Chi-Square Calculator

The chi-square ($͵^2$) statistic evaluates the magnitude of discrepancy between empirical observation and theoretical expectation. The core formula for a single category is expressed as:

͵^2 = (O - E)^2 / E

Where O represents the observed frequency and E represents the expected frequency. When working with multiple categories, the total chi-square statistic is the sum of the squared differences divided by the expected value for every individual category: ͵^2 = ∑ [ (O - E)^2 / E ]. A higher result indicates a greater divergence between what was recorded and what was anticipated, signaling that random chance is less likely to be the sole cause of the variation.

Imagine a genetics study where a researcher expects a litter of 100 mice to exhibit a specific coat color ratio, predicting 60 brown mice and 40 white mice. After the experiment concludes, the actual observed counts are 50 brown mice and 50 white mice. We calculate the chi-square component for each category to evaluate this shift.

Step 1: Brown Mice Category
Observed (O) = 50, Expected (E) = 60
(50 - 60)^2 / 60 = (-10)^2 / 60 = 100 / 60 = 1.667

Step 2: White Mice Category
Observed (O) = 50, Expected (E) = 40
(50 - 40)^2 / 40 = (10)^2 / 40 = 100 / 40 = 2.500

Step 3: Total Chi-Square Statistic
͵^2 = 1.667 + 2.500 = 4.167. This total statistic can then be compared against critical values on a chi-square distribution table to test the hypothesis.

Ensure your expected value for any single category is generally 5 or greater; small expected frequencies can distort the chi-square distribution and yield unreliable p-values. Always verify that your input data represents actual counts or frequencies rather than percentages, proportions, or normalized scores, as the formula relies strictly on raw numerical occurrences.