Chi-Squared Test Details

If we flip a coin 100 times and we get 60 heads and 40 tails, is the coin fair? The simplest way of testing this is using a z-score, which has the following formula.

z = \frac{x - np}{\sqrt{np(1-p)}}

We can then calculate the z-score as follows.

z = \frac{60 - (100)(0.5)}{\sqrt{(100)(0.5)(0.5)}} = \frac{10}{5} = 2

Using the normal curve (also known as a "bell curve"), we can use this z-score to make an educated guess as to whether the coin is fair or not.

What if we try to use a chi-squared test instead? The chi-squared statistic has the following formula.

\chi^{2} = \sum_{i} \frac{(O_{i} - E_{i})^2}{E_{i}}

For the coin flipping case, we get the following value.

\chi^{2} = \frac{(40 - 50)^2}{50} + \frac{(60 - 50)^2}{50} = 4

Notice how in this case, $\chi^2 = z^2$ . When the $\chi^2$ statistic has one degree of freedom, it is equivalent to the square of the $z$ statistic.

Let's verify this relationship with several examples. The table below shows various coin flip scenarios and compares $z^2$ with $\chi^2$ :

Pr(Heads)	Heads	Tails	z²	χ²
0.10	60	40	277.78	277.78
0.20	60	40	100.00	100.00
0.30	60	40	42.86	42.86
0.40	60	40	16.67	16.67
0.50	60	40	4.00	4.00
0.60	60	40	0.00	0.00
0.70	60	40	4.76	4.76
0.80	60	40	25.00	25.00
0.90	60	40	100.00	100.00

← Previous

Changelog

Block Longest Run Test