Chi-Square Test: Definition and Example

Introduction

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

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

• Chi-square test
  • goodness of fit test
  • test for independence
  • distribution table

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:

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:

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.

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:

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:

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:

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

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:

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

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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