The chi-square is one of the most popular statistics because it is easy to calculate and interpret. There are two kinds of chi-square tests. The first is called a one-way analysis, and the second is called a two-way analysis. The purpose of both is to determine whether the observed frequencies (counts) markedly differ from the frequencies that we would expect by chance.

The observed cell frequencies are organized in rows and columns like a spreadsheet. This table of observed cell frequencies is called a contingency table, and the chi-square test if part of a contingency table analysis.

The chi-square statistic is the sum of the contributions from each of the individual cells. Every cell in a table contributes something to the overall chi-square statistic. If a given cell differs markedly from the expected frequency, then the contribution of that cell to the overall chi-square is large. If a cell is close to the expected frequency for that cell, then the contribution of that cell to the overall chi-square is low. A large chi-square statistic indicates that somewhere in the table, the observed frequencies differ markedly from the expected frequencies. It does not tell which cell (or cells) are causing the high chi-square...only that they are there. When a chi-square is high, you must visually examine the table to determine which cell(s) are responsible.

In order to help find the cause of a significant chi-square statistic, a standardized residual analysis is performed. The difference between an observed cell frequency and the expected cell frequency is called a residual. The residuals for all the cells can be standardized and a probability can be calculated for each cell residual. The probability indicates how "odd" that cell is. That is, how likely it is that you'd see the observed frequency of the cell by chance. Cells with the lowest probabilities are the likely cause of the significant chi-square. After clicking the submit button, the probabilities of the standardized residuals are shown in a table below the results. If you find an entire row or column with low probabilities, that row or column is "unique" when compared to the others and interpretation is relatively easy. However, if you find isolated cells with low probabilities, more careful interpretations must be entertained.

When there are exactly two rows and two columns, the chi-square statistic becomes inaccurate, and Yates correction for continuity will be applied. if any cell has an expected frequency less than 5 or the total N is less than 50, the result will be shown as the "Corrected Chi-square".

If there is only one column or one row (a one-way chi-square test), the degrees of freedom is the number of cells minus one. For a two-way chi-square (where there are both rows and columns), the degrees of freedom is the number or rows minus one times the number of columns minus one.

Using the chi-square statistic and its associated degrees of freedom, the software reports the probability that the differences between the observed and expected frequencies occurred by chance. Generally, a probability of .05 or less is considered to be a significant difference.

The software will create a table of rows and columns that allow you to enter the counts for each cell. After you've finished entering the data, the program will print the chi-square, degrees of freedom and probability of chance.

Use caution when interpreting the chi-square statistic if any of the cell frequencies are less than five. Also, use caution when the total for all cells is less than 50.

Example

A drug manufacturing company conducted a survey of customers. The research question is: Is there a significant relationship between packaging preference (size of the bottle purchased) and economic status? There were four packaging sizes: small, medium, large, and jumbo. Economic status was: lower, middle, and upper. The following data was collected.

You could make a statement like this: A two-way chi-square test was performed to examine the relationship between packaging preference and economic status. The probability of the chi-square was not significant at the .05 alpha level, X²(6)=9.743, p=.136. Thus, we conclude that there was no statistically significant relationship.