Chi-Square Test: Definition and Example

Chi-Square Test: Definition and Example

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Vikram
Vikram Singh
Assistant Manager - Content
Updated on Dec 9, 2022 08:46 IST

Introduction

Chi-square test is a statistically significant test for Hypothesis Testing.

2022_02_chi-square-test_featured-image.jpg

There are 3 steps in Hypothesis Testing:

  • State Null and Alternate Hypothesis
  • Perform Statistical Test
  • Accept and reject the Null Hypothesis

In this article, we will discuss the Chi-square test.

Table of Content

What is Chi-Square test?

Statistical method which is used to find the difference or correlation between the observed and expected categorical variables in the dataset.

Example: Food delivery company wants to find the relationship between gender, location and food choices of peoples India.

It is used to determine that the difference between 2 categorical variables are:

  • Due to chance or
  • Due to relationship

Mathematical Formula:

2022_02_chi-square-formula.jpg

Types of Chi-square Test:

  • goodness of fit test
  • test for independence

Goodness of fit test:

  • Number of variable = 1
  • Used to determine, whether the variable(sample) belongs to population or not
  • Degree of freedom:
2022_02_degree-of-freedom_goodness_of_fit.jpg

To know more about sample and population and degree of freedom, read the article Basics of Statistics for Data Science and z-test

Example:

Problem Statement: 

The observed and expected frequency of numbers appearing on dice.

2022_02_problem-1_chi-square-test.jpg

Using chi-square test at 5% significance level determine whether,

Observed frequencies are different from expected frequency or not.

Solution:

Step-1: State Null and Alternate Hypothesis:

Null Hypothesis: 

There is no difference between observed and expected frequency of outcome of rolling dice

Alternate Hypothesis:

 There is a difference between observed and expected frequency of outcome of rolling dice

Step-2: Significance level and Degree of Freedom:

Significance level = 5%

Degree of Freedom = 6-1 = 5

Corresponding chi-square value = 11.07

Step-3: Find the chi-square value:

2022_02_solution_chi-square-test.jpg

Step-4: Comparing with the significance level:

From, step-2 and step – 3, we have:

0.1186 < 11.07

So, we have to accept the Null Hypothesis

There is no difference between observed and expected frequency of outcome of rolling dice.

Test for independence

  • Number of variables = 2
  • Used to determine, whether the variables are different or same
  • Degree of Freedom:
2022_02_degree-of-freedom_independence.jpg

Example:

Problem Statement: Election commission decides to find the relationship between Gender and casting vote.

A sample of 10,000 people voters were taken, the result are summarized as:

2022_02_example_chi_square_test.jpg

Solution:

Step-1: State Null and Alternate Hypothesis

Null Hypothesis: Gender is independent of voting.

Alternate Hypothesis: Gender and Voting are independent.

Step-2: Significance level and Degree of Freedom

Significance level = 5%

Degree of Freedom = (2-1) x (2-1) = 1

Corresponding chi-square value = 3.84

Step-3: Find the chi-square value

2022_02_solution_2_chi-square-test.jpg

Step-4: Comparing with the significance level

From step-2 and step-3, we have,

6.6 > 3.84

Hence, rejecting the null hypothesis.

i.e. Gender and Voting are independent of each other.

Distribution Table:

2022_02_p-value_chi-square-test.jpg
df p = 0.75 p = 0.90 p = 0.95 p = 0.975 p = 0.99
1 1.32 2.71 3.84 5.02 6.64
2 2.77 4.60 5.99 7.37 9.21
3 4.10 6.24 7.80 9.33 11.31
4 5.38 7.77 9.48 11.14 13.27
5 6.62 9.23 11.07 12.83 15.08
6 7.84 10.64 12.59 14.44 16.81
7 9.04 12.02 14.07 16.01 18.48
8 10.22 13.36 15.51 17.54 20.09
9 11.39 14.68 16.92 19.02 21.67
10 12.5 15.9 18.3 20.5 23.2
11 13.7 17.3 19.7 21.9 24.7
12 14.8 18.6 21.0 23.3 26.2
13 16.0 19.8 22.4 24.7 27.7
14 17.1 21.1 23.7 26.1 29.1
15 18.2 22.3 25.0 27.5 30.6
16 19.4 23.5 26.3 28.8 32.0
17 20.5 24.8 27.6 30.2 33.4
18 21.6 26.0 28.9 31.5 34.8
19 22.7 27.2 30.1 32.9 36.2
20 23.8 28.4 31.4 34.2 37.6
Chi-square distribution table

Conclusion:

Chi-square is a statistically significant test for the hypothesis testing (null and alternative hypotheses) when the variables are categorical.

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Vikram Singh
Assistant Manager - Content

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio