How to Use a Chi Square
Two key questions in many types of research are whether two variables are correlated, and if so, the strength (or significance) of that relationship. Is there a significant correlation, for example, between gender or ethnicity and political affiliation? The chi-square test is a widely used method for measuring if a significant relationship exists between two nominal or categorical variables, such as gender and political affiliation.
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Instructions
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Begin with a hypothesis before you start your data analysis. A common hypothesis in much research is that there is no correlation between the two variables of interest. The chi (rhymes with "my") square test measures the level of deviance from a given hypothesis. The larger the chi-square statistic, the less well the hypothesis fits the data. For example, suppose we are looking at a set of data that asked 125 registered voters (65 women and 60 men) their political party affiliation (Democratic or Republican). Suppose we know from previous research that 55 percent of voters identified themselves as Democrats. Our working hypothesis is that this 55 percent will be evenly distributed between men and women.
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Calculate the expected values based on your hypothesized model of political affiliation by gender. Based on 125 voters, we expect that 55 percent (69 voters) will identify themselves as Democrats. By gender, we expect that 36 women and 33 men will express a preference for the Democratic Party, leaving 29 women and 27 men favoring the Republican Party. Organize your data in a 2-by-2 matrix (two rows and two columns). Let party affiliation be the column variables and gender be the row variables.
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Compare the actual values from your data with the expected values you estimated in Step 2. For this example, let's say that among the 65 women, 44 percent identified themselves as Democrats and 21 as Republicans, while 36 men claimed a Democratic affiliation and 24 preferred the Republican Party.
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Calculate the chi-square statistic, which is the sum of the squared differences between the observed and expected values (also known as the residuals), divided by the expected values. You will need this for the four possible combinations of gender and political affiliation specified in your model. If you're using a computer, many statistical and spreadsheet programs can calculate the chi-square statistic for you. In our example, the sum of squared differentials divided by expected values is 4.59.
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Determine whether the chi-square statistic you calculated in Step 4 is statistically significant. To do this, you need to know two things: the degrees of freedom and the significance level. Degrees of freedom is the number of rows in your table minus one, times the number of columns minus one. Significance level refers to the probability that the observed correlation could have occurred by chance alone. Many researchers prefer a .05 significance level, meaning there is only a 5 percent likelihood that the observed relationship is pure chance. In our example, we have only 1 degree of freedom. Using your statistics book (usually in the appendix), look up the chi-square value that corresponds to the significance level and degrees of freedom. For our example, the chi-square value for 1 degree of freedom and .05 significance level is 3.84. Our value of 4.59 is greater, meaning there is a statistically significant relationship between gender and political affiliation, with women being significantly more likely to identify themselves as Democrats.
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Tips & Warnings
Remember that statistical significance is not the same as substantive significance. Chi-square values indicate the probability that an observed relationship between two nominal variables could have occurred merely by chance.
