Chi-Square Goodness-of-Fit Calculator

What you can do

• Input observed frequency counts by category.
• Provide positive expected counts for the same categories.
• Account for parameters estimated from data when determining degrees of freedom.
• Read the statistic and p-value together with the null hypothesis.

Formula and interpretation

The Pearson goodness-of-fit statistic is χ² = Σ(Oᵢ−Eᵢ)²/Eᵢ, summing across categories. With k categories and p model parameters estimated from the observed counts, the usual degrees of freedom are k − 1 − p. The chi-square approximation depends on sufficiently large expected counts and a suitable sampling model.

Worked goodness-of-fit example

Let observed counts be [20,30,50] and expected counts be [25,25,50]. The statistic is (20−25)²/25 + (30−25)²/25 + (50−50)²/50 = 1 + 1 + 0 = 2. With three categories and no fitted parameters, df = 2; the upper-tail p-value is about 0.368.

How to use the workspace

1. Open Research Mode and find Chi-square Goodness of Fit.
2. Enter 20,30,50 and 25,25,50 in matching category order.
3. Enter how many parameters were estimated using the same data, if any.
4. Interpret the p-value against a previously chosen α without treating it as the probability that the null is true.

Assumptions and common errors

• Counts, not percentages: use observed and expected frequencies on the same total.
• Expected counts: small expected cells can make the approximation unreliable.
• Test type: this workflow is goodness-of-fit, not a general contingency-table independence test.

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