Chi-Square Distribution (X²) – Tutorial With Examples

03.04.23 Types of distributions Time to read: 5min

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The chi-square distribution is a standard application in hypothesis testing. It is a constant probability dispersal whose shape is based on the parameter k.

This blog post discusses more about the chi-square distribution.

Chi-Square Distribution – In a Nutshell

  • The chi-square distribution is a constant probability dispersal commonly applied in hypothesis testing.
  • The parameter k determines the outline of a chi-square distribution. The parameter k represents the freedom degree.
  • Apart from the chi-square distribution, other commonly applied distributions are normal distribution and Poisson distributions.

Definition: Chi-square distribution

The chi-square distribution is a constant hypothetical dispersal of values for a population. It is commonly applied in statistical hypothesis tests.1

The parameter k, which denotes the degrees of freedom, governs the outline of a chi-square distribution. The chi-square distribution applies to theoretical distributions. In contrast, normal and Poisson distributions apply in real-world distributions.2

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Chi-square distribution vs. standard normal distribution

The chi-square distribution is closely related to the standard regular dispersal; hence its application in hypothesis testing.

The standard normal distribution is a normal distribution with a mean and variance of zero and one, respectively.


Assessing chi-square distribution from standard normal distribution when k = 1

  • You pick a casual model of an ordinary normal distribution, denoted as (Z).
  • Squaring all the sample values would give you the chi-distribution with parameters of 1 (k = 1).


X21= (Z)2


Assessing chi-square distribution from standard normal distribution when k = 2

  • You pick models from two typical normal distributions labeled (Z1 and Z2).
  • Every time you sample a set of values, you square and add them together to get the chi-square distribution with parameters equal to two (k = 2).


X22= (Z1)2+ (Z2)2

Evaluating the k degrees of freedom

Generally, if you are ample from k autonomous average nominal distributions and square and quantify their values, you will yield a chi-square distribution in the existence of k degrees of freedom.


X2k= (Z2)2+XXX+(Zk)2

Chi-square distribution formula

Chi-square assessments are types of hypothesis tests whose test statistics tail a chi-square distribution in the null proposition. The most common chi-square test is Pearson’s chi-square assessment, which was also the first to be discovered.1

The table explains Pearson’s chi-square test figure:

Formula Description
X2= (Ʃ (0-E)2)/E X2 = chi-square test statistic
Ʃ = “take the summation of”
O = observed frequency
E = anticipated frequency

The test measurement of a sampled population usually follows a chi-square distribution if the null hypothesis is correct. This applies where you sample a population severally and find the Pearson’s test statistic for every trial.

Chi-square distribution shape

Graphs of the chi-square probability compactness function illustrate how the chi-square distribution alters as k increases. A concentration function defines a continuous prospect dispersal.

K = 1 or K = 2

The outline of a chi-square distribution alters depending on the value of k. When k is ½, the shape curves back into a “J.”

This implies a high probability of the X2 being close to zero.

K greater than 2

When the parameter is better than two, the chi-square distribution appears hump-shaped. This implies that it starts out low followed by an increase, then a decrease. This denotes a low probability of X2 being close to zero.1

In contrast, when the parameter (k) is slightly more than two, the chi distribution will be longer on the right peak side than on the left.2

The chi distribution tends to resemble the normal distribution as k increases. In some instances (when k is 90 or more) the normal distribution applies as an estimate of a chi-square distribution.

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Properties of a chi-square distribution

Chi-square distribution usually has some standard properties. Here are its properties:

Property Symbol
Type (Discrete/ continuous) Continuous
Variance 2k
Mean K
Standard deviation Square root of 2k
Mode K – 2 (if k less than 2)
Range Zero to infinity
Symmetry Rightly-skewed, increasing as k increases

Chi-square distribution example

The chi-square distribution is applied in many statistical and theoretical tests. Here are its most frequent applications:

Pearson’s chi-square test

This is a statistical test for definite statistics used to determine the significance of the difference between the data and your expectations.

There are two categories of Pearson’s chi-square tests:2

  • Chi-square test of good fit
  • Chi-square independence test


Design a study to test the most popular t-shirt color sold by a particular company. Gather sales data for different colors over a specific period to find the F value. The chi-square test of good fit will help you determine if the F values differ significantly from equal frequencies. Compare Pearson’s chi-square test to the chi-square distribution to calculate the probability that the frequencies occur coincidentally.

Population variance inferences

The chi-square distribution comes in handy when making extrapolations about standard deviation or variance.

It helps with hypothesis testing to determine if the population variance equates to a specific value or to calculate its confidence intervals.


In a study to determine if all staff members with the equal seniority level get similar wages, compare the chi-square test statistic to the chi-square distribution to determine whether to reject or retain the null hypothesis.

F distribution

The chi-square distribution helps with defining the incidence distribution, especially in ANOVAs.


Pick random samples from the distribution and divide them by k. Repeat it with a separate distribution and find the ratio of the values. The ratio is the F distribution.

Non-central chi-square distribution

This is an overall kind of chi-square distribution used in some forms of power analyses. It features an extra lambda and non-central parameter, which changes its shape. Its peak shifts to the right and increases as the variance grows. The lambda parameter defines the mean figure of the normal dispersal.3


The curve shifts from downward to hump-shaped. The more the k increases, the more right-skewed it gets.

The chi-square distribution is a constant probability dispersal commonly applied in hypothesis testing.

There are two types. They are:

  • the chi-square test of good fit
  • the chi-square freedom test

The chi-square distribution is used to define the quantity of a squared random variable.3


1 ScienceDirect. “Chi-Square Distribution.” Accessed March 22, 2023.

2 Statistics Knowledge Portal. “The Chi-Square Distribution.” JMP. Accessed March 22, 2023.

3 Primas, Mia “Chi-Square Distribution: Definition & Examples.” December 07, 2018.