9231 · 4.3
χ²-tests
The χ²-test is a way to measure the 'surprise' level between what you actually see (observed data) and what you thought you'd see (expected data). If the surprise is big enough, you reject your initial theory.
Need to know
What you need to know
- The null hypothesis, $H_0$, states that the data follows the specified distribution.
- The alternative hypothesis, $H_1$, states that the data does not follow the specified distribution.
- The test is always one-tailed, as we are interested in whether the difference between O and E is significantly large. We therefore always look at the upper tail of the χ² distribution.
Explanation
Observed vs. Expected: The Showdown
- First, state your initial belief (the null hypothesis) and calculate the frequencies you'd expect to see if that belief were true.
- Next, use the χ² formula, $\chi^2 = \sum \frac{(O - E)^2}{E}$, to calculate a single test statistic that summarises the total difference between your observed (O) and expected (E) counts.
- Then, determine the 'degrees of freedom' (a measure of how many values can vary) and look up the corresponding critical value in a table for your chosen significance level.
- Finally, compare your calculated χ² statistic with the critical value. If your statistic is larger, you reject your initial belief; otherwise, you don't have enough evidence to reject it.