The chi-square goodness-of-fit test checks whether observed category counts match a distribution you expected.
Why it matters
It tests claims such as "births are equally likely on every weekday" or "blood groups follow national proportions".
Key ideas
- One categorical variable
- Observed counts O vs expected counts E
- χ² = Σ (O − E)² ÷ E
- df = categories − 1
- Expected count should be at least 5 in most cells
- Large χ² → poor fit
Real-life research example
120 customers choose among 4 pharmacies. If equally popular, each expects 30. Observed: 45, 30, 25, 20.
When to use
- Compare one sample's proportions with a known pattern
When NOT to use
- Don't use with percentages instead of counts
- Don't use with tiny expected counts
Mini example
O = 45, 30, 25, 20; E = 30 each → χ² = 7.5 + 0 + 0.83 + 3.33 = 11.67, df = 3, p ≈ 0.009
Quick takeaway
Counts vs expected counts → a fit test.
Try it yourself
Enter observed counts 45, 30, 25, 20 and equal expected proportions. Then try 31, 29, 30, 30.
Model answer
The first set gives a large χ² and small p (poor fit); the second is almost exactly equal, so χ² is near 0 and p is large.