ChiSquare Goodness of Fit Test  Formula, Guide & Examples
A chisquare (Χ^{2}) goodness of fit test is a type of Pearson’s chisquare test. You can use it to test whether the observed distribution of a categorical variable differs from your expectations.
The chisquare goodness of fit test tells you how well a statistical model fits a set of observations. It’s often used to analyze genetic crosses.
What is the chisquare goodness of fit test?
A chisquare (Χ^{2}) goodness of fit test is a goodness of fit test for a categorical variable. Goodness of fit is a measure of how well a statistical model fits a set of observations.
 When goodness of fit is high, the values expected based on the model are close to the observed values.
 When goodness of fit is low, the values expected based on the model are far from the observed values.
The statistical models that are analyzed by chisquare goodness of fit tests are distributions. They can be any distribution, from as simple as equal probability for all groups, to as complex as a probability distribution with many parameters.
Hypothesis testing
The chisquare goodness of fit test is a hypothesis test. It allows you to draw conclusions about the distribution of a population based on a sample. Using the chisquare goodness of fit test, you can test whether the goodness of fit is “good enough” to conclude that the population follows the distribution.
With the chisquare goodness of fit test, you can ask questions such as: Was this sample drawn from a population that has…
 Equal proportions of male and female turtles?
 Equal proportions of red, blue, yellow, green, and purple jelly beans?
 90% righthanded and 10% lefthanded people?
 Offspring with an equal probability of inheriting all possible genotypic combinations (i.e., unlinked genes)?
 A Poisson distribution of floods per year?
 A normal distribution of bread prices?
After weeks of hard work, your dog food experiment is complete and you compile your data in a table:
Flavor  Observed  Expected 
Garlic Blast  22  25 
Blueberry Delight  30  25 
Minty Munch  23  25 
To help visualize the differences between your observed and expected frequencies, you also create a bar graph:
The president of the dog food company looks at your graph and declares that they should eliminate the Garlic Blast and Minty Munch flavors to focus on Blueberry Delight. “Not so fast!” you tell him.
You explain that your observations were a bit different from what you expected, but the differences aren’t dramatic. They could be the result of a real flavor preference or they could be due to chance.
To put it another way: You have a sample of 75 dogs, but what you really want to understand is the population of all dogs. Was this sample drawn from a population of dogs that choose the three flavors equally often?
Chisquare goodness of fit test hypotheses
Like all hypothesis tests, a chisquare goodness of fit test evaluates two hypotheses: the null and alternative hypotheses. They’re two competing answers to the question “Was the sample drawn from a population that follows the specified distribution?”
 Null hypothesis (H_{0}): The population follows the specified distribution.
 Alternative hypothesis (H_{a}): The population does not follow the specified distribution.
These are general hypotheses that apply to all chisquare goodness of fit tests. You should make your hypotheses more specific by describing the “specified distribution.” You can name the probability distribution (e.g., Poisson distribution) or give the expected proportions of each group.
When to use the chisquare goodness of fit test
The following conditions are necessary if you want to perform a chisquare goodness of fit test:
 You want to test a hypothesis about the distribution of one categorical variable. If your variable is continuous, you can convert it to a categorical variable by separating the observations into intervals. This process is known as data binning.
 The sample was randomly selected from the population.
 There are a minimum of five observations expected in each group.
How to calculate the test statistic (formula)
The test statistic for the chisquare (Χ^{2}) goodness of fit test is Pearson’s chisquare:
Formula  Explanation 


The larger the difference between the observations and the expectations (O − E in the equation), the bigger the chisquare will be.
To use the formula, follow these five steps:
Step 1: Create a table
Create a table with the observed and expected frequencies in two columns.
Flavor  Observed  Expected 
Garlic Blast  22  25 
Blueberry Delight  30  25 
Minty Munch  23  25 
Step 2: Calculate O − E
Add a new column called “O − E”. Subtract the expected frequencies from the observed frequency.
Flavor  Observed  Expected  O − E 
Garlic Blast  22  25  22 − 25 = −3 
Blueberry Delight  30  25  5 
Minty Munch  23  25  −2 
Step 3: Calculate (O − E)^{2}
Add a new column called “(O − E)^{2}”. Square the values in the previous column.
Flavor  Observed  Expected  O − E  (O − E)^{2} 
Garlic Blast  22  25  −3  (−3)^{2} = 9 
Blueberry Delight  30  25  5  25 
Minty Munch  23  25  −2  4 
Step 4: Calculate (O − E)^{2} / E
Add a final column called “(O − E)² / E“. Divide the previous column by the expected frequencies.
Flavor  Observed  Expected  O − E  (O − E)^{2}  (O − E)² / E 
Garlic Blast  22  25  −3  9  9/25 = 0.36 
Blueberry Delight  30  25  5  25  1 
Minty Munch  23  25  −2  4  0.16 
Step 5: Calculate Χ^{2}
Add up the values of the previous column. This is the chisquare test statistic (Χ^{2}).
Flavor  Observed  Expected  O − E  (O − E)^{2}  (O − E)^{2 }/ E 
Garlic Blast  22  25  −3  9  9/25 = 0.36 
Blueberry Delight  30  25  5  25  1 
Minty Munch  23  25  −2  4  0.16 
Χ^{2} = 0.36 + 1 + 0.16 = 1.52
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How to perform the chisquare goodness of fit test
The chisquare statistic is a measure of goodness of fit, but on its own it doesn’t tell you much. For example, is Χ2 = 1.52 a low or high goodness of fit?
To interpret the chisquare goodness of fit, you need to compare it to something. That’s what a chisquare test is: comparing the chisquare value to the appropriate chisquare distribution to decide whether to reject the null hypothesis.
To perform a chisquare goodness of fit test, follow these five steps (the first two steps have already been completed for the dog food example):
Step 1: Calculate the expected frequencies
Sometimes, calculating the expected frequencies is the most difficult step. Think carefully about which expected values are most appropriate for your null hypothesis.
In general, you’ll need to multiply each group’s expected proportion by the total number of observations to get the expected frequencies.
Step 2: Calculate chisquare
Calculate the chisquare value from your observed and expected frequencies using the chisquare formula.
Step 3: Find the critical chisquare value
Find the critical chisquare value in a chisquare critical value table or using statistical software. The critical value is calculated from a chisquare distribution. To find the critical chisquare value, you’ll need to know two things:
 The degrees of freedom (df): For chisquare goodness of fit tests, the df is the number of groups minus one.
 Significance level (α): By convention, the significance level is usually .05.
Step 4: Compare the chisquare value to the critical value
Compare the chisquare value to the critical value to determine which is larger.
Step 5: Decide whether the reject the null hypothesis
 If the Χ2 value is greater than the critical value, then the difference between the observed and expected distributions is statistically significant (p < α).
 The data allows you to reject the null hypothesis and provides support for the alternative hypothesis.
 If the Χ2 value is less than the critical value, then the difference between the observed and expected distributions is not statistically significant (p > α).
 The data doesn’t allow you to reject the null hypothesis and doesn’t provide support for the alternative hypothesis.