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# vartest

Chi-square variance test

## Description

example

h = vartest(x,v) returns a test decision for the null hypothesis that the data in vector x comes from a normal distribution with variance v, using the chi-square variance test. The alternative hypothesis is that x comes from a normal distribution with a different variance. The result h is 1 if the test rejects the null hypothesis at the 5% significance level, and 0 otherwise.

example

h = vartest(x,v,Name,Value) performs the chi-square variance test with additional options specified by one or more name-value pair arguments. For example, you can change the significance level or conduct a one-sided test.

example

[h,p] = vartest(___) also returns the p-value of the test, p, using any of the input arguments in the previous syntaxes.

example

[h,p,ci,stats] = vartest(___) also returns the confidence interval for the true variance, ci, and the structure stats containing information about the test statistic.

## Examples

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### Test for a Specified Variance

Load the sample data. Create a vector containing the first column of the students' exam grades matrix.

```load examgrades;
```

Test the null hypothesis that the data comes from a distribution with a variance of 25.

`[h,p,ci,stats] = vartest(x,25)`
```h =
1

p =
0

ci =
59.8936
99.7688

stats =
chisqstat: 361.9597
df: 119```

The returned value h = 1 indicates that vartest rejects the null hypothesis at the default 5% significance level. ci shows the lower and upper boundaries of the 95% confidence interval for the true variance, and suggests that the true variance is greater than 25.

### One-Sided Hypothesis Test

Load the sample data. Create a vector containing the first column of the students' exam grades matrix.

```load examgrades;
```

Test the null hypothesis that the data comes from a distribution with a variance of 25, against the alternative hypothesis that the variance is greater than 25.

`[h,p] = vartest(x,25,'Tail','right')`
```h =
1

p =
2.4269e-26```

The returned value of h = 1 indicates that vartest rejects the null hypothesis at the default 5% significance level, in favor of the alternative hypothesis that the variance is greater than 25.

## Input Arguments

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### x — Sample datavector | matrix | multidimensional array

Sample data, specified as a vector, matrix, or multidimensional array. For matrices, vartest performs separate tests along each column of x, and returns a row vector of results. For multidimensional arrays, vartest works along the first nonsingleton dimension of x.

Data Types: single | double

### v — Hypothesized variancenonnegative scalar value

Hypothesized variance, specified as a nonnegative scalar value.

Data Types: single | double

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Tail','right','Alpha',0.01 specifies a right-tailed hypothesis test at the 1% significance level.

### 'Alpha' — Significance level0.05 (default) | scalar value in the range (0,1)

Significance level of the hypothesis test, specified as the comma-separated pair consisting of 'Alpha' and a scalar value in the range (0,1).

Example: 'Alpha',0.01

Data Types: single | double

### 'Dim' — Dimensionfirst nonsingleton dimension (default) | positive integer value

Dimension of the input matrix to test along, specified as the comma-separated pair consisting of 'Dim' and a positive integer value. For example, specifying 'Dim',1 tests the data in each column for equality to the hypothesized variance, while 'Dim',2 tests the data in each row.

Example: 'Dim',2

Data Types: single | double

### 'Tail' — Type of alternative hypothesis'both' (default) | 'right' | 'left'

Type of alternative hypothesis to evaluate, specified as the comma-separated pair consisting of 'Tail' and one of the following.

 'both' Test the alternative hypothesis that the population variance is not v. 'right' Test the alternative hypothesis that the population variance is greater than v. 'left' Test the alternative hypothesis that the population variance is less than v.

Example: 'Tail','right'

## Output Arguments

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### h — Hypothesis test result1 | 0

Hypothesis test result, returned as a logical value.

• If h = 1, this indicates the rejection of the null hypothesis at the Alpha significance level.

• If h = 0, this indicates a failure to reject the null hypothesis at the Alpha significance level.

### p — p-valuescalar value in the range [0,1]

p-value of the test, returned as a scalar value in the range [0,1]. p is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. Small values of p cast doubt on the validity of the null hypothesis.

### ci — Confidence intervalvector

Confidence interval for the true variance, returned as a two-element vector containing the lower and upper boundaries of the 100 × (1 – Alpha)% confidence interval.

### stats — Test statisticsstructure

Test statistics for the chi-square variance test, returned as a structure containing:

• chisqstat — Value of the test statistic.

• df — Degrees of freedom of the test.

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### Chi-Square Variance Test

The chi-square variance test is used to test if the variance of a population is equal to a hypothesized value.

The test statistic is

$T=\left(n-1\right){\left(\frac{s}{{\sigma }_{0}}\right)}^{2},$

where n is the sample size, s is the sample standard deviation, and σ0 is the hypothesized standard deviation. The denominator is the ratio of the sample standard deviation to the hypothesized standard deviation. The further this ratio deviates from 1, the more likely you are to reject the null hypothesis. The test statistic T has a chi-square distribution with n – 1 degrees of freedom under the null hypothesis.

### Multidimensional Array

A multidimensional array has more than two dimensions. For example, if x is a 1-by-3-by-4 array, then x is a three-dimensional array.

### First Nonsingleton Dimension

The first nonsingleton dimension is the first dimension of an array whose size is not equal to 1. For example, if x is a 1-by-2-by-3-by-4 array, then the second dimension is the first nonsingleton dimension of x.