One-Way Tables

Prerequisites
Chi Square Distribution, Basic Concepts of Probability, Significance Testing

Learning Objectives

1. Describe what it means for there to be theoretically-expected frequencies
2. Compute expected frequencies
3. Compute Chi Square
4. Determine the degrees of freedom

The Chi Square distribution can be used to test whether observed data differ significantly from theoretically expectations. For example, for a fair six-sided die, the probability of of any given outcome on a single roll would be 1/6. The data in Table 1 were obtained by rolling a six-sided die 36 times. However, as can be seen in Table 1, some outcomes occurred more frequently than others. For example a "3" came up nine times whereas a "4" came up only two times. Are these data consistent with the hypothesis that the die is a fair die? Naturally, we do not expect the sample frequencies of the six possible outcomes throws to be the same since chance differences will occur. So, the finding that the frequencies differ does not mean that the die is not fair. One way to test whether the die is fair is to conduct a significance test. The null hypothesis is that the die is fair. This hypothesis is tested by computing the probability of obtaining frequencies as discrepant or more discrepant from a uniform distribution of frequencies as obtained in the sample. If this probability is sufficiently low, then the null hypothesis that the die is fair can be rejected.

Table 1. Outcome Frequencies
from a Six-Sided Die
 Outcome Frequency 1 2 3 4 5 6 8 5 9 2 7 5

The first step in conducting the significance test is to compute the expected frequency for each outcome given that the null hypothesis is true. For example, the expected frequency of a "1" is 6 since the probability of a "1" coming up is 1/6 and there were a total of 36 rolls of the die.

Expected frequency = (1/6)(36) = 6

Note that the expected frequencies are expected only in a theoretical sense. We do not really "expect" the observed frequencies to match the "expected frequencies" exactly.

The calculation continues as follows. Letting E be the expected frequency of an outcome and O be the observed frequency of that outcome, compute

for each outcome. Table 2 shows these calculations.

Next we add up all the values in Column 4 of Table 2.

This sampling distribution of

is approximately distributed as Chi Square on k-1 degrees of freedom where k is the number of categories. Therefore, for this problem the test statistic is

which means the value of Chi Square with 5 degrees of freedom is 5.333.

From a Chi Square calculator it can be determined that the probability of a Chi Square of 5.333 or larger is 0.377. Therefore, the null hypothesis that the die is fair cannot be rejected.

This Chi Square test can also be used to test other deviations between expected and observed frequencies. The following example shows a test of whether the variable "University GPA" in the SAT and College GPA case study is normally distributed.

The second column of Table 3 shows the proportions of a normal distribution falling between various limits. The expected frequencies (E) are calculated by multiplying the number of scores (105) by the proportion. The final column shows the observed number of scores in each range. It is clear that the observed frequencies vary greatly from the expected frequencies. Note that if the distribution were normal then there would have been only about 35 scores between -1 and 0 whereas 60 were observed.

Table 3. Expected and Observed Scores for 105 University GPA Scores.
 Range Proportion E O Above 1 0.159 16.695 19 0 to 1 0.341 35.805 17 -1 to 0 0.341 35.805 60 Below -1 0.159 16.695 9

The test of whether the observed scores deviate significantly from the expected is computed using the familiar calculation.

The subscript "3" means there are three degrees of freedom. As before, the degrees of freedom is the number of outcomes, which is four in this example. The Chi Square distribution calculator shows that p < 0.001 for this Chi Square. Therefore, the null hypothesis that the scores are normally distributed can be rejected.