Math · Introductory statistics · Worked example
Test whether a die is fair
A die is rolled 120 times, giving 25 ones, 17 twos, 15 threes, 23 fours, 24 fives and 16 sixes. Is this consistent with a fair die at α = 0.05?
Find the expected counts
A fair die gives each face probability 1/6.
Add the contributions
The squared differences from 20 are 25, 9, 25, 9, 16 and 16. Each is divided by the expected count, 20.
Find the p-value
With 6 faces there are 5 degrees of freedom, and P(χ² ≥ 5) ≈ 0.416.
Decide
0.416 > 0.05, so fail to reject H₀: the counts are consistent with a fair die. That does not prove the die is fair; small biases need many more rolls to detect.
Result
χ² = 5.0 with 5 degrees of freedom and p ≈ 0.42: no evidence against a fair die.
Your turn
With counts 35, 10, 15, 23, 24 and 13 from 120 rolls, find χ².
Show the answer and explanation
χ² = 21.2, with p ≈ 0.0007: strong evidence the die is not fair.
The squared differences from 20 are 225, 100, 25, 9, 16 and 49, which add to 424, and 424/20 = 21.2.
Keep exploring
In Statistics, Distributions shows P(X ≥ 5) ≈ 0.41588 for chi-square with 5 degrees of freedom. The table view gives the 5% critical value, 11.0705: χ² would have to exceed it to reject.
Return to the concept →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Find the p-value in Statistics Check χ² in Math Open worked example on a board Statistics formulas in Math ReferenceYour existing work stays on this device. Examples open as editable copies.