# Chi-Square (Χ²) Distributions | Definition & Examples

A chi-square (Χ2) distribution is a continuous probability distribution that is used in many hypothesis tests.

The shape of a chi-square distribution is determined by the parameter k. The graph below shows examples of chi-square distributions with different values of k. ## What is a chi-square distribution?

Chi-square (Χ2) distributions are a family of continuous probability distributions. They’re widely used in hypothesis tests, including the chi-square goodness of fit test and the chi-square test of independence.

The shape of a chi-square distribution is determined by the parameter k, which represents the degrees of freedom.

Very few real-world observations follow a chi-square distribution. The main purpose of chi-square distributions is hypothesis testing, not describing real-world distributions.

In contrast, most other widely used distributions, like normal distributions or Poisson distributions, can describe useful things such as newborns’ birth weights or disease cases per year, respectively.

### Relationship to the standard normal distribution

Chi-square distributions are useful for hypothesis testing because of their close relationship to the standard normal distribution. The standard normal distribution, which is a normal distribution with a mean of zero and a variance of one, is central to many important statistical tests and theories.

Imagine taking a random sample of a standard normal distribution (Z). If you squared all the values in the sample, you would have the chi-square distribution with k = 1.

Χ21 = (Z)2

Now imagine taking samples from two standard normal distributions (Z1 and Z2). If each time you sampled a pair of values, you squared them and added them together, you would have the chi-square distribution with k = 2.

Χ22 = (Z1)2 + (Z2)2

More generally, if you sample from k independent standard normal distributions and then square and sum the values, you’ll produce a chi-square distribution with k degrees of freedom

Χ2k = (Z1)2 + (Z2)2 + … + (Zk)2

## Chi-square test statistics (formula)

Chi-square tests are hypothesis tests with test statistics that follow a chi-square distribution under the null hypothesis. Pearson’s chi-square test was the first chi-square test to be discovered and is the most widely used.

Pearson’s chi-square test statistic is:

Formula Explanation Where

• X² is the chi-square test statistic
• is the summation operator (it means “take the sum of”)
• is the observed frequency
• is the expected frequency

If you sample a population many times and calculate Pearson’s chi-square test statistic for each sample, the test statistic will follow a chi-square distribution if the null hypothesis is true.

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## The shape of chi-square distributions

We can see how the shape of a chi-square distribution changes as the degrees of freedom (k) increase by looking at graphs of the chi-square probability density function. A probability density function is a function that describes a continuous probability distribution.

### When k is one or two

When k is one or two, the chi-square distribution is a curve shaped like a backwards “J.” The curve starts out high and then drops off, meaning that there is a high probability that Χ² is close to zero. ### When k is greater than two

When k is greater than two, the chi-square distribution is hump-shaped. The curve starts out low, increases, and then decreases again. There is low probability that Χ² is very close to or very far from zero. The most probable value of Χ² is Χ² − 2.

When k is only a bit greater than two, the distribution is much longer on the right side of its peak than its left (i.e., it is strongly right-skewed). As k increases, the distribution looks more and more similar to a normal distribution. In fact, when k is 90 or greater, a normal distribution is a good approximation of the chi-square distribution. ## Properties of chi-square distributions

Chi-square distributions start at zero and continue to infinity. The chi-square distribution starts at zero because it describes the sum of squared random variables, and a squared number can’t be negative.

The mean (μ) of the chi-square distribution is its degrees of freedom, k. Because the chi-square distribution is right-skewed, the mean is greater than the median and mode. The variance of the chi-square distribution is 2k.

Properties of chi-square distributions
Property Value
Continuous or discrete Continuous
Mean k
Mode k − 2 (when k > 2)
Variance 2
Standard deviation Range 0 to ∞
Symmetry Asymmetrical (right-skewed), but increasingly symmetrical as k increases.

## Example applications of chi-square distributions

The chi-square distribution makes an appearance in many statistical tests and theories. The following are a few of the most common applications of the chi-square distribution.

### Pearson’s chi-square test

One of the most common applications of chi-square distributions is Pearson’s chi-square tests. Pearson’s chi-square tests are statistical tests for categorical data. They’re used to determine whether your data are significantly different from what you expected. There are two types of Pearson’s chi-square tests:

### Population variance inferences

The chi-square distribution can also be used to make inferences about a population’s variance (σ²) or standard deviation (σ). Using the chi-square distribution, you can test the hypothesis that a population variance is equal to a certain value using the test of a single variance or calculate confidence intervals for a population’s variance.

### F distribution definition

Chi-square distributions are important in defining the F distribution, which is used in ANOVAs.

Imagine you take random samples from a chi-square distribution, and then divide the sample by the k of the distribution. Next, you repeat the process with a different chi-square distribution. If you take the ratios of the values from the two distributions, you will have an F distribution.

## The non-central chi-square distribution

The non-central chi-square distribution is a more general version of the chi-square distribution. It’s used in some types of power analyses.

The non-central chi-square distribution has an extra parameter called λ (lambda) or the non-central parameter. This parameter changes the shape of the distribution, shifting the peak to the right and increasing the variance as λ increases. The λ parameter works by defining the mean of the normal distributions that underlie the chi-square distribution. For example, you can produce a non-central chi-square distribution with λ = 2 and k = 3 by squaring and summing values sampled from three normal distributions, each with a mean of two and a variance of one.

What happens to the shape of the chi-square distribution as the degrees of freedom (k) increase?

As the degrees of freedom (k) increases, the chi-square distribution goes from a downward curve to a hump shape. As the degrees of freedom increases further, the hump goes from being strongly right-skewed to being approximately normal.

What properties does the chi-square distribution have?

A chi-square distribution is a continuous probability distribution. The shape of a chi-square distribution depends on its degrees of freedom, k. The mean of a chi-square distribution is equal to its degrees of freedom (k) and the variance is 2k. The range is 0 to ∞. 