Chi-Square Test of Independence

Created:December 10, 2024

Last Updated:October 7, 2025

The calculator helps you determine whether there is a significant relationship between two categorical variables. Whether you have raw data with two categorical columns, a pre-computed cross-tabulation table (with counts or percentages), or want to manually enter your contingency table, this calculator handles all formats. It analyzes whether the observed frequency distribution differs significantly from the expected distribution, assuming the variables are independent. This test is widely used in fields like social sciences, market research, and healthcare to analyze survey data, clinical outcomes, and demographic relationships. Common applications include examining relationships between demographic factors and preferences, testing associations between treatments and outcomes, or analyzing connections between categorical variables in survey responses. for a quick example.

Calculator

1. Load Your Data

2. Select Data Type & Options

Data Input Type:

Two Categorical Columns (Raw Data)

Cross-tabulation Table (Pre-computed Frequencies)

Select first categorical variable:

Select second categorical variable:

Significance Level:

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Chi-Square Test of Independence

Definition

Chi-Square Test of Independence examines whether there is a significant association between two categorical variables. It tests whether the observed frequencies in a contingency table differ significantly from the frequencies we would expect if there were no relationship between the variables.

Formula

Test Statistic:

\chi^2 = \sum\frac{(O_{ij} - E_{ij})^2}{E_{ij}}

Where:

$O_{ij}$ = observed frequency in cell ( $i$ , $j$ )
$E_{ij}$ = expected frequency in cell ( $i$ , $j$ )
$E_{ij} = \frac{(\text{row }i\text{ total})(\text{column }j\text{ total})}{\text{grand total}}$

Modified Formula with Yates Continuity Correction (for 2x2 tables):

\chi^2 = \sum\frac{(|O_{ij} - E_{ij}| - 0.5)^2}{E_{ij}}

Key Assumptions

Random Sampling: Data must be randomly sampled

Independence: Observations must be independent

Sample Size: Expected frequenciesshould be

\geq 5

Mutual Exclusivity: Categories must be mutually exclusive

Practical Example

Step 1: State the Data

Contingency table of Gender and Product Preference:

	Like	Dislike	Total
Male	40	30	70
Female	30	50	80
Total	70	80	150

Step 2: State Hypotheses

$H_0$ : Gender and Preference are independent
$H_a$ : Gender and Preference are not independent
$\alpha = 0.05$

Step 3: Calculate Expected Frequencies

Male, Like: $E_{11} = \frac{70 \times 70}{150} = 32.67$
Male, Dislike: $E_{12} = \frac{70 \times 80}{150} = 37.33$
Female, Like: $E_{21} = \frac{80 \times 70}{150} = 37.33$
Female, Dislike: $E_{22} = \frac{80 \times 80}{150} = 42.67$

Step 4: Calculate Chi-Square Statistic

\chi^2 = \frac{(|40-32.67|-0.5)^2}{32.67} + \frac{(|30-37.33|-0.5)^2}{37.33} + \frac{(|30-37.33|-0.5)^2}{37.33} + \frac{(|50-42.67|-0.5)^2}{42.67} = 5.02

Step 5: Calculate Degrees of Freedom

$df = (r-1)(c-1) = (2-1)(2-1) = 1$

Step 6: Draw Conclusion

At $\alpha = 0.05$ with $df = 1$ , the critical value is $3.841$ . Since $\chi^2 = 5.02 \gt 3.841$ , we reject $H_0$ . There is sufficient evidence to conclude that Gender and Product Preference are not independent ( $p$ -value $= 0.008$ ).

Effect Size

Cramer's V measures the strength of association:

V = \sqrt{\frac{\chi^2}{n(\min(r,c)-1)}}

For our example:

V = \sqrt{\frac{5.02}{150(1)}} = 0.18

For $2\times 2$ tables:

Small effect: $V \approx 0.10$
Medium effect: $V \approx 0.30$
Large effect: $V \approx 0.50$

With $V = 0.18$ , this indicates a small effect size, suggesting a weak association between Gender and Product Preference in our sample.

Verification

Chi-Square Test of Independence

Created:December 10, 2024

Last Updated:October 7, 2025

Calculator

1. Load Your Data

2. Select Data Type & Options

Data Input Type:

Two Categorical Columns (Raw Data)

Cross-tabulation Table (Pre-computed Frequencies)

Select first categorical variable:

Select second categorical variable:

Significance Level:

Related Calculators

Chi-Square Goodness of Fit Test Calculator

One-Sample T-Test Calculator

Two-Sample T-Test Calculator

One-Way ANOVA Calculator

Learn More

Chi-Square Test of Independence

Definition

Formula

Test Statistic:

\chi^2 = \sum\frac{(O_{ij} - E_{ij})^2}{E_{ij}}

Where:

$O_{ij}$ = observed frequency in cell ( $i$ , $j$ )
$E_{ij}$ = expected frequency in cell ( $i$ , $j$ )
$E_{ij} = \frac{(\text{row }i\text{ total})(\text{column }j\text{ total})}{\text{grand total}}$

Modified Formula with Yates Continuity Correction (for 2x2 tables):

\chi^2 = \sum\frac{(|O_{ij} - E_{ij}| - 0.5)^2}{E_{ij}}

Key Assumptions

Random Sampling: Data must be randomly sampled

Independence: Observations must be independent

Sample Size: Expected frequenciesshould be

\geq 5

Mutual Exclusivity: Categories must be mutually exclusive

Practical Example

Step 1: State the Data

Contingency table of Gender and Product Preference:

	Like	Dislike	Total
Male	40	30	70
Female	30	50	80
Total	70	80	150

Step 2: State Hypotheses

$H_0$ : Gender and Preference are independent
$H_a$ : Gender and Preference are not independent
$\alpha = 0.05$

Step 3: Calculate Expected Frequencies

Male, Like: $E_{11} = \frac{70 \times 70}{150} = 32.67$
Male, Dislike: $E_{12} = \frac{70 \times 80}{150} = 37.33$
Female, Like: $E_{21} = \frac{80 \times 70}{150} = 37.33$
Female, Dislike: $E_{22} = \frac{80 \times 80}{150} = 42.67$

Step 4: Calculate Chi-Square Statistic

\chi^2 = \frac{(|40-32.67|-0.5)^2}{32.67} + \frac{(|30-37.33|-0.5)^2}{37.33} + \frac{(|30-37.33|-0.5)^2}{37.33} + \frac{(|50-42.67|-0.5)^2}{42.67} = 5.02

Step 5: Calculate Degrees of Freedom

$df = (r-1)(c-1) = (2-1)(2-1) = 1$

Step 6: Draw Conclusion

Effect Size

Cramer's V measures the strength of association:

V = \sqrt{\frac{\chi^2}{n(\min(r,c)-1)}}

For our example:

V = \sqrt{\frac{5.02}{150(1)}} = 0.18

For $2\times 2$ tables:

Small effect: $V \approx 0.10$
Medium effect: $V \approx 0.30$
Large effect: $V \approx 0.50$

With $V = 0.18$ , this indicates a small effect size, suggesting a weak association between Gender and Product Preference in our sample.

Verification

Dislike

Total

Male

Female

Total

150

Dislike

Total

Male

Female

Total

150

Chi-Square Test of Independence

Calculator

1. Load Your Data

2. Select Data Type & Options

Related Calculators

Chi-Square Goodness of Fit Test Calculator

One-Sample T-Test Calculator

Two-Sample T-Test Calculator

One-Way ANOVA Calculator

Learn More

Chi-Square Test of Independence

Definition

Formula

Key Assumptions

Practical Example

Step 1: State the Data

Step 2: State Hypotheses

Step 3: Calculate Expected Frequencies

Step 4: Calculate Chi-Square Statistic

Step 5: Calculate Degrees of Freedom

Step 6: Draw Conclusion

Effect Size

Verification

View Verification Details

Chi-Square Test of Independence

Calculator

1. Load Your Data

2. Select Data Type & Options

Related Calculators

Chi-Square Goodness of Fit Test Calculator

One-Sample T-Test Calculator

Two-Sample T-Test Calculator

One-Way ANOVA Calculator

Learn More

Chi-Square Test of Independence

Definition

Formula

Key Assumptions

Practical Example

Step 1: State the Data

Step 2: State Hypotheses

Step 3: Calculate Expected Frequencies

Step 4: Calculate Chi-Square Statistic

Step 5: Calculate Degrees of Freedom

Step 6: Draw Conclusion

Effect Size

Verification

View Verification Details

Chi-Square Test of Independence

Calculator

1. Load Your Data

2. Select Data Type & Options

Related Calculators

Chi-Square Goodness of Fit Test Calculator

One-Sample T-Test Calculator

Two-Sample T-Test Calculator

One-Way ANOVA Calculator

Learn More

Chi-Square Test of Independence

Definition

Formula

Key Assumptions

Practical Example

Step 1: State the Data

Step 2: State Hypotheses

Step 3: Calculate Expected Frequencies

Step 4: Calculate Chi-Square Statistic

Step 5: Calculate Degrees of Freedom

Step 6: Draw Conclusion

Effect Size

Verification

View Verification Details

Chi-Square Test of Independence

Calculator

1. Load Your Data

2. Select Data Type & Options

Related Calculators

Chi-Square Goodness of Fit Test Calculator

One-Sample T-Test Calculator

Two-Sample T-Test Calculator