# chi2gof

Chi-square goodness-of-fit test

## Description

returns
a test decision for the null hypothesis that the data in vector `h`

= chi2gof(`x`

)`x`

comes
from a normal distribution with a mean and variance estimated from `x`

,
using the chi-square goodness-of-fit
test. The alternative hypothesis is that the data does not
come from such a distribution. The result `h`

is `1`

if
the test rejects the null hypothesis at the 5% significance level,
and `0`

otherwise.

returns
a test decision for the chi-square goodness-of-fit test with additional
options specified by one or more name-value pair arguments. For example,
you can test for a distribution other than normal, or change the significance
level of the test.`h`

= chi2gof(`x`

,`Name,Value`

)

## Examples

### Test for Normal Distribution

Create a standard normal probability distribution object. Generate a data vector `x`

using random numbers from the distribution.

pd = makedist('Normal'); rng default; % for reproducibility x = random(pd,100,1);

Test the null hypothesis that the data in `x`

comes from a population with a normal distribution.

h = chi2gof(x)

h = 0

The returned value `h = 0`

indicates that `chi2gof`

does not reject the null hypothesis at the default 5% significance level.

### Test Hypothesis at Different Significance Level

Create a standard normal probability distribution object. Generate a data vector `x`

using random numbers from the distribution.

pd = makedist('Normal'); rng default; % for reproducibility x = random(pd,100,1);

Test the null hypothesis that the data in `x`

comes from a population with a normal distribution at the 1% significance level.

`[h,p] = chi2gof(x,'Alpha',0.01)`

h = 0

p = 0.3775

The returned value `h = 0`

indicates that `chi2gof`

does not reject the null hypothesis at the 1% significance level.

### Test for Weibull Distribution Using Probability Distribution Object

Load the light bulb lifetime sample data.

`load lightbulb`

Create a vector from the first column of the data matrix, which contains the lifetime in hours of the light bulbs.

x = lightbulb(:,1);

Test the null hypothesis that the data in `x`

comes from a population with a Weibull distribution. Use `fitdist`

to create a probability distribution object with `A`

and `B`

parameters estimated from the data.

pd = fitdist(x,'Weibull'); h = chi2gof(x,'CDF',pd)

h = 1

The returned value `h = 1`

indicates that `chi2gof`

rejects the null hypothesis at the default 5% significance level.

### Test for Poisson Distribution

Create six bins, numbered 0 through 5, to use for data pooling.

bins = 0:5;

Create a vector containing the observed counts for each bin and compute the total number of observations.

obsCounts = [6 16 10 12 4 2]; n = sum(obsCounts);

Fit a Poisson probability distribution object to the data and compute the expected count for each bin. Use the transpose operator `.'`

to transform `bins`

and `obsCounts`

from row vectors to column vectors.

pd = fitdist(bins','Poisson','Frequency',obsCounts'); expCounts = n * pdf(pd,bins);

Test the null hypothesis that the data in `obsCounts`

comes from a Poisson distribution with a lambda parameter equal to `lambdaHat`

.

[h,p,st] = chi2gof(bins,'Ctrs',bins,... 'Frequency',obsCounts, ... 'Expected',expCounts,... 'NParams',1)

h = 0

p = 0.4654

`st = `*struct with fields:*
chi2stat: 2.5550
df: 3
edges: [-0.5000 0.5000 1.5000 2.5000 3.5000 5.5000]
O: [6 16 10 12 6]
E: [7.0429 13.8041 13.5280 8.8383 6.0284]

The returned value `h = 0`

indicates that `chi2gof`

does not reject the null hypothesis at the default 5% significance level. The vector `E`

contains the expected counts for each bin under the null hypothesis, and `O`

contains the observed counts for each bin.

### Test for Normal Distribution Using Function Handle

Use the probability distribution function `normcdf`

as a function handle in the chi-square goodness-of-fit test (`chi2gof`

).

Test the null hypothesis that the sample data in the input vector `x`

comes from a normal distribution with parameters *µ* and *σ* equal to the mean (`mean`

) and standard deviation (`std`

) of the sample data, respectively.

rng('default') % For reproducibility x = normrnd(50,5,100,1); h = chi2gof(x,'cdf',{@normcdf,mean(x),std(x)})

h = 0

The returned result `h = 0`

indicates that `chi2gof`

does not reject the null hypothesis at the default 5% significance level.

## Input Arguments

`x`

— Sample data

vector

Sample data for the hypothesis test, specified as a vector.

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **`'NBins',8,'Alpha',0.01`

pools the
data into eight bins and conducts the hypothesis test at the 1% significance
level.

`NBins`

— Number of bins

`10`

(default) | positive integer value

Number of bins to use for the data pooling, specified as the
comma-separated pair consisting of `'NBins'`

and
a positive integer value. If you specify a value for `NBins`

,
do not specify a value for `Ctrs`

or `Edges`

.

**Example: **`'NBins',8`

**Data Types: **`single`

| `double`

`Ctrs`

— Bin centers

vector

Bin centers, specified as the comma-separated pair consisting
of `'Ctrs'`

and a vector of center values for each
bin. If you specify a value for `Ctrs`

, do not specify
a value for `NBins`

or `Edges`

.

**Example: **`'Ctrs',[1 2 3 4 5]`

**Data Types: **`single`

| `double`

`Edges`

— Bin edges

vector

Bin edges, specified as the comma-separated pair consisting
of `'Edges'`

and a vector of edge values for each
bin. If you specify a value for `Edges`

, do not specify
a value for `NBins`

or `Ctrs`

.

**Example: **`'Edges',[-2.5 -1.5 -0.5 0.5 1.5 2.5]`

**Data Types: **`single`

| `double`

`CDF`

— cdf of hypothesized distribution

probability distribution object | function handle | cell array

The cdf of the hypothesized distribution, specified as the comma-separated
pair consisting of `'CDF'`

and a probability distribution
object, function handle, or cell array.

If

`CDF`

is a probability distribution object, the degrees of freedom account for whether you estimate the parameters using`fitdist`

or specify them using`makedist`

.If

`CDF`

is a function handle, the distribution function must take`x`

as its only argument.If

`CDF`

is a cell array, the first element must be a function handle, and the remaining elements must be parameter values, one per cell. The function must take`x`

as its first argument, and the other parameters in the array as later arguments.

If you specify a value for `CDF`

, do not specify
a value for `Expected`

.

**Example: **`'CDF',pd_object`

**Data Types: **`single`

| `double`

`Expected`

— Expected counts

vector of nonnegative values

Expected counts for each bin, specified as the comma-separated
pair of `'Expected'`

and a vector of nonnegative
values. If `Expected`

depends on estimated parameters,
use `NParams`

to ensure that `chi2gof`

correctly
calculates the degrees of freedom. If you specify a value for `Expected`

,
do not specify a value for `CDF`

.

**Example: **```
'Expected',[19.1446 18.3789 12.3224 8.2432
4.1378]
```

**Data Types: **`single`

| `double`

`NParams`

— Number of estimated parameters

positive integer value

Number of estimated parameters used to describe the null distribution,
specified as the comma-separated pair consisting of `'NParams'`

and
a positive integer value. This value adjusts the degrees of freedom
of the test based on the number of estimated parameters used to compute
the cdf or expected counts.

The default value for `NParams`

depends on
how you specify the null distribution:

If you specify

`CDF`

as a probability distribution object,`NParams`

is equal to the number of estimated parameters used to create the object.If you specify

`CDF`

as a function name or handle, the default value of`NParams`

is`0`

.If you specify

`CDF`

as a cell array, the default value of`NParams`

is the number of parameters in the array.If you specify

`Expected`

, the default value of`NParams`

is`0`

.

**Example: **`'NParams',1`

**Data Types: **`single`

| `double`

`EMin`

— Minimum expected count per bin

`5`

(default) | nonnegative integer value

Minimum expected count per bin, specified as the comma-separated
pair consisting of `'EMin'`

and a nonnegative integer
value. If the bin at the extreme end of either tail has an expected
value less than `EMin`

, it is combined with a neighboring
bin until the count in each extreme bin is at least 5. If any interior
bins have a count less than 5, `chi2gof`

displays
a warning, but does not combine the interior bins. In that case, you
should use fewer bins, or provide bin centers or edges, to increase
the expected counts in all bins. Specify `EMin`

as `0`

to
prevent the combining of bins.

**Example: **`'EMin',0`

**Data Types: **`single`

| `double`

`Frequency`

— Frequency

vector of nonnegative integer values

Frequency of data values, specified as the comma-separated pair
consisting of `'Frequency'`

and a vector of nonnegative
integer values that is the same length as the vector `x`

.

**Example: **`'Frequency',[20 16 13 10 8]`

**Data Types: **`single`

| `double`

`Alpha`

— Significance level

`0.05`

(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`

## Output Arguments

`h`

— Hypothesis test result

`1`

| `0`

Hypothesis test result, returned as `1`

or `0`

.

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*-value

scalar 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.

`stats`

— Test statistics

structure

Test statistics, returned as a structure containing the following:

`chi2stat`

— Value of the test statistic.`df`

— Degrees of freedom of the test.`edges`

— Vector of bin edges after pooling.`O`

— Vector of observed counts for each bin.`E`

— Vector of expected counts for each bin.

## More About

### Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test determines if a data sample comes from a specified probability distribution, with parameters estimated from the data.

The test groups the data into bins, calculating the observed and expected counts for those bins, and computing the chi-square test statistic

$${\chi}^{2}={\displaystyle \sum _{i=1}^{N}{\left({O}_{i}-{E}_{i}\right)}^{2}}/{E}_{i}\text{\hspace{0.17em}},$$

where *O*_{i} are
the observed counts and *E*_{i} are
the expected counts based on the hypothesized distribution. The test
statistic has an approximate chi-square distribution when the counts
are sufficiently large.

## Algorithms

`chi2gof`

compares the value of the test statistic
to a chi-square distribution with degrees of freedom equal to *nbins* -
1 - *nparams*, where *nbins* is
the number of bins used for the data pooling and *nparams* is
the number of estimated parameters used to determine the expected
counts. If there are not enough degrees of freedom to conduct the
test, `chi2gof`

returns the *p*-value
as `NaN`

.

## Extended Capabilities

### GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

## Version History

**Introduced before R2006a**

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