Chi-Square (Goodness-of-Fit) Calculator
Enter your observed counts and (optionally) expected counts to compute the chi-square statistic, degrees of freedom, p-value, and decision at your chosen significance level.
What Is a “Qui Square” Test?
Many people search for a “qui square test calculator”, but the formal term is chi-square test (pronounced “kai-square”). It is a classic statistical method used to compare what you observed in real data against what you would expect under a hypothesis.
This page focuses on the chi-square goodness-of-fit test. It helps answer questions like: “Do my category counts follow the pattern I expected?” If the mismatch between observed and expected values is too large to be explained by chance, the test suggests your expected model may not fit.
When to Use This Calculator
- You have count data grouped into categories.
- You want to compare observed frequencies to expected frequencies.
- Your categories are mutually exclusive (each case belongs to one category).
- You need a quick decision using a p-value and a chosen significance level (α).
Examples
- Testing whether survey responses are evenly distributed across four options.
- Checking whether dice outcomes match a fair distribution over many rolls.
- Comparing observed genetic trait counts to Mendelian expected ratios.
Chi-Square Formula (Goodness-of-Fit)
The statistic is:
χ2 = Σ ((Oi - Ei)2 / Ei)
where:
- Oi = observed count in category i
- Ei = expected count in category i
- Σ = sum across all categories
Degrees of freedom are usually df = k - 1 for k categories (assuming no extra parameters are estimated from data).
How to Use the Calculator
Step 1: Enter observed counts
Type your category counts in order. Example: 12, 19, 15, 14.
Step 2: Enter expected counts (or leave blank)
If you already know your expected counts, enter them in the same order and with the same number of categories. If left blank, the calculator assumes equal expected counts.
Step 3: Choose α and calculate
Common choices are α = 0.05 or α = 0.01. The calculator reports χ2, df, critical value, p-value, and a decision.
How to Interpret the Output
- Small p-value (p < α): reject the null hypothesis. Observed and expected differ more than chance alone would suggest.
- Large p-value (p ≥ α): fail to reject the null hypothesis. Data are reasonably consistent with expected counts.
“Fail to reject” does not prove the model is true; it only means the sample does not provide strong evidence against it.
Assumptions and Good Practice
- Use raw counts, not percentages.
- Observations should be independent.
- Expected counts should usually be large enough (a common guideline is at least 5 per category).
- Categories should be clearly defined and non-overlapping.
Common Mistakes
- Entering proportions instead of counts.
- Mismatched number of observed and expected categories.
- Using negative values or leaving hidden blanks in the input list.
- Interpreting p-value as the probability the null hypothesis is true.
Quick FAQ
Is this calculator for independence tests too?
This tool is designed for one-way goodness-of-fit. A chi-square test of independence uses a two-way contingency table and different degrees-of-freedom logic.
Why did my expected values get adjusted?
If expected totals do not match observed totals, this calculator rescales expected counts to the same total before computing χ2. That keeps the comparison valid for goodness-of-fit interpretation.
Can I use decimal expected values?
Yes. Expected counts can be decimal values, as long as they are positive.