The chi-square test is a fundamental statistical tool that helps us determine whether there's a significant difference between what we expect to happen and what actually happens in our data. It's particularly useful when working with categorical data, where we want to test if observed frequencies differ significantly from expected frequencies. The test uses a simple but powerful formula that calculates the chi-square statistic by summing the squared differences between observed and expected values, divided by the expected values.
Expected frequencies represent what we would predict to happen if there's no relationship between our variables, assuming independence. Observed frequencies are what actually occurred in our data collection. For example, if we flip a fair coin 100 times, we expect 50 heads and 50 tails. But we might actually observe 45 heads and 55 tails. The chi-square test helps us determine if this difference is statistically significant or just due to random variation.
The chi-square calculation follows three clear steps. First, we calculate the squared difference between observed and expected values for each category. For our coin example, both heads and tails have a squared difference of 25. Second, we divide each squared difference by its expected value, giving us 0.5 for each category. Finally, we sum all these values to get our chi-square statistic of 1.0. We square the differences to eliminate negative values and to give more weight to larger deviations from what we expected.
Now let's apply our chi-square knowledge to a practical case study. A company wants to determine if customer product preferences are independent of age groups. We have three age categories: young, middle-aged, and senior customers, and three products: A, B, and C. Our null hypothesis states that product preference is independent of age group, meaning there's no relationship between them. The alternative hypothesis suggests that age does influence product preference. We'll use a significance level of 0.05 and analyze the data in this 3 by 3 contingency table with 200 total customers surveyed.
Let's work through the complete chi-square calculation. First, we calculate expected frequencies using the formula: row total times column total divided by grand total. For example, the expected frequency for young customers choosing product A is 60 times 60 divided by 200, which equals 18. We need 4 degrees of freedom, calculated as 3 minus 1 times 3 minus 1. The critical value at alpha equals 0.05 with 4 degrees of freedom is 9.488. After calculating all chi-square components and summing them, we get a chi-square statistic of 6.944. Since 6.944 is less than 9.488, we fail to reject the null hypothesis.