Chi-Square

The Single-Sample Chi-Square

The chi-square formula is an expression of the ratio of observed frequencies (fo) of the variable categories compared to the expected frequencies (fe) of those same categories:

$${\mathrm{\chi }}^2=\frac{(fo-fe){}^2}{fe}$$

The observed frequency is the number of times the category actually appeared in the data whereas the expected frequency is the number of times the category was expected to appear assuming category membership is random. The chi-square tests whether or not the observed frequencies differ from random, leading to the claim that the observed differences are not due to chance.

To calculate the statistic, one must know the total number of observations (N) and the number of categories (sometimes referred to as cells due to the frequent use of tables to display frequency data) the variable could take (k). For example, a public relations researcher wants to determine whether or not the dominant color of a promotional flyer influences a person’s decision to read the flyer. The researcher creates four colored flyers: red, blue, green, and white. The researcher then collects data by observing 100 people and which flyer they chose to read. For each person, the researcher identified which color flyer was read, resulting in the observed frequencies for the variable categories. Those frequencies can be found in Table 1.

Table 1 Sample Flyer Data

Before calculating the chi-square statistic, one must determine the expected frequencies of the categories. Because the expected frequencies are based on chance, they are calculated by determining the probability of obtaining each category. The laws of chance predict that with four categories, each color will have a 25% chance of being chosen. To determine the expected frequency value, with regard to the sample size, one divides the number of observations by the number of possible

...