Editing Pearson's chi-squared test (section)

===Fairness of dice===
A 6-sided die is thrown 60 times. The number of times it lands with 1, 2, 3, 4, 5 and 6 face up is 5, 8, 9, 8, 10 and 20, respectively. Is the die biased, according to the Pearson's chi-squared test at a significance level of 95% and/or 99%?

The null hypothesis is that the die is unbiased, hence each number is expected to occur the same number of times, in this case, {{sfrac|60|''n''}} = 10. The outcomes can be tabulated as follows:
{| class="wikitable" style="text-align:center;"
|-
! style="padding:0 1em;"|<math> i</math>
! style="padding:0 1em;"|<math> O_i</math>
! style="padding:0 1em;"|<math> E_i</math>
!<math> O_i - E_i</math>
!<math> (O_i - E_i)^2</math>
|-
| 1 ||  5 || 10 || &minus;5 ||  25 
|-
| 2 ||  8 || 10 || &minus;2 ||   4 
|-
| 3 ||  9 || 10 || &minus;1 ||   1 
|-
| 4 ||  8 || 10 || &minus;2 ||   4 
|-
| 5 || 10 || 10 ||        0 ||   0 
|-
| 6 || 20 || 10 ||       10 || 100 
|-
| colspan="4" |Sum
|134
|}

We then consult an ''[https://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm Upper-tail critical values of chi-square distribution]'' table, the tabular value refers to the sum of the squared variables each divided by the expected outcomes. For the present example, this means

<math display="block">\chi^2 = \frac{25}{10} + \frac{4}{10} + \frac{1}{10} + \frac{4}{10} + \frac{0}{10} + \frac{100}{10}
= 13.4 </math>

This is the experimental result whose unlikeliness (with a fair die) we wish to estimate. 
{| class="wikitable" style="text-align:center;font-size:90%;line-height:0.9;"
|-
! rowspan="2"|Degrees<br /> of<br />freedom
! colspan="5"|Probability less than the critical value
|-
!    0.90 ||    ''0.95'' ||   0.975 ||    ''0.99'' ||   0.999
|-
!5
| 9.236 || 11.070 || 12.833 || 15.086 || 20.515
|}

The experimental sum of 13.4 is between the critical values of 97.5% and 99% significance or confidence ([[p-value]]). Specifically, getting 20 rolls of 6, when the expectation is only 10 such values, is unlikely with a fair die.