The chi-square test is a versatile statistical method used to determine if there is a significant association between two categorical variables in a sample. It can be employed in various fields including biology, sociology, and market research. This article will guide you through the steps of calculating the chi-square test statistic using a contingency table.

Step 1: Set Up Your Contingency Table

A contingency table, also known as a cross-tabulation or crosstab, is used to display the relationship between two categorical variables. It consists of rows and columns representing each possible value of each variable, with the cell frequencies indicating how many observations fall into each category pairing.

Here’s an example of a 2×2 contingency table:

Variable B

Category 1   Category 2   Row Total

Variable A

Category 1         O11             O12           R1

Category 2         O21             O22           R2

Column Total      C1              C2            N (sample size)

Step 2: Calculate Expected Values

In order to calculate the chi-square test statistic, you must first find the expected values for each cell in your contingency table. Expected values are calculated using the following formula:

Eij = (Ri * Cj) / N

where Eij is the expected value for cell i,j, Ri is the row total for row i, Cj is the column total for column j, and N is the total number of observations.

For example:

E11 = (R1 * C1) / N

E12 = (R1 * C2) / N

E21 = (R2 * C1) / N

E22 = (R2 * C2) / N

Step 3: Compute Chi-Square Test Statistic

Now that you have your observed values and expected values, you can calculate the chi-square test statistic using the following formula:

χ² = Σ [(Oij – Eij)² / Eij]

where χ² represents the chi-square test statistic and Oij are the observed values in each cell.

For example:

χ² = ((O11-E11)² / E11) + ((O12-E12)² / E12) + ((O21-E21)² / E21) + ((O22-E22)² / E22)

Step 4: Determine Degrees of Freedom

To interpret the significance of your chi-square test statistic, you must first determine the degrees of freedom, calculated as:

df = (number of rows – 1) * (number of columns – 1)

For a 2×2 contingency table, this would result in:

df = (2-1) * (2-1) = 1

Step 5: Compare to Chi-Square Distribution

Finally, compare your chi-square test statistic to the critical value from the chi-square distribution table with your calculated degrees of freedom. If your test statistic is greater than the critical value at a pre-determined significance level (e.g., 0.05 or 0.01), you can reject the null hypothesis that there is no association between the two categorical variables.

In summary, calculating the chi-square test statistic involves setting up a contingency table, computing expected values, finding the difference between observed and expected values, squaring those differences, dividing by expected values and summing them up. Don’t forget to determine degrees of freedom and check against a chi-square distribution table to assess statistical significance.