Introduction

The chi-square goodness-of-fit test, also known as Pearson’s chi-square goodness-of-fit test, is one of the most useful non-parametric one-sample tests and is widely applied in the social and psychological sciences. Its main purpose is to examine whether the distribution of observations in a categorical variable follows a known or hypothetical distribution. This means we compare the observed frequencies derived from our data with the expected frequencies based on a theoretical hypothesis or prior data. For example, we may test whether the proportion of men and women in our sample matches the proportion in the general population, or whether the percentage of men and women who support a specific political party corresponds to our expectations.

Equal and Unequal Expected Proportions

When conducting this test, it is necessary to specify whether the expected proportions are equal or unequal. In the case of equal proportions, we assume that each category includes the same number of cases. For instance, if we are examining gender, we may assume that men and women are distributed equally, at fifty percent each. In the case of unequal proportions, the expected frequencies are determined by a theory or hypothesis and differ from one category to another, such as when we expect seventy percent of a party’s supporters to be men and thirty percent to be women. This distinction is crucial because it affects both how we enter the parameters into SPSS and how we interpret the results.

Assumptions of the Chi-Square Goodness-of-Fit Test

For the test to be valid, the data must satisfy certain assumptions. First, the variable being examined must be categorical, which means it may be dichotomous, nominal, or ordinal. Examples include gender, medication use, nationality, occupation, or Likert scales measuring attitudes and opinions. Second, independence of observations must be ensured, meaning that each case in the sample is independent and not related to another. Third, the categories of the variable must be mutually exclusive, which means that each participant can be classified in only one category. Finally, there must be at least five expected frequencies in each category, otherwise the estimation of the test is not reliable. SPSS provides a relevant warning when this condition is not met.

Procedure in SPSS (with Equal Expected Proportions)

To carry out the test in SPSS when equal expected proportions are assumed, a specific procedure is followed. First, the data must be entered, ensuring that the categorical variable is correctly coded. Next, from the menu we select Analyze, then Nonparametric Tests, then Legacy Dialogs, and finally Chi-Square. We move the variable of interest into the Test Variable List field and specify that the expected frequencies are equal. After making these selections, we run the test by pressing OK, and the program displays the corresponding tables.

Interpretation of Results

The SPSS output includes tables that present the observed and expected frequencies for each category, as well as the value of the chi-square statistic and the p-value. If the p-value is less than the chosen level of significance, usually 0.05, then we reject the null hypothesis and conclude that there is a statistically significant difference between the observed and expected distributions. If the p-value is greater than 0.05, then we do not have evidence to reject the null hypothesis and consider that the two distributions do not differ substantially.

Conclusion

The chi-square goodness-of-fit test is a valuable tool that allows us to check whether our data agree with a theoretical or hypothetical distribution. Proper understanding and application require that certain assumptions be satisfied and that the correct procedure in SPSS is followed. The distinction between equal and unequal expected proportions is critical, as it affects data entry and interpretation. When the assumptions are met and the results are interpreted correctly, this test can provide important insights into the distribution of categorical data and support valid research conclusions.