Variance
org.apache.commons.math3.stat.descriptive.moment

## Class Variance

• All Implemented Interfaces:
Serializable, StorelessUnivariateStatistic, UnivariateStatistic, WeightedEvaluation, MathArrays.Function

```public class Variance
extends AbstractStorelessUnivariateStatistic
implements Serializable, WeightedEvaluation```
Computes the variance of the available values. By default, the unbiased "sample variance" definitional formula is used:

variance = sum((x_i - mean)^2) / (n - 1)

where mean is the `Mean` and `n` is the number of sample observations.

The definitional formula does not have good numerical properties, so this implementation does not compute the statistic using the definitional formula.

• The `getResult` method computes the variance using updating formulas based on West's algorithm, as described in Chan, T. F. and J. G. Lewis 1979, Communications of the ACM, vol. 22 no. 9, pp. 526-531.
• The `evaluate` methods leverage the fact that they have the full array of values in memory to execute a two-pass algorithm. Specifically, these methods use the "corrected two-pass algorithm" from Chan, Golub, Levesque, Algorithms for Computing the Sample Variance, American Statistician, vol. 37, no. 3 (1983) pp. 242-247.
Note that adding values using `increment` or `incrementAll` and then executing `getResult` will sometimes give a different, less accurate, result than executing `evaluate` with the full array of values. The former approach should only be used when the full array of values is not available.

The "population variance" ( sum((x_i - mean)^2) / n ) can also be computed using this statistic. The `isBiasCorrected` property determines whether the "population" or "sample" value is returned by the `evaluate` and `getResult` methods. To compute population variances, set this property to `false.`

Note that this implementation is not synchronized. If multiple threads access an instance of this class concurrently, and at least one of the threads invokes the `increment()` or `clear()` method, it must be synchronized externally.

Serialized Form
• ### Constructor Summary

Constructors
Constructor and Description
`Variance()`
Constructs a Variance with default (true) `isBiasCorrected` property.
`Variance(boolean isBiasCorrected)`
Constructs a Variance with the specified `isBiasCorrected` property
```Variance(boolean isBiasCorrected, SecondMoment m2)```
Constructs a Variance with the specified `isBiasCorrected` property and the supplied external second moment.
`Variance(SecondMoment m2)`
Constructs a Variance based on an external second moment.
`Variance(Variance original)`
Copy constructor, creates a new `Variance` identical to the `original`
• ### Method Summary

Methods
Modifier and Type Method and Description
`void` `clear()`
Clears the internal state of the Statistic
`Variance` `copy()`
Returns a copy of the statistic with the same internal state.
`static void` ```copy(Variance source, Variance dest)```
Copies source to dest.
`double` `evaluate(double[] values)`
Returns the variance of the entries in the input array, or `Double.NaN` if the array is empty.
`double` ```evaluate(double[] values, double mean)```
Returns the variance of the entries in the input array, using the precomputed mean value.
`double` ```evaluate(double[] values, double[] weights)```
Returns the weighted variance of the entries in the the input array.
`double` ```evaluate(double[] values, double[] weights, double mean)```
Returns the weighted variance of the values in the input array, using the precomputed weighted mean value.
`double` ```evaluate(double[] values, double[] weights, double mean, int begin, int length)```
Returns the weighted variance of the entries in the specified portion of the input array, using the precomputed weighted mean value.
`double` ```evaluate(double[] values, double[] weights, int begin, int length)```
Returns the weighted variance of the entries in the specified portion of the input array, or `Double.NaN` if the designated subarray is empty.
`double` ```evaluate(double[] values, double mean, int begin, int length)```
Returns the variance of the entries in the specified portion of the input array, using the precomputed mean value.
`double` ```evaluate(double[] values, int begin, int length)```
Returns the variance of the entries in the specified portion of the input array, or `Double.NaN` if the designated subarray is empty.
`long` `getN()`
Returns the number of values that have been added.
`double` `getResult()`
Returns the current value of the Statistic.
`void` `increment(double d)`
Updates the internal state of the statistic to reflect the addition of the new value.
`boolean` `isBiasCorrected()`
`void` `setBiasCorrected(boolean biasCorrected)`
• ### Methods inherited from class org.apache.commons.math3.stat.descriptive.AbstractStorelessUnivariateStatistic

`equals, hashCode, incrementAll, incrementAll`
• ### Methods inherited from class org.apache.commons.math3.stat.descriptive.AbstractUnivariateStatistic

`evaluate, getData, setData, setData`
• ### Methods inherited from class java.lang.Object

`getClass, notify, notifyAll, toString, wait, wait, wait`
• ### Constructor Detail

• #### Variance

`public Variance()`
Constructs a Variance with default (true) `isBiasCorrected` property.
• #### Variance

`public Variance(SecondMoment m2)`
Constructs a Variance based on an external second moment. When this constructor is used, the statistic may only be incremented via the moment, i.e., `increment(double)` does nothing; whereas `m2.increment(value)` increments both `m2` and the Variance instance constructed from it.
Parameters:
`m2` - the SecondMoment (Third or Fourth moments work here as well.)
• #### Variance

`public Variance(boolean isBiasCorrected)`
Constructs a Variance with the specified `isBiasCorrected` property
Parameters:
`isBiasCorrected` - setting for bias correction - true means bias will be corrected and is equivalent to using the argumentless constructor
• #### Variance

```public Variance(boolean isBiasCorrected,
SecondMoment m2)```
Constructs a Variance with the specified `isBiasCorrected` property and the supplied external second moment.
Parameters:
`isBiasCorrected` - setting for bias correction - true means bias will be corrected
`m2` - the SecondMoment (Third or Fourth moments work here as well.)
• #### Variance

```public Variance(Variance original)
throws NullArgumentException```
Copy constructor, creates a new `Variance` identical to the `original`
Parameters:
`original` - the `Variance` instance to copy
Throws:
`NullArgumentException` - if original is null
• ### Method Detail

• #### increment

`public void increment(double d)`
Updates the internal state of the statistic to reflect the addition of the new value.

If all values are available, it is more accurate to use `evaluate(double[])` rather than adding values one at a time using this method and then executing `getResult()`, since `evaluate` leverages the fact that is has the full list of values together to execute a two-pass algorithm. See `Variance`.

Note also that when `Variance(SecondMoment)` is used to create a Variance, this method does nothing. In that case, the SecondMoment should be incremented directly.

Specified by:
`increment` in interface `StorelessUnivariateStatistic`
Specified by:
`increment` in class `AbstractStorelessUnivariateStatistic`
Parameters:
`d` - the new value.
• #### getResult

`public double getResult()`
Returns the current value of the Statistic.
Specified by:
`getResult` in interface `StorelessUnivariateStatistic`
Specified by:
`getResult` in class `AbstractStorelessUnivariateStatistic`
Returns:
value of the statistic, `Double.NaN` if it has been cleared or just instantiated.
• #### getN

`public long getN()`
Returns the number of values that have been added.
Specified by:
`getN` in interface `StorelessUnivariateStatistic`
Returns:
the number of values.
• #### clear

`public void clear()`
Clears the internal state of the Statistic
Specified by:
`clear` in interface `StorelessUnivariateStatistic`
Specified by:
`clear` in class `AbstractStorelessUnivariateStatistic`
• #### evaluate

```public double evaluate(double[] values)
throws MathIllegalArgumentException```
Returns the variance of the entries in the input array, or `Double.NaN` if the array is empty.

See `Variance` for details on the computing algorithm.

Returns 0 for a single-value (i.e. length = 1) sample.

Throws `MathIllegalArgumentException` if the array is null.

Does not change the internal state of the statistic.

Specified by:
`evaluate` in interface `UnivariateStatistic`
Specified by:
`evaluate` in interface `MathArrays.Function`
Overrides:
`evaluate` in class `AbstractStorelessUnivariateStatistic`
Parameters:
`values` - the input array
Returns:
the variance of the values or Double.NaN if length = 0
Throws:
`MathIllegalArgumentException` - if the array is null
`UnivariateStatistic.evaluate(double[])`
• #### evaluate

```public double evaluate(double[] values,
int begin,
int length)
throws MathIllegalArgumentException```
Returns the variance of the entries in the specified portion of the input array, or `Double.NaN` if the designated subarray is empty.

See `Variance` for details on the computing algorithm.

Returns 0 for a single-value (i.e. length = 1) sample.

Does not change the internal state of the statistic.

Throws `MathIllegalArgumentException` if the array is null.

Specified by:
`evaluate` in interface `UnivariateStatistic`
Specified by:
`evaluate` in interface `MathArrays.Function`
Overrides:
`evaluate` in class `AbstractStorelessUnivariateStatistic`
Parameters:
`values` - the input array
`begin` - index of the first array element to include
`length` - the number of elements to include
Returns:
the variance of the values or Double.NaN if length = 0
Throws:
`MathIllegalArgumentException` - if the array is null or the array index parameters are not valid
`UnivariateStatistic.evaluate(double[], int, int)`
• #### evaluate

```public double evaluate(double[] values,
double[] weights,
int begin,
int length)
throws MathIllegalArgumentException```

Returns the weighted variance of the entries in the specified portion of the input array, or `Double.NaN` if the designated subarray is empty.

Uses the formula

```   Σ(weights[i]*(values[i] - weightedMean)2)/(Σ(weights[i]) - 1)
```
where weightedMean is the weighted mean

This formula will not return the same result as the unweighted variance when all weights are equal, unless all weights are equal to 1. The formula assumes that weights are to be treated as "expansion values," as will be the case if for example the weights represent frequency counts. To normalize weights so that the denominator in the variance computation equals the length of the input vector minus one, use

```   `evaluate(values, MathArrays.normalizeArray(weights, values.length)); `
```

Returns 0 for a single-value (i.e. length = 1) sample.

Throws `IllegalArgumentException` if any of the following are true:

• the values array is null
• the weights array is null
• the weights array does not have the same length as the values array
• the weights array contains one or more infinite values
• the weights array contains one or more NaN values
• the weights array contains negative values
• the start and length arguments do not determine a valid array

Does not change the internal state of the statistic.

Throws `MathIllegalArgumentException` if either array is null.

Specified by:
`evaluate` in interface `WeightedEvaluation`
Parameters:
`values` - the input array
`weights` - the weights array
`begin` - index of the first array element to include
`length` - the number of elements to include
Returns:
the weighted variance of the values or Double.NaN if length = 0
Throws:
`MathIllegalArgumentException` - if the parameters are not valid
Since:
2.1
• #### evaluate

```public double evaluate(double[] values,
double[] weights)
throws MathIllegalArgumentException```

Returns the weighted variance of the entries in the the input array.

Uses the formula

```   Σ(weights[i]*(values[i] - weightedMean)2)/(Σ(weights[i]) - 1)
```
where weightedMean is the weighted mean

This formula will not return the same result as the unweighted variance when all weights are equal, unless all weights are equal to 1. The formula assumes that weights are to be treated as "expansion values," as will be the case if for example the weights represent frequency counts. To normalize weights so that the denominator in the variance computation equals the length of the input vector minus one, use

```   `evaluate(values, MathArrays.normalizeArray(weights, values.length)); `
```

Returns 0 for a single-value (i.e. length = 1) sample.

Throws `MathIllegalArgumentException` if any of the following are true:

• the values array is null
• the weights array is null
• the weights array does not have the same length as the values array
• the weights array contains one or more infinite values
• the weights array contains one or more NaN values
• the weights array contains negative values

Does not change the internal state of the statistic.

Throws `MathIllegalArgumentException` if either array is null.

Specified by:
`evaluate` in interface `WeightedEvaluation`
Parameters:
`values` - the input array
`weights` - the weights array
Returns:
the weighted variance of the values
Throws:
`MathIllegalArgumentException` - if the parameters are not valid
Since:
2.1
• #### evaluate

```public double evaluate(double[] values,
double mean,
int begin,
int length)
throws MathIllegalArgumentException```
Returns the variance of the entries in the specified portion of the input array, using the precomputed mean value. Returns `Double.NaN` if the designated subarray is empty.

See `Variance` for details on the computing algorithm.

The formula used assumes that the supplied mean value is the arithmetic mean of the sample data, not a known population parameter. This method is supplied only to save computation when the mean has already been computed.

Returns 0 for a single-value (i.e. length = 1) sample.

Throws `MathIllegalArgumentException` if the array is null.

Does not change the internal state of the statistic.

Parameters:
`values` - the input array
`mean` - the precomputed mean value
`begin` - index of the first array element to include
`length` - the number of elements to include
Returns:
the variance of the values or Double.NaN if length = 0
Throws:
`MathIllegalArgumentException` - if the array is null or the array index parameters are not valid
• #### evaluate

```public double evaluate(double[] values,
double mean)
throws MathIllegalArgumentException```
Returns the variance of the entries in the input array, using the precomputed mean value. Returns `Double.NaN` if the array is empty.

See `Variance` for details on the computing algorithm.

If `isBiasCorrected` is `true` the formula used assumes that the supplied mean value is the arithmetic mean of the sample data, not a known population parameter. If the mean is a known population parameter, or if the "population" version of the variance is desired, set `isBiasCorrected` to `false` before invoking this method.

Returns 0 for a single-value (i.e. length = 1) sample.

Throws `MathIllegalArgumentException` if the array is null.

Does not change the internal state of the statistic.

Parameters:
`values` - the input array
`mean` - the precomputed mean value
Returns:
the variance of the values or Double.NaN if the array is empty
Throws:
`MathIllegalArgumentException` - if the array is null
• #### evaluate

```public double evaluate(double[] values,
double[] weights,
double mean,
int begin,
int length)
throws MathIllegalArgumentException```
Returns the weighted variance of the entries in the specified portion of the input array, using the precomputed weighted mean value. Returns `Double.NaN` if the designated subarray is empty.

Uses the formula

```   Σ(weights[i]*(values[i] - mean)2)/(Σ(weights[i]) - 1)
```

The formula used assumes that the supplied mean value is the weighted arithmetic mean of the sample data, not a known population parameter. This method is supplied only to save computation when the mean has already been computed.

This formula will not return the same result as the unweighted variance when all weights are equal, unless all weights are equal to 1. The formula assumes that weights are to be treated as "expansion values," as will be the case if for example the weights represent frequency counts. To normalize weights so that the denominator in the variance computation equals the length of the input vector minus one, use

```   `evaluate(values, MathArrays.normalizeArray(weights, values.length), mean); `
```

Returns 0 for a single-value (i.e. length = 1) sample.

Throws `MathIllegalArgumentException` if any of the following are true:

• the values array is null
• the weights array is null
• the weights array does not have the same length as the values array
• the weights array contains one or more infinite values
• the weights array contains one or more NaN values
• the weights array contains negative values
• the start and length arguments do not determine a valid array

Does not change the internal state of the statistic.

Parameters:
`values` - the input array
`weights` - the weights array
`mean` - the precomputed weighted mean value
`begin` - index of the first array element to include
`length` - the number of elements to include
Returns:
the variance of the values or Double.NaN if length = 0
Throws:
`MathIllegalArgumentException` - if the parameters are not valid
Since:
2.1
• #### evaluate

```public double evaluate(double[] values,
double[] weights,
double mean)
throws MathIllegalArgumentException```

Returns the weighted variance of the values in the input array, using the precomputed weighted mean value.

Uses the formula

```   Σ(weights[i]*(values[i] - mean)2)/(Σ(weights[i]) - 1)
```

The formula used assumes that the supplied mean value is the weighted arithmetic mean of the sample data, not a known population parameter. This method is supplied only to save computation when the mean has already been computed.

This formula will not return the same result as the unweighted variance when all weights are equal, unless all weights are equal to 1. The formula assumes that weights are to be treated as "expansion values," as will be the case if for example the weights represent frequency counts. To normalize weights so that the denominator in the variance computation equals the length of the input vector minus one, use

```   `evaluate(values, MathArrays.normalizeArray(weights, values.length), mean); `
```

Returns 0 for a single-value (i.e. length = 1) sample.

Throws `MathIllegalArgumentException` if any of the following are true:

• the values array is null
• the weights array is null
• the weights array does not have the same length as the values array
• the weights array contains one or more infinite values
• the weights array contains one or more NaN values
• the weights array contains negative values

Does not change the internal state of the statistic.

Parameters:
`values` - the input array
`weights` - the weights array
`mean` - the precomputed weighted mean value
Returns:
the variance of the values or Double.NaN if length = 0
Throws:
`MathIllegalArgumentException` - if the parameters are not valid
Since:
2.1
• #### isBiasCorrected

`public boolean isBiasCorrected()`
Returns:
Returns the isBiasCorrected.
• #### setBiasCorrected

`public void setBiasCorrected(boolean biasCorrected)`
Parameters:
`biasCorrected` - The isBiasCorrected to set.
• #### copy

`public Variance copy()`
Returns a copy of the statistic with the same internal state.
Specified by:
`copy` in interface `StorelessUnivariateStatistic`
Specified by:
`copy` in interface `UnivariateStatistic`
Specified by:
`copy` in class `AbstractStorelessUnivariateStatistic`
Returns:
a copy of the statistic
• #### copy

```public static void copy(Variance source,
Variance dest)
throws NullArgumentException```
Copies source to dest.

Neither source nor dest can be null.

Parameters:
`source` - Variance to copy
`dest` - Variance to copy to
Throws:
`NullArgumentException` - if either source or dest is null