THE NUMORIX GUIDE
How to use the Chi-Square Calculator
Last reviewed September 14, 2026
What this calculator does
For each category compute (observed-expected)^2/expected, sum contributions to chi-square, set degrees of freedom to number of categories minus one, and approximate the upper-tail p-value from the chi-square distribution.
Formula and method
For each category compute (observed-expected)^2/expected, sum contributions to chi-square, set degrees of freedom to number of categories minus one, and approximate the upper-tail p-value from the chi-square distribution.
Variables and inputs
Observed Values default to 50,30,20 and Expected Values to 40,40,20; both are paired comma-separated counts or frequencies. Lists need the same length, at least two categories, and every expected value must be positive.
Worked example
For observed 50,30,20 and expected 40,40,20: contributions are (10^2/40)=2.5, (-10)^2/40=2.5, and 0; chi-square=5.0; df=3-1=2. The standard upper-tail p is about 0.0821, while this engine's approximation path displays about 0.8116, so verify that p-value independently.
How to interpret the result
A larger chi-square means observed counts are farther from the expected pattern. The p-value asks how unusual a deviation at least this large would be under the expected model.
Common mistakes to avoid
Do not compare counts with percentages unless they are put on a common scale. Expected values should represent a specified null model, not values chosen after seeing the observations.
Assumptions and limitations
The engine uses df = categories - 1 and a hand-coded approximation whose current CDF calculation can differ materially from the standard upper-tail value; it does not apply continuity corrections, estimate parameters, or check the usual expected-count conditions.
Practical use and checks
The Chi-Square Calculator compares observed category counts with expected counts from a specified model. Enter observed values 50, 30, 20 and expected values 40, 40, 20 as a check. The category contributions are 2.5, 2.5, and 0, so chi-square should be 5 with 2 degrees of freedom. The expected values must represent a hypothesis chosen before interpreting the result; they are not a second sample to compare casually with the observations. A larger statistic means the observed pattern is farther from the expected pattern in the calculator's count scale. The p-value is a tail probability under the null model, not the chance that the null hypothesis is true and not a measure of practical importance. Keep counts on the same basis, use categories that are mutually exclusive, and check that expected counts are large enough for the approximation. The implementation uses a hand-coded distribution approximation and fixed degrees-of-freedom logic, so verify the p-value with a trusted statistics package for a consequential test, especially near a decision threshold. It does not apply continuity corrections, estimate parameters, or establish causation. Report the sampling design, expected-count construction, degrees of freedom, effect size, and decision rule with the statistic.