What is Cramér's V and how do I interpret it?

Cramér's V is a measure of effect size for chi-square tests, ranging from 0 to 1. It indicates the strength of association between two categorical variables. Interpretation guidelines: V 0.5 indicates strong association. Unlike the p-value, Cramér's V is not affected by sample size.

How do I enter data into the chi-square calculator?

Enter your contingency table data with rows on separate lines and columns separated by spaces or commas. For example, for a 2×3 table: enter '10, 20, 30' on the first line and '15, 25, 35' on the second line. All rows must have the same number of columns. The calculator accepts both integer and decimal values.

Chi-Square Test Calculator

Q: When should I use Fisher's Exact Test instead of Chi-Square?

Fisher's Exact Test is preferred when expected frequencies are small (typically when any expected count is less than 5). The chi-square test is an approximation that becomes less accurate with small expected values. For 2×2 tables with small samples, Fisher's Exact Test provides exact p-values. This calculator warns you when expected frequencies are below 5.

Perform a chi-square test of independence to determine if there is a significant association between two categorical variables. Get chi-square statistic, p-value, expected frequencies, cell contributions, and Cramer's V effect size with step-by-step calculations.

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Quick Examples

Observed Frequencies (rows on separate lines, columns by spaces/commas)

Significance Level (α)

Decimal Precision

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About Chi-Square Test Calculator

The Chi-Square Test Calculator performs the chi-square test of independence to determine whether there is a statistically significant association between two categorical variables. This comprehensive tool calculates the chi-square statistic, p-value, degrees of freedom, expected frequencies, cell contributions, and effect size (Cramer's V), providing complete statistical analysis with step-by-step explanations.

What is the Chi-Square Test of Independence?

The chi-square test of independence is a non-parametric statistical test used to analyze the relationship between two categorical variables organized in a contingency table. It compares the observed frequencies (actual counts from your data) to the expected frequencies (what we would expect if the variables were truly independent).

The test evaluates the null hypothesis that the two variables are independent of each other. If the test produces a sufficiently large chi-square statistic (resulting in a small p-value), we reject the null hypothesis and conclude that there is a statistically significant association between the variables.

Chi-Square Statistic Formula

Chi-Square 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)
The sum is taken over all cells in the contingency table

Expected Frequency Formula

Expected Frequency

$$E_{ij} = \frac{R_i \times C_j}{N}$$

Where:

R_i = Total of row i
C_j = Total of column j
N = Grand total of all observations

Degrees of Freedom

$$df = (r - 1) \times (c - 1)$$

Where r is the number of rows and c is the number of columns. The degrees of freedom determine which chi-square distribution to use for calculating the p-value.

Interpreting Chi-Square Test Results

The P-Value

The p-value tells you the probability of observing a chi-square statistic as extreme as (or more extreme than) what you calculated, assuming the null hypothesis is true:

p-value ≤ α: Reject the null hypothesis. There is a statistically significant association between the variables.
p-value > α: Fail to reject the null hypothesis. There is insufficient evidence to conclude an association exists.

Common significance levels (α):

α Level	Confidence	Use Case
0.10	90%	Exploratory analysis, preliminary studies
0.05	95%	Standard for most research (most common)
0.01	99%	More rigorous testing, medical research
0.001	99.9%	Very strict criteria, high-stakes decisions

Effect Size: Cramer's V

While the p-value tells you whether an association exists, Cramer's V measures the strength of that association:

Cramer's V

$$V = \sqrt{\frac{\chi^2}{N \times (k - 1)}}$$

Where k = min(rows, columns). Interpretation guidelines:

Cramer's V	Association Strength
0.00 - 0.10	Negligible association
0.10 - 0.30	Weak association
0.30 - 0.50	Moderate association
0.50+	Strong association

Assumptions of the Chi-Square Test

Independence: Observations must be independent of each other
Sample size: Expected frequencies should generally be at least 5 in each cell
Random sampling: Data should come from a random sample
Categorical data: Both variables must be categorical (nominal or ordinal)

When expected frequencies are below 5, the chi-square approximation may be unreliable. For 2×2 tables, consider using Fisher's Exact Test instead. This calculator warns you when any expected frequency is below 5.

Common Applications

Medical research: Testing whether a treatment is associated with patient outcomes
Marketing: Analyzing relationships between demographics and purchasing behavior
Genetics: Testing whether traits follow expected inheritance patterns
Social sciences: Examining associations between survey responses
Quality control: Determining if defect rates vary across production lines
Education: Analyzing relationships between teaching methods and student performance

How to Use This Calculator

Enter your contingency table: Input observed frequencies with rows on separate lines and columns separated by spaces or commas
Select significance level: Choose α = 0.05 (95% confidence) for standard analysis, or adjust based on your requirements
Set decimal precision: Select the number of decimal places for results
Review results: Examine the chi-square statistic, p-value, conclusion, and effect size
Analyze tables: Compare observed vs expected frequencies and identify cells contributing most to the statistic

Frequently Asked Questions

What is the Chi-Square Test of Independence?

The Chi-Square Test of Independence is a statistical hypothesis test used to determine whether there is a significant association between two categorical variables. It compares observed frequencies in a contingency table to expected frequencies calculated under the assumption of independence. If the chi-square statistic is large enough (p-value below significance level), we reject the null hypothesis of independence.

How do I interpret the p-value in a chi-square test?

The p-value represents the probability of observing a chi-square statistic as extreme as (or more extreme than) the calculated value, assuming the null hypothesis is true. If p-value ≤ α (commonly 0.05), reject the null hypothesis and conclude there is a significant association. If p-value > α, fail to reject the null hypothesis.

What are degrees of freedom in a chi-square test?

Degrees of freedom (df) for a chi-square test of independence equals (r-1) × (c-1), where r is the number of rows and c is the number of columns. For example, a 3×4 table has df = (3-1) × (4-1) = 6.

What is Cramer's V and how do I interpret it?

Cramer's V measures effect size, ranging from 0 to 1. It indicates association strength: V < 0.1 is negligible, 0.1-0.3 is weak, 0.3-0.5 is moderate, and V > 0.5 is strong. Unlike the p-value, Cramer's V is not affected by sample size.

When should I use Fisher's Exact Test instead?

Use Fisher's Exact Test when expected frequencies are small (any expected count below 5). The chi-square test is an approximation that becomes less accurate with small expected values. For 2×2 tables with small samples, Fisher's Exact Test provides exact p-values.

How do I enter data into the calculator?

Enter your contingency table with rows on separate lines and columns separated by spaces or commas. For a 2×3 table: enter '10, 20, 30' on line one and '15, 25, 35' on line two. All rows must have the same number of columns.

Additional Resources

Reference this content, page, or tool as:

"Chi-Square Test Calculator" at https://MiniWebtool.com/chi-square-test-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 20, 2026

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About Chi-Square Test Calculator

What is the Chi-Square Test of Independence?

Chi-Square Statistic Formula

Expected Frequency Formula

Degrees of Freedom

Interpreting Chi-Square Test Results

The P-Value

Effect Size: Cramer's V

Assumptions of the Chi-Square Test

Common Applications

How to Use This Calculator

Frequently Asked Questions

What is the Chi-Square Test of Independence?

How do I interpret the p-value in a chi-square test?

What are degrees of freedom in a chi-square test?

What is Cramer's V and how do I interpret it?

When should I use Fisher's Exact Test instead?

How do I enter data into the calculator?

Additional Resources

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