/org/apache/commons/math3/stat/descriptive/AbstractUnivariateStatistic.html">
          AbstractUnivariateStatistic</a> and
          <a href="../apidocs/org/apache/commons/math3/stat/descriptive/AbstractStorelessUnivariateStatistic.html">
          AbstractStorelessUnivariateStatistic</a> respectively.
        </p>
        <p>
          Each statistic is implemented as a separate class, in one of the
          subpackages (moment, rank, summary) and each extends one of the abstract
          classes above (depending on whether or not value storage is required to
          compute the statistic). There are several ways to instantiate and use statistics.
          Statistics can be instantiated and used directly,  but it is generally more convenient
          (and efficient) to access them using the provided aggregates,
          <a href="../apidocs/org/apache/commons/math3/stat/descriptive/DescriptiveStatistics.html">
           DescriptiveStatistics</a> and
           <a href="../apidocs/org/apache/commons/math3/stat/descriptive/SummaryStatistics.html">
           SummaryStatistics.</a>
        </p>
        <p>
           <code>DescriptiveStatistics</code> maintains the input data in memory
           and has the capability of producing "rolling" statistics computed from a
           "window" consisting of the most recently added values.
        </p>
        <p>
           <code>SummaryStatistics</code> does not store the input data values
           in memory, so the statistics included in this aggregate are limited to those
           that can be computed in one pass through the data without access to
           the full array of values.
        </p>
        <p>
          <table>
            <tr><th>Aggregate</th><th>Statistics Included</th><th>Values stored?</th>
            <th>"Rolling" capability?</th></tr><tr><td>
            <a href="../apidocs/org/apache/commons/math3/stat/descriptive/DescriptiveStatistics.html">
            DescriptiveStatistics</a></td><td>min, max, mean, geometric mean, n,
            sum, sum of squares, standard deviation, variance, percentiles, skewness,
            kurtosis, median</td><td>Yes</td><td>Yes</td></tr><tr><td>
            <a href="../apidocs/org/apache/commons/math3/stat/descriptive/SummaryStatistics.html">
            SummaryStatistics</a></td><td>min, max, mean, geometric mean, n,
            sum, sum of squares, standard deviation, variance</td><td>No</td><td>No</td></tr>
          </table>
        </p>
        <p>
          <code>SummaryStatistics</code> can be aggregated using 
          <a href="../apidocs/org/apache/commons/math3/stat/descriptive/AggregateSummaryStatistics.html">
          AggregateSummaryStatistics.</a>  This class can be used to concurrently
          gather statistics for multiple datasets as well as for a combined sample
          including all of the data.
       </p>
        <p>
           <code>MultivariateSummaryStatistics</code> is similar to
           <code>SummaryStatistics</code> but handles n-tuple values instead of
           scalar values. It can also compute the full covariance matrix for the
           input data.
        </p>
        <p>
           Neither <code>DescriptiveStatistics</code> nor <code>SummaryStatistics</code>
           is thread-safe.
           <a href="../apidocs/org/apache/commons/math3/stat/descriptive/SynchronizedDescriptiveStatistics.html">
           SynchronizedDescriptiveStatistics</a> and
           <a href="../apidocs/org/apache/commons/math3/stat/descriptive/SynchronizedSummaryStatistics.html"> 
           SynchronizedSummaryStatistics</a>, respectively, provide thread-safe
           versions for applications that require concurrent access to statistical
           aggregates by multiple threads.
           <a href="../apidocs/org/apache/commons/math3/stat/descriptive/SynchronizedMultiVariateSummaryStatistics.html"> 
           SynchronizedMultivariateSummaryStatistics</a> provides thread-safe
           <code>MultivariateSummaryStatistics.</code>
        </p>
        <p>
          There is also a utility class,
          <a href="../apidocs/org/apache/commons/math3/stat/StatUtils.html">
          StatUtils</a>, that provides static methods for computing statistics
          directly from double[] arrays.
        </p>
        <p>
          Here are some examples showing how to compute Descriptive statistics.
          <dl>
          <dt>Compute summary statistics for a list of double values</dt>
          <br></br>
          <dd>Using the <code>DescriptiveStatistics</code> aggregate
          (values are stored in memory):
        <source>
// Get a DescriptiveStatistics instance
DescriptiveStatistics stats = new DescriptiveStatistics();

// Add the data from the array
for( int i = 0; i &lt; inputArray.length; i++) {
        stats.addValue(inputArray[i]);
}

// Compute some statistics
double mean = stats.getMean();
double std = stats.getStandardDeviation();
double median = stats.getPercentile(50);
        </source>
        </dd>
        <dd>Using the <code>SummaryStatistics</code> aggregate (values are
        <strong>not</strong> stored in memory):
       <source>
// Get a SummaryStatistics instance
SummaryStatistics stats = new SummaryStatistics();

// Read data from an input stream,
// adding values and updating sums, counters, etc.
while (line != null) {
        line = in.readLine();
        stats.addValue(Double.parseDouble(line.trim()));
}
in.close();

// Compute the statistics
double mean = stats.getMean();
double std = stats.getStandardDeviation();
//double median = stats.getMedian(); &lt;-- NOT AVAILABLE
        </source>
        </dd>
         <dd>Using the <code>StatUtils</code> utility class:
       <source>
// Compute statistics directly from the array
// assume values is a double[] array
double mean = StatUtils.mean(values);
double std = StatUtils.variance(values);
double median = StatUtils.percentile(values, 50);

// Compute the mean of the first three values in the array
mean = StatUtils.mean(values, 0, 3);
        </source>
        </dd>
        <dt>Maintain a "rolling mean" of the most recent 100 values from
        an input stream</dt>
        <br></br>
        <dd>Use a <code>DescriptiveStatistics</code> instance with
        window size set to 100
        <source>
// Create a DescriptiveStats instance and set the window size to 100
DescriptiveStatistics stats = new DescriptiveStatistics();
stats.setWindowSize(100);

// Read data from an input stream,
// displaying the mean of the most recent 100 observations
// after every 100 observations
long nLines = 0;
while (line != null) {
        line = in.readLine();
        stats.addValue(Double.parseDouble(line.trim()));
        if (nLines == 100) {
                nLines = 0;
                System.out.println(stats.getMean());
       }
}
in.close();
        </source>
        </dd>
        <dt>Compute statistics in a thread-safe manner</dt>
        <br/>
        <dd>Use a <code>SynchronizedDescriptiveStatistics</code> instance
        <source>
// Create a SynchronizedDescriptiveStatistics instance and
// use as any other DescriptiveStatistics instance
DescriptiveStatistics stats = new SynchronizedDescriptiveStatistics();
        </source>
        </dd>
        <dt>Compute statistics for multiple samples and overall statistics concurrently</dt>
        <br/>
        <dd>There are two ways to do this using <code>AggregateSummaryStatistics.</code> 
        The first is to use an <code>AggregateSummaryStatistics</code> instance
        to accumulate overall statistics contributed by <code>SummaryStatistics</code>
        instances created using
        <a href="../apidocs/org/apache/commons/math3/stat/descriptive/AggregateSummaryStatistics.html#createContributingStatistics()">
        AggregateSummaryStatistics.createContributingStatistics()</a>:
        <source>
// Create a AggregateSummaryStatistics instance to accumulate the overall statistics 
// and AggregatingSummaryStatistics for the subsamples
AggregateSummaryStatistics aggregate = new AggregateSummaryStatistics();
SummaryStatistics setOneStats = aggregate.createContributingStatistics();
SummaryStatistics setTwoStats = aggregate.createContributingStatistics();
// Add values to the subsample aggregates
setOneStats.addValue(2);
setOneStats.addValue(3);
setTwoStats.addValue(2);
setTwoStats.addValue(4);
...
// Full sample data is reported by the aggregate
double totalSampleSum = aggregate.getSum();
        </source>
        The above approach has the disadvantages that the <code>addValue</code> calls must be synchronized on the
        <code>SummaryStatistics</code> instance maintained by the aggregate and each value addition updates the
        aggregate as well as the subsample. For applications that can wait to do the aggregation until all values
        have been added, a static
        <a href="../apidocs/org/apache/commons/math3/stat/descriptive/AggregateSummaryStatistics.html#aggregate(java.util.Collection)">
          aggregate</a> method is available, as shown in the following example.
        This method should be used when aggregation needs to be done across threads.
        <source>
// Create SummaryStatistics instances for the subsample data
SummaryStatistics setOneStats = new SummaryStatistics();
SummaryStatistics setTwoStats = new SummaryStatistics();
// Add values to the subsample SummaryStatistics instances
setOneStats.addValue(2);
setOneStats.addValue(3);
setTwoStats.addValue(2);
setTwoStats.addValue(4);
...
// Aggregate the subsample statistics
Collection&lt;SummaryStatistics&gt; aggregate = new ArrayList&lt;SummaryStatistics&gt;();
aggregate.add(setOneStats);
aggregate.add(setTwoStats);
StatisticalSummary aggregatedStats = AggregateSummaryStatistics.aggregate(aggregate);

// Full sample data is reported by aggregatedStats
double totalSampleSum = aggregatedStats.getSum();
        </source>
        </dd>
        </dl>
       </p>
      </subsection>
      <subsection name="1.3 Frequency distributions">
        <p>
          <a href="../apidocs/org/apache/commons/math3/stat/Frequency.html">
          Frequency</a>
          provides a simple interface for maintaining counts and percentages of discrete
          values.
        </p>
        <p>
          Strings, integers, longs and chars are all supported as value types,
          as well as instances of any class that implements <code>Comparable.</code>
          The ordering of values used in computing cumulative frequencies is by
          default the <i>natural ordering,</i> but this can be overriden by supplying a
          <code>Comparator</code> to the constructor. Adding values that are not
          comparable to those that have already been added results in an
          <code>IllegalArgumentException.</code>
        </p>
        <p>
          Here are some examples.
          <dl>
          <dt>Compute a frequency distribution based on integer values</dt>
          <br></br>
          <dd>Mixing integers, longs, Integers and Longs:
          <source>
 Frequency f = new Frequency();
 f.addValue(1);
 f.addValue(new Integer(1));
 f.addValue(new Long(1));
 f.addValue(2);
 f.addValue(new Integer(-1));
 System.out.prinltn(f.getCount(1));   // displays 3
 System.out.println(f.getCumPct(0));  // displays 0.2
 System.out.println(f.getPct(new Integer(1)));  // displays 0.6
 System.out.println(f.getCumPct(-2));   // displays 0
 System.out.println(f.getCumPct(10));  // displays 1
          </source>
          </dd>
          <dt>Count string frequencies</dt>
          <br></br>
          <dd>Using case-sensitive comparison, alpha sort order (natural comparator):
          <source>
Frequency f = new Frequency();
f.addValue("one");
f.addValue("One");
f.addValue("oNe");
f.addValue("Z");
System.out.println(f.getCount("one")); // displays 1
System.out.println(f.getCumPct("Z"));  // displays 0.5
System.out.println(f.getCumPct("Ot")); // displays 0.25
          </source>
          </dd>
          <dd>Using case-insensitive comparator:
          <source>
Frequency f = new Frequency(String.CASE_INSENSITIVE_ORDER);
f.addValue("one");
f.addValue("One");
f.addValue("oNe");
f.addValue("Z");
System.out.println(f.getCount("one"));  // displays 3
System.out.println(f.getCumPct("z"));  // displays 1
          </source>
         </dd>
       </dl>
      </p>
      </subsection>
      <subsection name="1.4 Simple regression">
        <p>
         <a href="../apidocs/org/apache/commons/math3/stat/regression/SimpleRegression.html">
          SimpleRegression</a> provides ordinary least squares regression with
         one independent variable estimating the linear model:
         </p>
         <p>
           <code> y = intercept + slope * x  </code>
         </p>
         <p>
           or
         </p>
         <p>
           <code> y = slope * x </code>
         </p>
         <p>
           Standard errors for <code>intercept</code> and <code>slope</code> are
           available as well as ANOVA, r-square and Pearson's r statistics.
         </p>
         <p>
           Observations (x,y pairs) can be added to the model one at a time or they
           can be provided in a 2-dimensional array.  The observations are not stored
           in memory, so there is no limit to the number of observations that can be
           added to the model.
         </p>
         <p>
           <strong>Usage Notes</strong>: <ul>
           <li> When there are fewer than two observations in the model, or when
            there is no variation in the x values (i.e. all x values are the same)
            all statistics return <code>NaN</code>.  At least two observations with
            different x coordinates are requred to estimate a bivariate regression
            model.</li>
           <li> getters for the statistics always compute values based on the current
           set of observations -- i.e., you can get statistics, then add more data
           and get updated statistics without using a new instance.  There is no
           "compute" method that updates all statistics.  Each of the getters performs
           the necessary computations to return the requested statistic.</li>
           <li> The intercept term may be suppressed by passing <code>false</code> to the
           <a href="../apidocs/org/apache/commons/math3/stat/regression/SimpleRegression.html#SimpleRegression(boolean)">
           SimpleRegression(boolean)</a> constructor.  When the <code>hasIntercept</code>
           property is false, the model is estimated without a constant term and
           <code>getIntercept()</code> returns <code>0</code>.</li>
          </ul>
        </p>
        <p>
           <strong>Implementation Notes</strong>: <ul>
           <li> As observations are added to the model, the sum of x values, y values,
           cross products (x times y), and squared deviations of x and y from their
           respective means are updated using updating formulas defined in
           "Algorithms for Computing the Sample Variance: Analysis and
           Recommendations", Chan, T.F., Golub, G.H., and LeVeque, R.J.
           1983, American Statistician, vol. 37, pp. 242-247, referenced in
           Weisberg, S. "Applied Linear Regression". 2nd Ed. 1985.  All regression
           statistics are computed from these sums.</li>
           <li> Inference statistics (confidence intervals, parameter significance levels)
           are based on on the assumption that the observations included in the model are
           drawn from a <a href="http://mathworld.wolfram.com/BivariateNormalDistribution.html">
           Bivariate Normal Distribution</a></li>
          </ul>
        </p>
        <p>
        Here are some examples.
        <dl>
          <dt>Estimate a model based on observations added one at a time</dt>
          <dd>Instantiate a regression instance and add data points
          <source>
regression = new SimpleRegression();
regression.addData(1d, 2d);
// At this point, with only one observation,
// all regression statistics will return NaN

regression.addData(3d, 3d);
// With only two observations,
// slope and intercept can be computed
// but inference statistics will return NaN

regression.addData(3d, 3d);
// Now all statistics are defined.
         </source>
         </dd>
         <dd>Compute some statistics based on observations added so far
         <source>
System.out.println(regression.getIntercept());
// displays intercept of regression line

System.out.println(regression.getSlope());
// displays slope of regression line

System.out.println(regression.getSlopeStdErr());
// displays slope standard error
         </source>
         </dd>
         <dd>Use the regression model to predict the y value for a new x value
         <source>
System.out.println(regression.predict(1.5d)
// displays predicted y value for x = 1.5
         </source>
         More data points can be added and subsequent getXxx calls will incorporate
         additional data in statistics.
         </dd>
         <br></br>
         <dt>Estimate a model from a double[][] array of data points</dt>
          <dd>Instantiate a regression object and load dataset
          <source>
double[][] data = { { 1, 3 }, {2, 5 }, {3, 7 }, {4, 14 }, {5, 11 }};
SimpleRegression regression = new SimpleRegression();
regression.addData(data);
          </source>
          </dd>
          <dd>Estimate regression model based on data
         <source>
System.out.println(regression.getIntercept());
// displays intercept of regression line

System.out.println(regression.getSlope());
// displays slope of regression line

System.out.println(regression.getSlopeStdErr());
// displays slope standard error
         </source>
         More data points -- even another double[][] array -- can be added and subsequent
         getXxx calls will incorporate additional data in statistics.
         </dd>
<br></br>
         <dt>Estimate a model from a double[][] array of data points, <em>excluding</em> the intercept</dt>
          <dd>Instantiate a regression object and load dataset
          <source>
double[][] data = { { 1, 3 }, {2, 5 }, {3, 7 }, {4, 14 }, {5, 11 }};
SimpleRegression regression = new SimpleRegression(false);
//the argument, false, tells the class not to include a constant
regression.addData(data);
          </source>
          </dd>
          <dd>Estimate regression model based on data
         <source>
System.out.println(regression.getIntercept());
// displays intercept of regression line, since we have constrained the constant, 0.0 is returned

System.out.println(regression.getSlope());
// displays slope of regression line

System.out.println(regression.getSlopeStdErr());
// displays slope standard error

System.out.println(regression.getInterceptStdErr() );
// will return Double.NaN, since we constrained the parameter to zero
         </source>
         Caution must be exercised when interpreting the slope when no constant is being estimated. The slope 
         may be biased.
         </dd>

         </dl>
        </p>
      </subsection>
      <subsection name="1.5 Multiple linear regression">
        <p>
         <a href="../apidocs/org/apache/commons/math3/stat/regression/OLSMultipleLinearRegression.html">
         OLSMultipleLinearRegression</a> and
         <a href="../apidocs/org/apache/commons/math3/stat/regression/GLSMultipleLinearRegression.html">
         GLSMultipleLinearRegression</a> provide least squares regression to fit the linear model:
         </p>
         <p>
           <code> Y=X*b+u </code>
         </p>
         <p>
         where Y is an n-vector <b>regressand</b>, X is a [n,k] matrix whose k columns are called
         <b>regressors</b>, b is k-vector of <b>regression parameters</b> and u is an n-vector 
         of <b>error terms</b> or <b>residuals</b>.
         </p>
         <p>
          <a href="../apidocs/org/apache/commons/math3/stat/regression/OLSMultipleLinearRegression.html">
          OLSMultipleLinearRegression</a> provides Ordinary Least Squares Regression, and 
          <a href="../apidocs/org/apache/commons/math3/stat/regression/GLSMultipleLinearRegression.html">
          GLSMultipleLinearRegression</a> implements Generalized Least Squares.  See the javadoc for these
          classes for details on the algorithms and forumlas used.
         </p>
         <p>
           Data for OLS models can be loaded in a single double[] array, consisting of concatenated rows of data, each containing
           the regressand (Y) value, followed by regressor values; or using a double[][] array with rows corresponding to
           observations. GLS models also require a double[][] array representing the covariance matrix of the error terms.  See
           <a href="../apidocs/org/apache/commons/math3/stat/regression/AbstractMultipleLinearRegression.html#newSampleData(double[], int, int)">
           AbstractMultipleLinearRegression#newSampleData(double[],int,int)</a>,  
           <a href="../apidocs/org/apache/commons/math3/stat/regression/OLSMultipleLinearRegression.html#newSampleData(double[], double[][])">
           OLSMultipleLinearRegression#newSampleData(double[], double[][])</a> and 
           <a href="../apidocs/org/apache/commons/math3/stat/regression/GLSMultipleLinearRegression.html#newSampleData(double[], double[][], double[][])">
           GLSMultipleLinearRegression#newSampleData(double[],double[][],double[][])</a> for details.
         </p>
         <p>
           <strong>Usage Notes</strong>: <ul>
           <li> Data are validated when invoking any of the newSample, newX, newY or newCovariance methods and
           <code>IllegalArgumentException</code> is thrown when input data arrays do not have matching dimensions
           or do not contain sufficient data to estimate the model. 
           </li>
           <li> By default, regression models are estimated with intercept terms.  In the notation above, this implies that the
           X matrix contains an initial row identically equal to 1.  X data supplied to the newX or newSample methods should not
           include this column - the data loading methods will create it automatically.  To estimate a model without an intercept
           term, set the <code>noIntercept</code> property to <code>true.</code></li>
          </ul>
        </p>
        <p>
        Here are some examples.
        <dl>
         <dt>OLS regression</dt>
          <br></br>
          <dd>Instantiate an OLS regression object and load a dataset:
          <source>
OLSMultipleLinearRegression regression = new OLSMultipleLinearRegression();
double[] y = new double[]{11.0, 12.0, 13.0, 14.0, 15.0, 16.0};
double[] x = new double[6][];
x[0] = new double[]{0, 0, 0, 0, 0};
x[1] = new double[]{2.0, 0, 0, 0, 0};
x[2] = new double[]{0, 3.0, 0, 0, 0};
x[3] = new double[]{0, 0, 4.0, 0, 0};
x[4] = new double[]{0, 0, 0, 5.0, 0};
x[5] = new double[]{0, 0, 0, 0, 6.0};          
regression.newSample(y, x);
          </source>
          </dd>
          <dd>Get regression parameters and diagnostics:
         <source>
double[] beta = regression.estimateRegressionParameters();       

double[] residuals = regression.estimateResiduals();

double[][] parametersVariance = regression.estimateRegressionParametersVariance();

double regressandVariance = regression.estimateRegressandVariance();

double rSquared = regression.calculateRSquared();

double sigma = regression.estimateRegressionStandardError();
         </source>
         </dd>
         <dt>GLS regression</dt>
          <br></br>
          <dd>Instantiate a GLS regression object and load a dataset:
          <source>
GLSMultipleLinearRegression regression = new GLSMultipleLinearRegression();
double[] y = new double[]{11.0, 12.0, 13.0, 14.0, 15.0, 16.0};
double[] x = new double[6][];
x[0] = new double[]{0, 0, 0, 0, 0};
x[1] = new double[]{2.0, 0, 0, 0, 0};
x[2] = new double[]{0, 3.0, 0, 0, 0};
x[3] = new double[]{0, 0, 4.0, 0, 0};
x[4] = new double[]{0, 0, 0, 5.0, 0};
x[5] = new double[]{0, 0, 0, 0, 6.0};          
double[][] omega = new double[6][];
omega[0] = new double[]{1.1, 0, 0, 0, 0, 0};
omega[1] = new double[]{0, 2.2, 0, 0, 0, 0};
omega[2] = new double[]{0, 0, 3.3, 0, 0, 0};
omega[3] = new double[]{0, 0, 0, 4.4, 0, 0};
omega[4] = new double[]{0, 0, 0, 0, 5.5, 0};
omega[5] = new double[]{0, 0, 0, 0, 0, 6.6};
regression.newSampleData(y, x, omega); 
          </source>
          </dd>
         </dl>
        </p>
      </subsection>    
      <subsection name="1.6 Rank transformations">
      <p>
         Some statistical algorithms require that input data be replaced by ranks.
         The <a href="../apidocs/org/apache/commons/math3/stat/ranking/package-summary.html">
         org.apache.commons.math3.stat.ranking</a> package provides rank transformation.
         <a href="../apidocs/org/apache/commons/math3/stat/ranking/RankingAlgorithm.html">
         RankingAlgorithm</a> defines the interface for ranking.  
         <a href="../apidocs/org/apache/commons/math3/stat/ranking/NaturalRanking.html">
         NaturalRanking</a> provides an implementation that has two configuration options.
         <ul>
         <li><a href="../apidocs/org/apache/commons/math3/stat/ranking/TiesStrategy.html">
         Ties strategy</a> deterimines how ties in the source data are handled by the ranking</li>
         <li><a href="../apidocs/org/apache/commons/math3/stat/ranking/NaNStrategy.html">
         NaN strategy</a> determines how NaN values in the source data are handled.</li>
         </ul>
      </p>
      <p>
         Examples:
         <source>
NaturalRanking ranking = new NaturalRanking(NaNStrategy.MINIMAL,
TiesStrategy.MAXIMUM);
double[] data = { 20, 17, 30, 42.3, 17, 50,
                  Double.NaN, Double.NEGATIVE_INFINITY, 17 };
double[] ranks = ranking.rank(exampleData);
         </source>
         results in <code>ranks</code> containing <code>{6, 5, 7, 8, 5, 9, 2, 2, 5}.</code>
         <source>
new NaturalRanking(NaNStrategy.REMOVED,TiesStrategy.SEQUENTIAL).rank(exampleData);   
         </source>
         returns <code>{5, 2, 6, 7, 3, 8, 1, 4}.</code>
      </p>
      <p>
        The default <code>NaNStrategy</code> is NaNStrategy.MAXIMAL.  This makes <code>NaN</code>
        values larger than any other value (including <code>Double.POSITIVE_INFINITY</code>). The
        default <code>TiesStrategy</code> is <code>TiesStrategy.AVERAGE,</code> which assigns tied
        values the average of the ranks applicable to the sequence of ties.  See the 
        <a href="../apidocs/org/apache/commons/math3/stat/ranking/NaturalRanking.html">
        NaturalRanking</a> for more examples and <a href="../apidocs/org/apache/commons/math3/stat/ranking/TiesStrategy.html">
        TiesStrategy</a> and <a href="../apidocs/org/apache/commons/math3/stat/ranking/NaNStrategy.html">NaNStrategy</a>
        for details on these configuration options.
       </p>
      </subsection>  
      <subsection name="1.7 Covariance and correlation">
        <p>
          The <a href="../apidocs/org/apache/commons/math3/stat/correlation/package-summary.html">
          org.apache.commons.math3.stat.correlation</a> package computes covariances
          and correlations for pairs of arrays or columns of a matrix.
          <a href="../apidocs/org/apache/commons/math3/stat/correlation/Covariance.html">
          Covariance</a> computes covariances, 
          <a href="../apidocs/org/apache/commons/math3/stat/correlation/PearsonsCorrelation.html">
          PearsonsCorrelation</a> provides Pearson's Product-Moment correlation coefficients and
          <a href="../apidocs/org/apache/commons/math3/stat/correlation/SpearmansCorrelation.html">
          SpearmansCorrelation</a> computes Spearman's rank correlation.
        </p>
        <p>
          <strong>Implementation Notes</strong>
          <ul>
          <li>
            Unbiased covariances are given by the formula <br></br>
            <code>cov(X, Y) = sum [(x<sub>i</sub> - E(X))(y<sub>i</sub> - E(Y))] / (n - 1)</code>
            where <code>E(X)</code> is the mean of <code>X</code> and <code>E(Y)</code>
           is the mean of the <code>Y</code> values. Non-bias-corrected estimates use 
           <code>n</code> in place of <code>n - 1.</code>  Whether or not covariances are
           bias-corrected is determined by the optional parameter, "biasCorrected," which
           defaults to <code>true.</code>      
          </li>
          <li>
          <a href="../apidocs/org/apache/commons/math3/stat/correlation/PearsonsCorrelation.html">
          PearsonsCorrelation</a> computes correlations defined by the formula <br></br>
          <code>cor(X, Y) = sum[(x<sub>i</sub> - E(X))(y<sub>i</sub> - E(Y))] / [(n - 1)s(X)s(Y)]</code><br/>
          where <code>E(X)</code> and <code>E(Y)</code> are means of <code>X</code> and <code>Y</code>
          and <code>s(X)</code>, <code>s(Y)</code> are standard deviations.
          </li>
          <li>
          <a href="../apidocs/org/apache/commons/math3/stat/correlation/SpearmansCorrelation.html">
          SpearmansCorrelation</a> applies a rank transformation to the input data and computes Pearson's
          correlation on the ranked data.  The ranking algorithm is configurable. By default, 
          <a href="../apidocs/org/apache/commons/math3/stat/ranking/NaturalRanking.html">
          NaturalRanking</a> with default strategies for handling ties and NaN values is used.
          </li> 
          </ul>
        </p>
        <p>
        <strong>Examples:</strong>
        <dl>
          <dt><strong>Covariance of 2 arrays</strong></dt>
          <br></br>
          <dd>To compute the unbiased covariance between 2 double arrays,
          <code>x</code> and <code>y</code>, use:
          <source>
new Covariance().covariance(x, y)
          </source>
          For non-bias-corrected covariances, use
          <source>
covariance(x, y, false)
          </source>
          </dd>
          <br></br>
          <dt><strong>Covariance matrix</strong></dt>
          <br></br>
          <dd> A covariance matrix over the columns of a source matrix <code>data</code>
          can be computed using
          <source>
new Covariance().computeCovarianceMatrix(data)
          </source>
          The i-jth entry of the returned matrix is the unbiased covariance of the ith and jth
          columns of <code>data.</code> As above, to get non-bias-corrected covariances,
          use 
         <source>
computeCovarianceMatrix(data, false)
         </source>
          </dd>
           <br></br>
          <dt><strong>Pearson's correlation of 2 arrays</strong></dt>
          <br></br>
          <dd>To compute the Pearson's product-moment correlation between two double arrays
          <code>x</code> and <code>y</code>, use:
          <source>
new PearsonsCorrelation().correlation(x, y)
          </source>
          </dd>
          <br></br>
          <dt><strong>Pearson's correlation matrix</strong></dt>
          <br></br>
          <dd> A (Pearson's) correlation matrix over the columns of a source matrix <code>data</code>
          can be computed using
          <source>
new PearsonsCorrelation().computeCorrelationMatrix(data)
          </source>
          The i-jth entry of the returned matrix is the Pearson's product-moment correlation between the
          ith and jth columns of <code>data.</code> 
          </dd>
           <br></br>
          <dt><strong>Pearson's correlation significance and standard errors</strong></dt>
          <br></br>
          <dd> To compute standard errors and/or significances of correlation coefficients
          associated with Pearson's correlation coefficients, start by creating a
          <code>PearsonsCorrelation</code> instance
          <source>
PearsonsCorrelation correlation = new PearsonsCorrelation(data);
          </source>
          where <code>data</code> is either a rectangular array or a <code>RealMatrix.</code>
          Then the matrix of standard errors is
          <source>
correlation.getCorrelationStandardErrors();
          </source>
          The formula used to compute the standard error is <br/>
          <code>SE<sub>r</sub> = ((1 - r<sup>2</sup>) / (n - 2))<sup>1/2</sup></code><br/>
           where <code>r</code> is the estimated correlation coefficient and 
          <code>n</code> is the number of observations in the source dataset.<br/><br/>
          <strong>p-values</strong> for the (2-sided) null hypotheses that elements of
          a correlation matrix are zero populate the RealMatrix returned by
          <source>
correlation.getCorrelationPValues()
          </source>
          <code>getCorrelationPValues().getEntry(i,j)</code> is the
          probability that a random variable distributed as <code>t<sub>n-2</sub></code> takes
           a value with absolute value greater than or equal to <br></br>
           <code>|r<sub>ij</sub>|((n - 2) / (1 - r<sub>ij</sub><sup>2</sup>))<sup>1/2</sup></code>,
           where <code>r<sub>ij</sub></code> is the estimated correlation between the ith and jth
           columns of the source array or RealMatrix. This is sometimes referred to as the 
           <i>significance</i> of the coefficient.<br/><br/>
           For example, if <code>data</code> is a RealMatrix with 2 columns and 10 rows, then 
           <source>
new PearsonsCorrelation(data).getCorrelationPValues().getEntry(0,1)
           </source>
           is the significance of the Pearson's correlation coefficient between the two columns
           of <code>data</code>.  If this value is less than .01, we can say that the correlation
           between the two columns of data is significant at the 99% level.
          </dd>
           <br></br>
          <dt><strong>Spearman's rank correlation coefficient</strong></dt>
          <br></br>
          <dd>To compute the Spearman's rank-moment correlation between two double arrays
          <code>x</code> and <code>y</code>:
          <source>
new SpearmansCorrelation().correlation(x, y)
          </source>
          This is equivalent to 
          <source>
RankingAlgorithm ranking = new NaturalRanking();
new PearsonsCorrelation().correlation(ranking.rank(x), ranking.rank(y))
          </source>
          </dd>
           <br></br>
        </dl>
        </p>
      </subsection>
            <subsection name="1.8 Statistical tests">
        <p>
          The <a href="../apidocs/org/apache/commons/math3/stat/inference/">
          org.apache.commons.math3.stat.inference</a> package provides implementations for
          <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm">
          Student's t</a>,
          <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm">
          Chi-Square</a>, 
          <a href="http://en.wikipedia.org/wiki/G-test">G Test</a>,
          <a href="http://www.itl.nist.gov/div898/handbook/prc/section4/prc43.htm">
          One-Way ANOVA</a>,
          <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc35.htm">
          Mann-Whitney U</a> and
          <a href="http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test">
          Wilcoxon signed rank</a> test statistics as well as
          <a href="http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue">
          p-values</a> associated with <code>t-</code>,
          <code>Chi-Square</code>, <code>G</code>, <code>One-Way ANOVA</code>, <code>Mann-Whitney U</code>
          and <code>Wilcoxon signed rank</code> tests. The respective test classes are
          <a href="../apidocs/org/apache/commons/math3/stat/inference/TTest.html">
          TTest</a>,
          <a href="../apidocs/org/apache/commons/math3/stat/inference/ChiSquareTest.html">
          ChiSquareTest</a>,
          <a href="../apidocs/org/apache/commons/math3/stat/inference/GTest.html">
          GTest</a>,
          <a href="../apidocs/org/apache/commons/math3/stat/inference/OneWayAnova.html">
          OneWayAnova</a>,
          <a href="../apidocs/org/apache/commons/math3/stat/inference/MannWhitneyUTest.html">
          MannWhitneyUTest</a>, and
          <a href="../apidocs/org/apache/commons/math3/stat/inference/WilcoxonSignedRankTest.html">
          WilcoxonSignedRankTest</a>.          
          The <a href="../apidocs/org/apache/commons/math3/stat/inference/TestUtils.html">
          TestUtils</a> class provides static methods to get test instances or
          to compute test statistics directly.  The examples below all use the
          static methods in <code>TestUtils</code> to execute tests.  To get
          test object instances, either use e.g., <code>TestUtils.getTTest()</code>
          or use the implementation constructors directly, e.g. <code>new TTest()</code>.
        </p>
        <p>
          <strong>Implementation Notes</strong>
          <ul>
          <li>Both one- and two-sample t-tests are supported.  Two sample tests
          can be either paired or unpaired and the unpaired two-sample tests can
          be conducted under the assumption of equal subpopulation variances or
          without this assumption.  When equal variances is assumed, a pooled
          variance estimate is used to compute the t-statistic and the degrees
          of freedom used in the t-test equals the sum of the sample sizes minus 2.
          When equal variances is not assumed, the t-statistic uses both sample
          variances and the
          <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/gifs/nu3.gif">
          Welch-Satterwaite approximation</a> is used to compute the degrees
          of freedom.  Methods to return t-statistics and p-values are provided in each
          case, as well as boolean-valued methods to perform fixed significance
          level tests.  The names of methods or methods that assume equal
          subpopulation variances always start with "homoscedastic."  Test or
          test-statistic methods that just start with "t" do not assume equal
          variances. See the examples below and the API documentation for
          more details.</li>
          <li>The validity of the p-values returned by the t-test depends on the
          assumptions of the parametric t-test procedure, as discussed
          <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">
          here</a></li>
          <li>p-values returned by t-, chi-square and Anova tests are exact, based
           on numerical approximations to the t-, chi-square and F distributions in the
           <code>distributions</code> package. </li>
          <li>The G test implementation provides two p-values:
           <code>gTest(expected, observed)</code>, which is the tail probability beyond
           <code>g(expected, observed)</code> in the ChiSquare distribution with degrees
           of freedom one less than the common length of input arrays and 
           <code>gTestIntrinsic(expected, observed)</code> which is the same tail
           probability computed using a ChiSquare distribution with one less degeree
           of freedom. </li>
          <li>p-values returned by t-tests are for two-sided tests and the boolean-valued
           methods supporting fixed significance level tests assume that the hypotheses
           are two-sided.  One sided tests can be performed by dividing returned p-values
           (resp. critical values) by 2.</li>
           <li>Degrees of freedom for G- and chi-square tests are integral values, based on the
           number of observed or expected counts (number of observed counts - 1).</li>
          </ul>
          </p>
          <p>
        <strong>Examples:</strong>
        <dl>
          <dt><strong>One-sample <code>t</code> tests</strong></dt>
          <br></br>
          <dd>To compare the mean of a double[] array to a fixed value:
          <source>
double[] observed = {1d, 2d, 3d};
double mu = 2.5d;
System.out.println(TestUtils.t(mu, observed));
          </source>
          The code above will display the t-statisitic associated with a one-sample
           t-test comparing the mean of the <code>observed</code> values against
           <code>mu.</code>
          </dd>
          <dd>To compare the mean of a dataset described by a
          <a href="../apidocs/org/apache/commons/math3/stat/descriptive/StatisticalSummary.html">
          StatisticalSummary</a>  to a fixed value:
          <source>
double[] observed ={1d, 2d, 3d};
double mu = 2.5d;
SummaryStatistics sampleStats = new SummaryStatistics();
for (int i = 0; i &lt; observed.length; i++) {
    sampleStats.addValue(observed[i]);
}
System.out.println(TestUtils.t(mu, observed));
</source>
           </dd>
           <dd>To compute the p-value associated with the null hypothesis that the mean
            of a set of values equals a point estimate, against the two-sided alternative that
            the mean is different from the target value:
            <source>
double[] observed = {1d, 2d, 3d};
double mu = 2.5d;
System.out.println(TestUtils.tTest(mu, observed));
           </source>
          The snippet above will display the p-value associated with the null
          hypothesis that the mean of the population from which the
          <code>observed</code> values are drawn equals <code>mu.</code>
          </dd>
          <dd>To perform the test using a fixed significance level, use:
          <source>
TestUtils.tTest(mu, observed, alpha);
          </source>
          where <code>0 &lt; alpha &lt; 0.5</code> is the significance level of
          the test.  The boolean value returned will be <code>true</code> iff the
          null hypothesis can be rejected with confidence <code>1 - alpha</code>.
          To test, for example at the 95% level of confidence, use
          <code>alpha = 0.05</code>
          </dd>
          <br></br>
          <dt><strong>Two-Sample t-tests</strong></dt>
          <br></br>
          <dd><strong>Example 1:</strong> Paired test evaluating
          the null hypothesis that the mean difference between corresponding
          (paired) elements of the <code>double[]</code> arrays
          <code>sample1</code> and <code>sample2</code> is zero.
          <p>
          To compute the t-statistic:
          <source>
TestUtils.pairedT(sample1, sample2);
          </source>
           </p>
           <p>
           To compute the p-value:
           <source>
TestUtils.pairedTTest(sample1, sample2);
           </source>
           </p>
           <p>
           To perform a fixed significance level test with alpha = .05:
           <source>
TestUtils.pairedTTest(sample1, sample2, .05);
           </source>
           </p>
           The last example will return <code>true</code> iff the p-value
           returned by <code>TestUtils.pairedTTest(sample1, sample2)</code>
           is less than <code>.05</code>
           </dd>
           <dd><strong>Example 2: </strong> unpaired, two-sided, two-sample t-test using
           <code>StatisticalSummary</code> instances, without assuming that
           subpopulation variances are equal.
           <p>
           First create the <code>StatisticalSummary</code> instances.  Both
           <code>DescriptiveStatistics</code> and <code>SummaryStatistics</code>
           implement this interface.  Assume that <code>summary1</code> and
           <code>summary2</code> are <code>SummaryStatistics</code> instances,
           each of which has had at least 2 values added to the (virtual) dataset that
           it describes.  The sample sizes do not have to be the same -- all that is required
           is that both samples have at least 2 elements.
           </p>
           <p><strong>Note:</strong> The <code>SummaryStatistics</code> class does
           not store the dataset that it describes in memory, but it does compute all
           statistics necessary to perform t-tests, so this method can be used to
           conduct t-tests with very large samples.  One-sample tests can also be
           performed this way.
           (See <a href="#1.2 Descriptive statistics">Descriptive statistics</a> for details
           on the <code>SummaryStatistics</code> class.)
           </p>
           <p>
          To compute the t-statistic:
          <source>
TestUtils.t(summary1, summary2);
          </source>
           </p>
           <p>
           To compute the p-value:
           <source>
TestUtils.tTest(sample1, sample2);
           </source>
           </p>
           <p>
           To perform a fixed significance level test with alpha = .05:
           <source>
TestUtils.tTest(sample1, sample2, .05);
           </source>
           </p>
           <p>
           In each case above, the test does not assume that the subpopulation
           variances are equal.  To perform the tests under this assumption,
           replace "t" at the beginning of the method name with "homoscedasticT"
           </p>
           </dd>
           <br></br>
          <dt><strong>Chi-square tests</strong></dt>
          <br></br>
          <dd>To compute a chi-square statistic measuring the agreement between a
          <code>long[]</code> array of observed counts and a <code>double[]</code>
          array of expected counts, use:
          <source>
long[] observed = {10, 9, 11};
double[] expected = {10.1, 9.8, 10.3};
System.out.println(TestUtils.chiSquare(expected, observed));
          </source>
          the value displayed will be
          <code>sum((expected[i] - observed[i])^2 / expected[i])</code>
          </dd>
          <dd> To get the p-value associated with the null hypothesis that
          <code>observed</code> conforms to <code>expected</code> use:
          <source>
TestUtils.chiSquareTest(expected, observed);
          </source>
          </dd>
          <dd> To test the null hypothesis that <code>observed</code> conforms to
          <code>expected</code> with <code>alpha</code> siginficance level
          (equiv. <code>100 * (1-alpha)%</code> confidence) where <code>
          0 &lt; alpha &lt; 1 </code> use:
          <source>
TestUtils.chiSquareTest(expected, observed, alpha);
          </source>
          The boolean value returned will be <code>true</code> iff the null hypothesis
          can be rejected with confidence <code>1 - alpha</code>.
          </dd>
          <dd>To compute a chi-square statistic statistic associated with a
          <a href="http://www.itl.nist.gov/div898/handbook/prc/section4/prc45.htm">
          chi-square test of independence</a> based on a two-dimensional (long[][])
          <code>counts</code> array viewed as a two-way table, use:
          <source>
TestUtils.chiSquareTest(counts);
          </source>
          The rows of the 2-way table are
          <code>count[0], ... , count[count.length - 1]. </code><br></br>
          The chi-square statistic returned is
          <code>sum((counts[i][j] - expected[i][j])^2/expected[i][j])</code>
          where the sum is taken over all table entries and
          <code>expected[i][j]</code> is the product of the row and column sums at
          row <code>i</code>, column <code>j</code> divided by the total count.
          </dd>
          <dd>To compute the p-value associated with the null hypothesis that
          the classifications represented by the counts in the columns of the input 2-way
          table are independent of the rows, use:
          <source>
 TestUtils.chiSquareTest(counts);
          </source>
          </dd>
          <dd>To perform a chi-square test of independence with <code>alpha</code>
          siginficance level (equiv. <code>100 * (1-alpha)%</code> confidence)
          where <code>0 &lt; alpha &lt; 1 </code> use:
          <source>
TestUtils.chiSquareTest(counts, alpha);
          </source>
          The boolean value returned will be <code>true</code> iff the null
          hypothesis can be rejected with confidence <code>1 - alpha</code>.
          </dd>
          <br></br>
          <dt><strong>G tests</strong></dt>
          <br></br>
          <dd>G tests are an alternative to chi-square tests that are recommended
          when observed counts are small and / or incidence probabillities for 
          some cells are small. See Ted Dunning's paper,
          <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.14.5962">
          Accurate Methods for the Statistics of Surprise and Coincidence</a> for
          background and an empirical analysis showing now chi-square
          statistics can be misldeading in the presence of low incidence probabilities.
          This paper also derives the formulas used in computing G statistics and the
          root log likelihood ratio provided by the <code>GTest</code> class.</dd>
          <dd>
          <dd>To compute a G-test statistic measuring the agreement between a
          <code>long[]</code> array of observed counts and a <code>double[]</code>
          array of expected counts, use:
          <source>
double[] expected = new double[]{0.54d, 0.40d, 0.05d, 0.01d};
long[] observed = new long[]{70, 79, 3, 4};
System.out.println(TestUtils.g(expected, observed));
          </source>
          the value displayed will be
          <code>2 * sum(observed[i]) * log(observed[i]/expected[i])</code>
          </dd>
          <dd> To get the p-value associated with the null hypothesis that
          <code>observed</code> conforms to <code>expected</code> use:
          <source>
TestUtils.gTest(expected, observed);
          </source>
          </dd>
          <dd> To test the null hypothesis that <code>observed</code> conforms to
          <code>expected</code> with <code>alpha</code> siginficance level
          (equiv. <code>100 * (1-alpha)%</code> confidence) where <code>
          0 &lt; alpha &lt; 1 </code> use:
          <source>
TestUtils.gTest(expected, observed, alpha);
          </source>
          The boolean value returned will be <code>true</code> iff the null hypothesis
          can be rejected with confidence <code>1 - alpha</code>.
          </dd>
          <dd>To evaluate the hypothesis that two sets of counts come from the
          same underlying distribution, use long[] arrays for the counts and
          <code>gDataSetsComparison</code> for the test statistic
          <source>
long[] obs1 = new long[]{268, 199, 42};
long[] obs2 = new long[]{807, 759, 184};
System.out.println(TestUtils.gDataSetsComparison(obs1, obs2)); // G statistic
System.out.println(TestUtils.gTestDataSetsComparison(obs1, obs2)); // p-value
          </source>
          </dd>
          <dd>For 2 x 2 designs, the <code>rootLogLikelihoodRaio</code> method
          computes the
          <a href="http://tdunning.blogspot.com/2008/03/surprise-and-coincidence.html">
          signed root log likelihood ratio.</a>  For example, suppose that for two events
          A and B, the observed count of AB (both occurring) is 5, not A and B (B without A)
          is 1995, A not B is 0; and neither A nor B is 10000.  Then
          <source>
new GTest().rootLogLikelihoodRatio(5, 1995, 0, 100000);
          </source>
          returns the root log likelihood associated with the null hypothesis that A 
          and B are independent.
          </dd>
          <br></br>
          <dt><strong>One-Way Anova tests</strong></dt>
          <br></br>
          <source>
double[] classA =
   {93.0, 103.0, 95.0, 101.0, 91.0, 105.0, 96.0, 94.0, 101.0 };
double[] classB =
   {99.0, 92.0, 102.0, 100.0, 102.0, 89.0 };
double[] classC =
   {110.0, 115.0, 111.0, 117.0, 128.0, 117.0 };
List classes = new ArrayList();
classes.add(classA);
classes.add(classB);
classes.add(classC);
          </source>
          Then you can compute ANOVA F- or p-values associated with the
          null hypothesis that the class means are all the same
          using a <code>OneWayAnova</code> instance or <code>TestUtils</code>
          methods:
          <source>
double fStatistic = TestUtils.oneWayAnovaFValue(classes); // F-value
double pValue = TestUtils.oneWayAnovaPValue(classes);     // P-value
          </source>
          To test perform a One-Way Anova test with signficance level set at 0.01
          (so the test will, assuming assumptions are met, reject the null
          hypothesis incorrectly only about one in 100 times), use
          <source>
TestUtils.oneWayAnovaTest(classes, 0.01); // returns a boolean
                                          // true means reject null hypothesis
          </source>
          </dd>
        </dl>
        </p>
      </subsection>
    </section>
  </body>
</document>
