Chi-square (χ²) test

Chi-square (χ²) test

Chi-square (χ²) test is a statistical test commonly used to determine whether there is a significant association between categorical variables. It is used when the data being analyzed consists of frequencies or counts for different categories. Chi-square test is used in various fields such as biology, social science, medicine, and business to analyze data and determine if there is a significant difference between observed and expected frequencies within categorical data.

Table of Contents

1. Introduction to the Chi-square (χ²) Test
2. Formulas for Calculating 
   • Goodness of Fit (One-Way) 
   • Independence (Two-Way) 
3. Chi-square Interpretation
   • Comparison with Critical Values
   • Degrees of Freedom Explanation
4. Example: Application Test
   • Step-by-Step Calculation 
   • Interpretation of Results
5. Frequently Asked Questions (FAQ)

Formula to Calculate Chi-square:

The formula to calculate the chi-square statistic depends on the type of data being analyzed. Here are the formulas for two common scenarios:

1. Goodness of Fit (One-Way) Chi-squar Test:

χ2=Σ((Oi​−Ei​)2/Ei​)

Where:

1. • Oi​ = Observed frequency in each category
   • Ei​ = Expected frequency in each category
2. Independence (Two-Way) Chi-square Test:

   χ2=Σ((Oi​j−Ei​j)2/Ei​j)

Where:

1. • Oi​j = Observed frequency in each cell of the contingency table
   • Ei​j = Expected frequency in each cell of the contingency table

Chi-square Interpretation:

After calculating the chi-square statistic, it is compared to a critical value from the chi-square distribution with a certain degree of freedom (df) to determine if the observed frequencies deviate significantly from the expected frequencies. If the calculated chi-square value is greater than the critical value, then there is evidence to reject the null hypothesis, suggesting a significant association between the variables.

Degrees of Freedom (df): Degrees of freedom in a chi-square test depend on the number of categories and the number of variables involved in the analysis. For a one-way chi-square test, df = (number of categories – 1), and for a two-way chi-square test, df = (number of rows – 1) * (number of columns – 1).

Critical Values (α = 0.01 to 0.05): Critical values for different levels of significance (α) can be found in the chi-square distribution table. For example, for a significance level of 0.05 and df = 5, the critical value is 11.070.

 Table  values for the chi-square distribution for significance levels of 0.01, 0.05, and 0.10 across degrees of freedom 0 to 10:

df	α = 0.01	α = 0.05 (Common)	α = 0.10
0	6.635	3.841	2.706
1	7.879	6.635	4.605
2	9.550	9.210	6.251
3	11.345	11.345	7.815
4	13.277	13.277	9.488
5	15.086	15.086	11.070
6	16.812	16.812	12.592
7	18.475	18.475	14.067
8	20.090	20.090	15.507
9	21.666	21.666	16.919
10	23.209	23.209	18.307

 

 Example:

A survey on voting preferences among males and females was conducted, and the results are summarized in the table below:

	BJP	CONGRESS	OTHER
Male	50	13	5
Female	60	18	4

Test whether there is a relationship between gender and voting preference. Use a significance level of 0.05.

To calculate the test statistic, we’ll follow these steps:

Step 1: Define the Hypotheses

• H0​: There is no link between gender and political party preference.
• H1​: There is a link between gender and political party preference.

Step 2: Calculate the Expected Values:

For each cell in the table, we calculate the expected value using the formula:

Given the observed values:

• For Male BJP: Observed (O) = 50
• For Male CONGRESS: Observed (O) = 13
• For Male OTHER: Observed (O) = 5
• For Female BJP: Observed (O) = 60
• For Female CONGRESS: Observed (O) = 18
• For Female OTHER: Observed (O) = 4

We’ll calculate the expected values:

Step 3: Calculate (O−E)2 / E ​ for Each Cell in the Table:

For each cell, we’ll calculate (O−E)² / E ​, where O is the observed value and E is the expected value.:

	BJP	CONGRESS	OTHER
Male	0.5246	0.0014	0.1736
Female	0.2585	0.0027	0.1261

Step 4: Calculate the Test Statistic χ2:

We’ll sum up all the values from the previous step to obtain

.

the calculated test statistic \( χ^2 \) value is approximately 1.0869. To interpret this result:

If the significance level is chosen as 0.05 and the degrees of freedom are 2, comparing the calculated \( χ^2 \) value to the critical value from the chi-square distribution table, we would find that 1.0869 is less than the critical value.

Therefore, we would fail to reject the null hypothesis. This suggests that there is no significant association between gender and political party preference based on the observed data.

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Frequently Asked Questions (FAQ)

1. What is the Chi-square (χ²) test used for?

   • The Chi-square test is a statistical method utilized to determine if there is a significant association between categorical variables. It assesses whether observed frequencies within different categories deviate significantly from expected frequencies.

2. In what fields is the Chi-square test commonly applied?

   • The Chi-square test finds application across various domains such as biology, social sciences, medicine, and business. It is particularly useful when analyzing categorical data.

3. What are the key components of the Chi-square test formula?

   • The formula involves calculating the difference between observed and expected frequencies, squaring this difference, and dividing it by the expected frequency. For a one-way Chi-square test, the formula is χ²=Σ((Oi​−Ei​)²/Ei​), where Oi​ represents observed frequencies and Ei​ represents expected frequencies.

4. How is the significance of the Chi-square test determined?

   • The significance of the Chi-square test is assessed by comparing the calculated Chi-square statistic to a critical value from the Chi-square distribution table. If the calculated value exceeds the critical value, there is evidence to reject the null hypothesis, indicating a significant association between variables.

5. What are degrees of freedom in the context of the Chi-square test?

   • Degrees of freedom (df) are determined by the number of categories and variables involved in the analysis. For one-way Chi-square tests, df = (number of categories – 1), while for two-way tests, df = (number of rows – 1) * (number of columns – 1).