Goodness-of-fit Test

The idea behind the chi-square goodness-of-fit test is to see if the sample comes from the population with the claimed distribution. Another way of looking at that is to ask if the frequency distribution fits a specific pattern.

Two values are involved, an observed value, which is the frequency of a category from a sample, and the expected frequency, which is calculated based upon the claimed distribution. The derivation of the formula is very similar to that of the variance which was done earlier.

The idea is that if the observed frequency is really close to the claimed (expected) frequency, then the square of the deviations will be small. The square of the deviation is divided by the expected frequency to weight frequencies. A difference of 10 may be very significant if 12 was the expected frequency, but a difference of 10 isn't very significant at all if the expected frequency was 1200.

If the sum of these weighted squared deviations is small, the observed frequencies are close to the expected frequencies and there would be no reason to reject the claim that it came from that distribution. Only when the sum is large is the a reason to question the distribution. Therefore, the chi-square goodness-of-fit test is always a right tail test.

The test statistic has a chi-square distribution when the following assumptions are met

The following are properties of the goodness-of-fit test

Interpreting the Claim

There are four ways you might be given a claim.

  1. The values occur with equal frequency. Other words for this are "uniform", "no preference", or "no difference". To find the expected frequencies, total the observed frequencies and divide by the number of categories. This quotient is the expected frequency for each category.
  2. Specific proportions or probabilities are given. To find the expected frequencies, multiply the total of the observed frequencies by the probability for each category.
  3. The expected frequencies are given to you. In this case, you don't have to do anything.
  4. A specific distribution is claimed. For example, "The data is normally distributed". To work a problem like this, you need to group the data and find the frequency for each class. Then, find the probability of being within that class by converting the scores to z-scores and looking up the probabilities. Finally, multiply the probabilities by the total observed frequency. (It's not really as bad as it sounds).

Using the TI-82

You can use the lists on the TI-82 to perform the chi-square goodness-of-fit test.

Table of contents
James Jones