Chi-Square Calculator
Calculate a chi-square statistic from observed and expected values.
Chi-square formula
Chi-square is the sum of (observed − expected)² / expected across categories. It measures how far observed counts differ from expected counts.
What the Chi-Square Statistic Shows
A chi-square statistic summarizes how far observed counts differ from expected counts. It is widely used for goodness-of-fit tests and tests of association in contingency tables.
Chi-Square Example
If two categories are expected to contain 50 observations each but the observed counts are 60 and 40, the differences from expectation contribute to the chi-square statistic after each squared difference is divided by its expected count.
Interpreting the Test
The calculated statistic is interpreted with the appropriate degrees of freedom. Expected cell counts should also be large enough for the usual chi-square approximation to be reliable.
Frequently Asked Questions
Can chi-square be negative?
No. The statistic is built from squared differences and is therefore zero or positive.
What does a larger chi-square value mean?
It indicates greater disagreement between observed and expected counts.
Does chi-square measure effect size?
No. It tests discrepancy; measures such as Cramer’s V are used for association strength.
Chi-Square Assumptions
Chi-square procedures compare observed and expected frequencies. Observations should be appropriately categorized and independent, and very small expected cell counts can make the usual approximation unreliable. The statistic depends on sample size and degrees of freedom, so interpret it with the relevant distribution rather than in isolation.
Methodology, Assumptions and Interpretation
A chi-square statistic compares observed counts with expected counts. Expected cell frequencies should be sufficiently large for the usual approximation, and the test indicates evidence of association or fit rather than the size or practical importance of an effect.