Statistical
Chi-Square Goodness-of-Fit statistic formula: Observed vs Expected counts
Compute the Chi-Square Goodness-of-Fit test statistic SUMPRODUCT((O - E)^2 / E), degrees of freedom, and critical-value hypothesis decision.
Test whether observed category counts match expected shares
The Chi-Square Goodness-of-Fit statistic measures whether observed categorical counts O_i deviate from expected counts E_i beyond random sampling noise: Chi^2 = SUM((O_i - E_i)^2 / E_i) with k - 1 degrees of freedom.
Important rule: feed RAW COUNTS (not percentages!) into Observed and Expected, and verify that SUM(Observed) equals SUM(Expected) with every expected cell >= 5.
| Variant (A) | Observed (B) | Expected (C) | (O - E)^2 / E |
|---|---|---|---|
| Plan A | 58 | 50 | 64 / 50 = 1.28 |
| Plan B | 42 | 50 | 64 / 50 = 1.28 |
| Plan C | 65 | 50 | 225 / 50 = 4.50 |
| Plan D | 35 | 50 | 225 / 50 = 4.50 (Total Chi^2 = 11.56) |
Single-cell Chi-Square statistic and critical-value gate
At df = 3 (4 categories - 1) and alpha = 0.05, the critical Chi-Square threshold is 7.815. Because 11.56 > 7.815, we reject equal preference.
=ROUND(SUMPRODUCT((B2:B5 - C2:C5)^2 / C2:C5), 2)Decision rule against critical threshold
Compare the test statistic against the alpha = 0.05 critical value in D2.
=IF(SUMPRODUCT((B2:B5 - C2:C5)^2 / C2:C5) > D2, "Reject Null", "Fail to Reject")Inputs and interpretation
Use independent observations in mutually exclusive, exhaustive categories and nonnegative observed counts. Expected counts are positive and sum to the same total. The k-1 degrees of freedom here assumes probabilities fixed before seeing the data; fitted parameters reduce degrees of freedom. Expected counts of at least 5 are a practical approximation guideline, not a mathematical requirement for computing the statistic. The 7.815 threshold is rounded and applies only to df=3 at alpha=0.05. Failure to reject does not prove the null.
Try the example
Use the original values or change them to test your understanding.