Why does a chi square test work?

Answer 1

A very superficial argument goes like this:

You have a null hypothesis under which your variable has some distribution. On the basis of this distribution you expect certain values (frequencies) in certain intervals. The intervals may be numeric or categoric. But what you observe are different values. You could look at the differences between your observed and expected values but then, in total, they would all cancel out. So you look at their squares. Also, an observed value of 15 where you expected 10 (difference = 5) is, relatively speaking, much bigger than an observed value of 1005 where you were expecting 1000 (diff still = 5). So you divide by the expected value.

Thus, for each interval you have (O-E)2/E. You add all these together and that is your chi-square test statistic. Call it C.

If your data are consistent with the null hypothesis, then the observed values will be close to the expected values so that the absolute value of (O-E) and therefore its square will be small. So under the null hypothesis, the test statistic will be small.

If C is small, the likelihood is that the observations are consistent with the null hypothesis. And in that case you accept the null hypothesis. As C gets larger, the chance of observing that large a value (or larger) when the null hypothesis is true decreases. Finally, for really large values of C, the chances of getting that big a value (or bigger), still under the null hypothesis, are so smaller than some pre-determined limit that you set - for example less than 5% for 95% confidence or 1% for 99% confidence etc. At that stage you decide that there is so little chance that the data are cnsistent with the null hypothesis that you must reject it and accept the alternative.

Rather than calculate the probability of observing a value of C or larger, you would look up tables of critical values of C at the 5%, 1% etc levels.

Finally, a word about degrees of freedom. If the data are classified one-way into n categories, the sum of the n expected values and the n observed values is the same. So, once you have n-1 of these the nth is determined. So you only have n-1 degrees of freedom. Similar arguments apply to 2-way, 3-way etc classifications.

For more detail I suggest you get hold of a decent textbook.

Actually