Chi-Square Goodness of Fit Calculator

The goodness of fit test compares observed frequencies to expected frequencies derived from a hypothesized distribution. The null hypothesis states that the observed data follow the expected distribution.

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 \(i\). The degrees of freedom are \(df = k - 1\), where \(k\) is the number of categories. If parameters were estimated from the data, subtract an additional degree of freedom for each estimated parameter.

Under the null hypothesis, the statistic follows a chi-square distribution with \(k - 1\) degrees of freedom. Large values indicate poor fit between observed and expected distributions. Each category should have an expected frequency of at least 5 for the approximation to be reliable.

The test is one-tailed — only large chi-square values provide evidence against the null hypothesis, since the statistic measures total discrepancy between observed and expected frequencies.

To interpret the goodness of fit results:

• Chi-Square Value: Compare to the critical value at your chosen α level. For example, with df = 2 and α = 0.05, the critical value is 5.991. If χ² exceeds this, reject the null hypothesis that the data follow the expected distribution.
• Individual Contributions: Each (O−E)²/E term shows which categories contribute most to the overall chi-square. Large individual contributions indicate where the data deviate most from expectations.
• Total Consistency: Verify that total observed and total expected frequencies are approximately equal, as they should be for a properly specified model.