Enter your observed counts and expected counts (or ratio). You get χ², the degrees of freedom, the critical value at p = 0.05 or 0.01 and the conclusion, with each step shown as you would write it.
χ² with 1 degree of freedom
χ² = 1.33
1.33 is less than the critical value of 3.84 at p = 0.05, so fail to reject the null hypothesis: the data fit the expected values.
| Category | o | e | (o − e)²/e |
|---|---|---|---|
| Tall | 290 | 300 | 0.33 |
| Short | 110 | 100 | 1 |
| Sum | 400 | 400 | 1.33 |
Degrees of freedom: 2 outcomes − 1 = 1. Expected counts use the observed total of 400, split by the ratio.
| Formula | χ² = Σ (o − e)² / e |
|---|---|
| Degrees of freedom | number of possible outcomes − 1 |
| Critical value | From the reference sheet's table, at p = 0.05 or 0.01 |
| Decision | Reject the null hypothesis if χ² is greater than the critical value |
If χ² is greater than the critical value, the difference is unlikely to be due to chance alone, so you reject the null hypothesis. If it is smaller, you fail to reject it: the data fit the prediction.
It depends on the degrees of freedom and the p-value the question sets. At p = 0.05 the reference sheet gives 3.84 for 1 degree of freedom, 5.99 for 2 and 7.81 for 3; at p = 0.01 it gives 6.63, 9.21 and 11.34.
The number of categories minus 1. A 9:3:3:1 dihybrid cross has 4 categories, so 3 degrees of freedom.
Yes. The reference sheet gives the formula and a table of critical values.