46
3
100
100
0
25
1
4
16
25
46
3
100
100
0
25
1
4
16
25
The Chi-Square Calculator performs a goodness-of-fit test by comparing observed frequencies to expected frequencies across categories. The chi-square (χ²) test is one of the most versatile non-parametric tests in statistics, used to determine whether observed data differs significantly from what would be expected under a specific hypothesis. It was introduced by Karl Pearson in 1900 and remains foundational in genetics, market research, quality control, and social sciences.
This calculator accepts four observed and four expected values, computes the chi-square statistic, reports the degrees of freedom (k - 1 = 3 for four categories), and shows the individual contribution of each cell to the overall statistic. The cell contributions reveal which categories deviate most from expectation, providing insight beyond the aggregate test statistic. Compare the result to chi-square critical values to determine statistical significance.
The chi-square statistic is calculated as:
$$\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}$$
Where O_i is the observed frequency and E_i is the expected frequency for category i. Each term $$\frac{(O_i - E_i)^2}{E_i}$$ represents the contribution of that category to the overall chi-square value. Larger contributions indicate categories where the observed data deviates most from expectation.
The degrees of freedom for a goodness-of-fit test are df = k - 1, where k is the number of categories. With 4 categories, df = 3. Critical values at common significance levels for df = 3 are: 6.251 (α = 0.10), 7.815 (α = 0.05), and 11.345 (α = 0.01). If χ² exceeds the critical value, reject the null hypothesis that the observed frequencies match the expected distribution.
If the chi-square statistic is below the critical value, there is insufficient evidence to reject the null hypothesis, meaning the observed data is consistent with the expected distribution. If it exceeds the critical value, the observed data differs significantly from what was expected. Examine the individual cell contributions to identify which categories drive the discrepancy. A rule of thumb: each expected frequency should be at least 5 for the chi-square approximation to be reliable.
Inputs
Results
χ² = 0.88 with df = 3. Critical value at α = 0.05 is 7.815, so we fail to reject the null: the die appears fair.
Inputs
Results
χ² = 36.0 far exceeds the critical value of 7.815, indicating preferences are not equally distributed. Categories 1 and 4 contribute most.
The chi-square test is used to determine whether observed categorical data matches an expected distribution (goodness-of-fit test) or whether two categorical variables are independent (test of independence). Common applications include testing genetic ratios, evaluating survey response distributions, checking quality control categories, and analyzing contingency tables.
The chi-square test relies on a continuous distribution to approximate discrete count data. When expected frequencies are below 5, the approximation becomes unreliable and the test may produce misleading results. If you have small expected values, consider combining adjacent categories or using Fisher's exact test as an alternative.
This calculator is configured for 4 categories. For more categories, the same formula applies: sum (O-E)²/E for each category and use df = k-1 degrees of freedom. You would need a calculator supporting more input fields or statistical software for larger analyses.
A goodness-of-fit test (this calculator) compares observed frequencies against a theoretically expected distribution for one variable. A test of independence uses a contingency table to determine whether two variables are related. Both use the same chi-square formula but differ in how expected frequencies are calculated and how degrees of freedom are determined.
Expected values come from the null hypothesis. If you hypothesize equal distribution across 4 categories with 100 total observations, each expected value is 25. If you hypothesize a specific ratio (e.g., 9:3:3:1 in genetics), multiply each proportion by the total count. The expected values must sum to the same total as the observed values.
For df = 3 (four categories), common critical values are: 6.251 (α = 0.10), 7.815 (α = 0.05), 11.345 (α = 0.01), and 16.266 (α = 0.001). If your chi-square statistic exceeds the critical value at your chosen significance level, reject the null hypothesis. Consult a chi-square table for other degrees of freedom.
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