Chi-squared test

Definition

The chi-squared test is a statistical method used to determine if there is a significant association between categorical variables. It compares the observed frequencies of events in a contingency table with the frequencies that would be expected if the variables were independent. By analyzing these frequencies, it helps to identify whether the relationship between the variables is stronger than would be expected by chance.

5 Must Know Facts For Your Next Test

1. The chi-squared test calculates a statistic that follows a chi-squared distribution under the null hypothesis, allowing for inference about the relationship between categorical variables.
2. A high chi-squared statistic indicates that the observed frequencies differ significantly from expected frequencies, suggesting a potential association between the variables.
3. The significance level (usually set at 0.05) determines whether to reject or fail to reject the null hypothesis based on the chi-squared statistic and its corresponding p-value.
4. The test can be applied in two main forms: the chi-squared test for independence and the chi-squared goodness-of-fit test, each serving different purposes in hypothesis testing.
5. It's important to ensure that expected frequencies are sufficiently large (usually at least 5) to maintain the validity of the chi-squared approximation.

Review Questions

• How does the chi-squared test help assess the independence of two categorical variables?
  • The chi-squared test assesses independence by comparing observed counts in a contingency table against expected counts under the assumption that the two variables are independent. If there is a significant difference between these counts, it suggests that knowledge of one variable provides information about the other, thus indicating dependence. This is crucial in identifying whether relationships exist beyond what would occur by chance.
• In what scenarios would you choose to use a chi-squared goodness-of-fit test over a chi-squared test for independence?
  • A chi-squared goodness-of-fit test is used when you want to determine how well an observed frequency distribution fits an expected distribution for a single categorical variable. In contrast, a chi-squared test for independence examines the relationship between two categorical variables. Therefore, you would use the goodness-of-fit when checking if a sample fits a theoretical distribution, while you would use the independence test when exploring relationships between different groups.
• Evaluate how assumptions related to sample size and expected frequencies impact the results of a chi-squared test.
  • Assumptions about sample size and expected frequencies are critical for ensuring valid results from a chi-squared test. If the sample size is too small or if many expected frequencies are below 5, this can lead to inaccurate estimates of statistical significance. Such violations may cause the chi-squared distribution to not approximate correctly, leading to misleading conclusions about independence or associations between variables. Properly meeting these assumptions ensures more reliable outcomes from hypothesis testing.

Related terms

Contingency Table: A table used to display the frequency distribution of variables, showing how the categories of one variable relate to those of another.

Null Hypothesis:

A statement that there is no effect or no relationship between variables, which is tested against the alternative hypothesis in statistical analysis.

Degrees of Freedom:

The number of independent values or quantities which can be assigned to a statistical distribution, often calculated as the number of categories minus one.