Have you ever gone shopping to buy yourself new clothes and noticed how the prices differed between stores? Have you wondered if lottery numbers are evenly distributed, or if some numbers occur with a higher frequency? Or, have you thought about if a coffee vending machine dispenses the same amount of coffee each time? You can answer these questions by using a specific hypothesis test.
Millions of flashcards designed to help you ace your studies
Sign up for freeWhat is the standard deviation of a \( \chi^{2}_{k} \) distribution?
A chi-square distribution with \(4\) degrees of freedom has a \(95\%\) critical value of \(9.49\).
A chi-square distribution with \(18\) degrees of freedom has a \(10\%\) critical value of \(25.99\).
What is the mode of a \( \chi^{2}_{k} \) distribution?
What is the variance of a \( \chi^{2}_{k} \) distribution?
A chi-square distribution is a _____ distribution that becomes increasingly ____ as its degrees of freedom, \(k\), increases.
What is the mean of a \( \chi^{2}_{k} \) distribution?
What distribution does\[ Q = \chi^{2}_{6} + \chi^{2}_{11} \]follow?
The chi-square distribution is the basis for three chi-square hypothesis tests:
What is the standard deviation of a \( \chi^{2}_{k} \) distribution?
A chi-square distribution with \(4\) degrees of freedom has a \(95\%\) critical value of \(9.49\).
A chi-square distribution with \(18\) degrees of freedom has a \(10\%\) critical value of \(25.99\).
What is the mode of a \( \chi^{2}_{k} \) distribution?
What is the variance of a \( \chi^{2}_{k} \) distribution?
A chi-square distribution is a _____ distribution that becomes increasingly ____ as its degrees of freedom, \(k\), increases.
What is the mean of a \( \chi^{2}_{k} \) distribution?
What distribution does\[ Q = \chi^{2}_{6} + \chi^{2}_{11} \]follow?
The chi-square distribution is the basis for three chi-square hypothesis tests:
Achieve better grades quicker with Premium
Geld-zurück-Garantie, wenn du durch die Prüfung fällst
Review generated flashcards
Start learning or create your own AI flashcards
Team Chi-Square Distribution Teachers
Figure 1. The chi-square distribution is useful for finding a relationship between two things; like clothing prices at different stores.
In this article, you will learn about a new type of distribution to answer questions such as these – the chi-square distribution. You will study the chi-square distribution formula, the properties of a chi-square distribution, chi-square distribution tables, and work through several chi-square distribution examples. You will also be introduced to the major applications of the chi-square distribution, including:
the chi-square test for goodness of fit – which tells you if data fit a certain distribution, like in the lottery number example.
the chi-square test for homogeneity – which tells you if two populations have the same distribution, like in the clothing example.
the chi-square test for independence – which tests for variability, like in the coffee machine example.
each of which has an article of its own.
What happens when you square a normally distributed random variable? You know the probability distribution of the random variable itself, but what does it tell you about the distribution of the squared random variable? That question led to the discovery of the chi-square distribution, and it turns out to be useful in a wide variety of contexts.
A Chi-Square \( (\chi^{2}) \) Distribution is a continuous probability distribution of the sum of squared, independent, standard normal random variables that is widely used in hypothesis tests.
The chi-square distribution is the basis for three chi-square tests:
the chi-square test for goodness of fit – allowing you to compare observed probability distributions to expected distributions,
the chi-square test for independence and the chi-square test for homogeneity;
the chi-square test for independence allows you to test the independence of categorical variables, and
the chi-square test for homogeneity allows you to test if two categorical variables follow the same probability distribution
the test of a single variance – allowing you to estimate the sampling distribution of the variance.
The basic shape of a chi-square distribution is determined by its degrees of freedom, denoted by \(k\).
The degrees of freedom, \(k\), are the number of values that are free to vary.
Let's take a look at an example.
Say you have \(4\) numbers that add up to \(1\):
\[ X_{1} + X_{2} + X_{3} + X_{4} = 1 \]
How many of the \(X\) values are free to vary?
Solution:
The answer is \(3\) because if you know \(3\) of the numbers, then you can solve for the \(4^{th}\) one:
\[ X_{4} = 1 - (X_{1} + X_{2} + X_{3}) \]
So, this example has \(3\) degrees of freedom.
In practice, the degrees of freedom, \(k\), is often one less than the number of observations.
The following graph illustrates examples of chi-square distributions with differing values of \(k\).
Figure 2. A comparison of Chi-Square Distributions with varying degrees of freedom.
Because very few real-world observations follow a chi-square distribution, the main purpose of a chi-square distribution is hypothesis testing.
The reason a chi-square distribution is useful for hypothesis testing is because of how closely it is related to the standard normal distribution: a normal distribution whose mean is \(0\) and variance is \(1\). Let's walk through this relationship.
Say you take a random sample of a standard normal distribution, \(Z\). If you square all the values in your sample, you now have the chi-square distribution with one degree of freedom, or \( k = 1 \). So, mathematically, you represent this as:
\[ \chi_{1}^{2} = Z^{2} \]
Now, say you want to take random samples from \(2\) independent standard normal distributions, \( Z_{1} \) and \( Z_{2} \). If you square each sample and add them together every time you sample a pair of values, you have the chi-square distribution with two degrees of freedom, or \( k = 2 \). You represent this mathematically as:
\[ \chi_{2}^{2} = (Z_{1})^{2} + (Z_{2})^{2} \]
Continuing with this pattern, in general, if you take random samples from \(k\) independent standard normal distributions and then square and sum those values, you get a chi-square distribution with \(k\) degrees of freedom. Again, this is represented mathematically as:
\[ \chi_{k}^{2} = (Z_{1})^{2} + (Z_{2})^{2} + \ldots + (Z_{k})^{2} \]
In summary, a common use of a chi-square distribution is to find the sum of squared, normally distributed, random variables. So, if \( Z_{i} \) represents a normally distributed random variable, then:
\[ \sum_{i=1}^{k} Z_{i}^{2} \sim \chi^{2}_{k} \]
Chi-square tests are hypothesis tests whose test statistics follow a chi-square distribution under the null hypothesis. The first, and most widely used, chi-square test to be discovered was Pearson's chi-square test.
Pearson's chi-square distribution formula (a.k.a. statistic, or test statistic) is
\[ \chi^{2} = \sum \frac{(O-E)^{2}}{E} \]
where,
\( \chi^{2} \) is the chi-square test statistic
\( \sum \) is the summation operator
\( O \) is the observed frequency
\( E \) is the expected frequency
If you take many samples from a population and calculate Pearson's chi-square test statistic for each sample, the test statistic will follow a chi-square distribution; provided the null hypothesis is true.
The mean of a chi-square distribution is the degrees of freedom:\[ \mu \left[ \chi^{2} \right] = k. \]
The variance of a chi-square distribution is twice the degrees of freedom:\[ \sigma^{2} \left[ \chi^{2} \right] = 2k. \]
The mode of a chi-square distribution is the degrees of freedom minus two (when \( k \geq 2 \)):\[ \text{mode} \left[ \chi^{2} \right] = k - 2, \text{ if } k \geq 2 \]
The standard deviation of a chi-square distribution is the square-root of twice the degrees of freedom:
\[ \sigma \left[ \chi^{2} \right] = \sqrt{2k} \]
The chi-square distribution has several properties that make it easy to work with and well-suited for hypothesis testing:
A chi-square distribution is a continuous distribution.
A chi-square distribution is defined by a single parameter: the degrees of freedom, \(k\).
The sum of independent chi-square random variables is also a chi-square random variable, with the degrees of freedom of the sum being the sum of the degrees of freedom:\[ \chi^{2}_{k_{1}} + \chi^{2}_{k_{2}} + \sim \chi^{2}_{k_{1} + k_{2}} \]
A chi-square distribution is never negative. This is easiest to see in the ratio-of-variances formula. Since both the top and bottom of the fraction are positive, the ratio can never be negative. In other words:
\[ \frac{(n-1)s^{2}}{\sigma^{2}} \geq 0 \]
This means the range is:\[ \text{range} \left[ \chi^{2} \right] = 0 \to \infty \]
The constraint that a chi-square distributed random variable can never be negative means that a chi-square distribution cannot be symmetrical. It is a non-symmetric distribution. However, a chi-square distribution becomes increasingly symmetrical as \(k\) increases.
The shape of a chi-square distribution depends on the degrees of freedom, \(k\). As the value of \(k\) increases, the chi-square distribution more closely resembles the bell-curve of a normal distribution. This is because, while a chi-square distribution can never be negative, it goes all the way to infinity in the positive direction. In statistical terms, you say that a chi-square distribution is skewed to the right because the right tail is longer than the left.
The skewness of a chi-square distribution is equal to:
\[ \text{Skewness} \left[ \chi^{2} \right] = \sqrt{\frac{8}{k}} \]
This means the mean of a chi-square distribution is greater than the median and the mode. As \(k\) gets increasingly large, the number under the square root gets closer and closer to zero, so the skewness of the distribution approaches zero as \(k\) approaches infinity.
When a chi-square distribution has only one or two degrees of freedom ( \( k = 1 \) or \( k = 2 \) ) it is shaped like a backwards "J".
Figure 3. Graphs of a Chi-Square Distribution when it has one and two degrees of freedom.
Because of the shape of these chi-square distributions, it means that there is a high probability that \( \chi^{2} \) is close to zero.
When a chi-square distribution has three or more degrees of freedom ( \( k \geq 3 \) ), it takes on a bump-shape that has a peak in the middle that more closely resembles a normal distribution. This means there is a low probability that \( \chi^{2} \) is either very close to or very far from zero.
When \(k\) is only slightly larger than \(2\), the chi-square distribution has a much longer right tail than left tail; that is, it is strongly right-skewed.
Figure 4. Graphs of a Chi-Square Distribution when it has three and five degrees of freedom.
Remember that the mean of a chi-square distribution is the degrees of freedom, then notice that the peak is always to the left of the mean. Also notice that the left tail ends at zero, but the right tail goes on forever. The peak of the distribution can never truly be in the middle; it always must be left of center (because half of infinity is also infinite).
As the degrees of freedom, \(k\), gets larger and larger, the skew of the distribution gets smaller and smaller. As the degrees of freedom approaches infinity, the distribution approaches that of a normal distribution.
Figure 5. A Chi-Square Distribution that has \(90\) degrees of freedom. At this point, you can use a normal distribution as a good approximation of the chi-square distribution.
In fact, when \( k \geq 90 \), you can consider a normal distribution as a good approximation of the chi-square distribution.
Nowadays, many calculators and any statistical software can calculate a chi-square distribution. But before that software was ubiquitous, people needed an easy way to approximate that value. That’s why they created the chi-square table. The chi-square distribution table is a reference tool you can use to find chi-square critical values.
A chi-square critical value is a threshold for statistical significance for hypothesis tests. It also defines confidence intervals for certain parameters.
Below is an example of a chi-square distribution table for \(1-5\) degrees of freedom.
| Percentage Points of the Chi-Square Distribution | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Degrees of Freedom (k) | Probability of a Larger Value of X2; Significance Level (α) | ||||||||
| 0.99 | 0.95 | 0.90 | 0.75 | 0.50 | 0.25 | 0.10 | 0.05 | 0.01 | |
| 1 | 0.000 | 0.004 | 0.016 | 0.102 | 0.455 | 1.32 | 2.71 | 3.84 | 6.63 |
| 2 | 0.020 | 0.103 | 0.211 | 0.575 | 1.386 | 2.77 | 4.61 | 5.99 | 9.21 |
| 3 | 0.115 | 0.352 | 0.584 | 1.212 | 2.366 | 4.11 | 6.25 | 7.81 | 11.34 |
| 4 | 0.297 | 0.711 | 1.064 | 1.923 | 3.357 | 5.39 | 7.78 | 9.49 | 13.28 |
| 5 | 0.554 | 1.145 | 1.610 | 2.675 | 4.351 | 6.63 | 9.24 | 11.07 | 15.09 |
Table 1: Table data, example chi-square distribution.
The leftmost column tells you the degrees of freedom, \(k\). The top row tells you the complement of the probability of a larger value of \( \chi^{2} \) you’re interested in; that is, if you want a probability of \(0.9\), you look in the column labelled \(0.1\). The number in each cell is the critical value at which the chi-square distribution has that much probability remaining to the right.
Let's walk through an example.
Say you have a chi-square distribution with \(6\) degrees of freedom, and you want to know the value at which your chi-square distribution reaches \(5\%\) probability. How can you use a chi-square distribution table to do this?
Solution:
These tables are most important for hypothesis testing. If your test statistic is greater than the number in the appropriate cell, that means you have found evidence to reject the null hypothesis. See the article on Chi-Square Tests for more information.
What are some common applications of the chi-square distribution? Well, the chi-square distribution appears in many statistical tests and theories. Below are some of the most common.
A major motivation for the chi-square distribution is drawing inference about the population standard deviation \( (\sigma) \) or variance \( (\sigma^{2}) \) from a relatively small sample. Using a chi-square distribution, you can test the hypothesis that a population's variance is equal to a specific value by using the test of a single variance. Alternatively, you could calculate confidence intervals for the population's variance.
A large union strives to make sure that all employees who are at the same level of seniority get paid similar salaries. Their goal: a standard deviation in hourly salary that is less than \($3\).
Is this enough evidence to conclude that the true standard deviation of all employees who are at the same level of seniority is less than \($3\)?
Solution:
To find out, the union should use the test of a single variance to determine if the standard deviation is significantly different from \($3\), using:
a null hypothesis of \( H_{0}: \sigma^{2} \geq 3^{2} \) and
an alternative hypothesis of \( H_{a}: \sigma^{2} < 3^{2} \).
Then, by comparing a chi-square test statistic to the appropriate chi-square distribution, the union can decide if it is appropriate to reject the null hypothesis.
Pearson's chi-square tests are some of the most common applications of chi-square distributions. You use these tests to determine if your data are significantly different from what you expect. The two types of Pearson's chi-square tests are:
the chi-square goodness of fit test and
the chi-square test for independence.
Say a t-shirt company wants to know if all colors of their t-shirts are equally popular. To find out, they record the number of sales per shirt color for a week. This data is represented in the table below:
| Sales per Shirt Color | |
|---|---|
| Color | Frequency |
| Black | 80 |
| Blue | 90 |
| Gray | 70 |
| Red | 60 |
| White | 100 |
Table 2. Sales per shirt color data.
Since the company sold \(400\) t-shirts, \(80\) sales per color would mean that the colors were equally popular. Based on the numbers in the table, you know that the company did not sell \(80\) of each color of t-shirt. However, this is only a one-week sample, so you should expect that the numbers won't be equal due to chance.
But, does this sample give enough confidence to conclude that the frequency of t-shirt sales truly differs between colors?
Solution:
This is where a chi-square goodness of fit test comes in. It could test whether the observed frequencies are significantly different from equal frequencies.
If you tell the company to compare the Pearson chi-square test statistic to the appropriate chi-square distribution, then the company can determine the probability of these t-shirt sale values happening because of chance.
Chi-square distributions are also integral in defining the \(F\) distribution, a distribution used in Analysis of Variance (ANOVAs).
Now, let's work through some examples!
Square and add up \(15\) standard normal random variables. What distribution does this sum follow?
Solution:
A squared standard normal random variable follows a chi-square distribution with \(1\) degree of freedom. The sum of chi-square random variables also follow a chi-square distribution, with the degrees of freedom of the sum being the sum of the individual degrees of freedom. Let’s follow this process.
Building on the previous example:
What are the
of the distribution from the previous example?
Solution:
Here is an example using a chi-square table.
Using a chi-square table, find the \(90\%\), \(95\%\), and \(99\%\) critical values for a chi-square distribution with \(8\) degrees of freedom.
Solution:
All you have to do for this question is read the table.
Table 3. Chi-square distribution example.
| Percentage Points of the Chi-Square Distribution | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Degrees of Freedom (k) | Probability of a Larger Value of X2; Significance Level (α) | ||||||||
| 0.99 | 0.95 | 0.90 | 0.75 | 0.50 | 0.25 | 0.10 | 0.05 | 0.01 | |
| 6 | 0.872 | 1.635 | 2.204 | 3.455 | 5.348 | 7.84 | 10.64 | 12.59 | 16.81 |
| 7 | 1.239 | 2.167 | 2.833 | 4.255 | 6.346 | 9.04 | 12.02 | 14.07 | 18.48 |
| 8 | 1.647 | 2.733 | 3.490 | 5.071 | 7.344 | 10.22 | 13.36 | 15.51 | 20.09 |
| 9 | 2.088 | 3.325 | 4.168 | 5.899 | 8.343 | 11.39 | 14.68 | 16.92 | 21.67 |
the chi-square test for goodness of fit,
the chi-square test for independence and the chi-square test for homogeneity, and
the test of a single variance.
\[ \chi^{2} = \sum \frac{(O-E)^{2}}{E} \]
\[ \sum_{i=1}^{k} z_{i}^{2} \sim \chi^{2}_{k} \]
Sign up for free to gain access to all our flashcards.
What is chi squared distribution?
A chi squared distribution is a probability distribution that represents the sum of squared, independent standard normal random variables.
What is the chi-square distribution used for?
The chi-square distribution can be used to model the sum of squared, independent standard normal random variables, including modeling variance. The chi-square distribution can also represent the ratio of observed variance to theoretical variance. For these two reasons, the chi-square distribution is used in many statistical hypothesis tests.
How is chi-square distribution calculated?
The chi-square distribution is a special form of gamma distribution, with parameters a=(k/2) and b=(1/2), where k is the degrees of freedom. Because the formula requires difficult calculus to evaluate, it is most often calculated using statistical software or on a calculator.
How to draw chi square distribution graph?
The shape of a chi-square distribution is determined by the degrees of freedom. With only one of two degrees of freedom, the probability density function (PDF) starts high and quickly decays toward zero, like an exponential distribution. With higher degrees of freedom, the chi-square PDF begins to look bell-shaped. The distribution can never be negative, so start with a point at 0. The peak of the distribution is to the left of the mean. A chi-square distribution has an infinite right tail.
What is an example of a chi-square distribution?
A common example of a chi-square distribution is the variance of a normal distribution. This is especially useful for drawing a confidence interval for the sampling variance.
At StudySmarter, we have created a learning platform that serves millions of students. Meet the people who work hard to deliver fact based content as well as making sure it is verified.
Lily Hulatt is a Digital Content Specialist with over three years of experience in content strategy and curriculum design. She gained her PhD in English Literature from Durham University in 2022, taught in Durham University’s English Studies Department, and has contributed to a number of publications. Lily specialises in English Literature, English Language, History, and Philosophy.
Get to know Lily
Gabriel Freitas is an AI Engineer with a solid experience in software development, machine learning algorithms, and generative AI, including large language models’ (LLMs) applications. Graduated in Electrical Engineering at the University of São Paulo, he is currently pursuing an MSc in Computer Engineering at the University of Campinas, specializing in machine learning topics. Gabriel has a strong background in software engineering and has worked on projects involving computer vision, embedded AI, and LLM applications.
Get to know Gabriel