Chi-square goodness-of-fit tests using Python

by Amrita Mitra | Feb 9, 2023 | Statistics For Machine Learning

What is the chi-square goodness-of-fit test?

The chi-square goodness-of-fit test is a non-parametric hypothesis test. This test is used to determine whether a variable is likely to follow a specified distribution. For example, let’s say we rolled a die 60 times. Now, if the die is a fair die, then we are expected to get each of the numbers 1, 2, 3, 4, 5, and 6 60/6=10 times. But let’s say we got the numbers 1, 2, 3, 4, 5, and 6 the following number of times: 5, 12, 7, 9, 14, 13. So, we can see a difference between the expected values and the obtained values. But is the difference statistically significant?

In other words, we want to know whether the categorical variable indeed follows a specific distribution, a discrete uniform distribution in our case, or whether the observed difference arises by chance only. To know that, we can perform a chi-square goodness-of-fit test.

Please note that the chi-square goodness-of-fit test should follow certain assumptions. They are:

1. We need to take random samples from the population.

2. We need to perform the chi-square goodness-of-fit test on categorical or nominal data.

3. A minimum of five observations is expected for data categories.

How is the test statistic calculated in the chi-square goodness-of-fit test?

In the chi-square goodness-of-fit test, the test statistic is calculated using the following formula:

$\chi^{2}=\sum_{i=1}^{n}\frac{(O_{i}-E_{i})^{2}}{E_{i}}$

n = the number of categories of the categorical variable = 6 in our case
O_i = the number of times the ith number appears on the face of the die.
E_i = the expected number of times the ith number should appear on the face of the die

So, in our case, we got the numbers 1, 2, 3, 4, 5, and 6 the following number of times: 5, 12, 7, 9, 14, 13. And each number …

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