## Class Statistics

- java.lang.Object
- edu.rit.numeric.Statistics

public class Statisticsextends Object

Class Statistics provides static methods for doing statistical tests.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.

### Method Summary

Methods Modifier and Type Method and Description `static double`

**bernoulliChiSquarePvalue**(double chisqr)Returns the p-value of a Bernoulli chi-square statistic.`static double`

**bernoulliChiSquareTest**(long total, long measured)Do a Bernoulli chi-square test on the given data.`static double`

**binomialKsTest**(int[] data, int n)Do a Kolmogorov-Smirnov (K-S) test on the given data.`static double`

**chiSquarePvalue**(double N, double chisqr)Returns the p-value of a chi-square statistic.`static double`

**chiSquareTest**(double[] measured, double[] expected)Do a chi-square test on the given data.`static double`

**ksPvalue**(double N, double D)Returns the p-value of a K-S statistic.`static double`

**ksTest**(double[] data)Do a Kolmogorov-Smirnov (K-S) test on the given data.`static double`

**ksTest**(double[] data, Function cdf)Do a Kolmogorov-Smirnov (K-S) test on the given data.`static double`

**normalPvalue**(double x, double mean, double stddev)Returns the p-value of a statistic drawn from a normal distribution.`static double`

**ySquarePvalue**(double N, double ysqr)Returns the p-value of a Y-square statistic.`static double`

**ySquareTest**(int N, double[] measured, double[] expected)Do a Y-square test on the given data.

### Method Detail

#### chiSquareTest

public static double chiSquareTest(double[] measured, double[] expected)

Do a chi-square test on the given data. The null hypothesis is that the data was drawn from the distribution given by`expected`. The`measured`and`expected`arrays must be the same length.- Parameters:
`measured`

- Measured count in each bin.`expected`

- Expected count in each bin.- Returns:
- Chi-square statistic.

#### chiSquarePvalue

public static double chiSquarePvalue(double N, double chisqr)

Returns the p-value of a chi-square statistic.- Parameters:
`N`

- Degrees of freedom.`chisqr`

- Chi-square statistic.- Returns:
- P-value.

#### bernoulliChiSquareTest

public static double bernoulliChiSquareTest(long total, long measured)

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).`total`is the total number of trials.`measured`is the number of trials yielding one of the outcomes. (`total`−`measured`) is the number of trials yielding the other outcome.- Parameters:
`total`

- Total number of trials.`measured`

- Number of trials yielding one of the outcomes.- Returns:
- Chi-square statistic.

#### bernoulliChiSquarePvalue

public static double bernoulliChiSquarePvalue(double chisqr)

Returns the p-value of a Bernoulli chi-square statistic.- Parameters:
`chisqr`

- Chi-square statistic.- Returns:
- P-value.

#### ySquareTest

public static double ySquareTest(int N, double[] measured, double[] expected)

Do a Y-square test on the given data. The null hypothesis is that the data was drawn from the distribution given by`expected`. The`measured`and`expected`arrays must be the same length.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:

L. Lucy. Hypothesis testing for meagre data sets.

*Monthly Notices of the Royal Astronomical Society,*318(1):92-100, October 2000.- Parameters:
`N`

- Degrees of freedom.`measured`

- Measured count in each bin.`expected`

- Expected count in each bin.- Returns:
- Y-square statistic.

#### ySquarePvalue

public static double ySquarePvalue(double N, double ysqr)

Returns the p-value of a Y-square statistic.- Parameters:
`N`

- Degrees of freedom.`ysqr`

- Y-square statistic.- Returns:
- P-value.

#### ksTest

public static double ksTest(double[] data)

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.The values in the

`data`array must all be in the range 0.0 through 1.0 and must be in ascending numerical order. The`ksTest()`method does not sort the data itself because the process that produced the data might already have sorted the data. If necessary, call`Arrays.sort(data)`before calling`ksTest(data)`.- Parameters:
`data`

- Data array.- Returns:
- K-S statistic.

#### ksTest

public static double ksTest(double[] data, Function cdf)

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 Function.`cdf.f(x)`must return the value of the cumulative distribution function at*x*, in the range 0.0 through 1.0.The values in the

`data`array must all be in the domain of`cdf`and must be in ascending numerical order. The`ksTest()`method does not sort the data itself because the process that produced the data might already have sorted the data. If necessary, call`Arrays.sort(data)`before calling`ksTest(data,cdf)`.- Parameters:
`data`

- Data array.`cdf`

- Cumulative distribution function.- Returns:
- K-S statistic.

#### binomialKsTest

public static double binomialKsTest(int[] data, int n)

Do a Kolmogorov-Smirnov (K-S) test on the given data. The null hypothesis is that the data was drawn from a binomial random variable*X*that is the sum of*n*equiprobable Bernoulli random variables. For 0 ≤*k*≤*n*, the probability that*X*equals*k*isPr[ *X*=*k*] = 2^{−n}*n*! /*k*! / (*n*−*k*)!The values in the

`data`array must all be in the range 0 ..*n*and must be in ascending numerical order. The`binomialKsTest()`method does not sort the data itself because the process that produced the data might already have sorted the data. If necessary, call`Arrays.sort(data)`before calling`binomialKsTest(data,n)`.*Note:*To prevent roundoff error, the internal calculations are done using exact rational arithmetic. The final K-S statistic is then converted to a double-precision floating-point number and is returned.- Parameters:
`data`

- Data array.`n`

- Number of Bernoulli random variables.- Returns:
- K-S statistic.
- Throws:
`IllegalArgumentException`

- (unchecked exception) Thrown if`n`≤ 0.

#### ksPvalue

public static double ksPvalue(double N, double D)

Returns the p-value of a K-S statistic.- Parameters:
`N`

- Number of data points.`D`

- K-S statistic.- Returns:
- P-value.

#### normalPvalue

public static double normalPvalue(double x, double mean, double stddev)

Returns the p-value of a statistic drawn from a normal distribution.- Parameters:
`x`

- Statistic.`mean`

- Mean of the normal distribution.`stddev`

- Standard deviation of the normal distribution.- Returns:
- P-value.

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