Chi-Square Test Cheat Sheet

The Chi-square test is a statistical tool used to analyze categorical data by comparing observed frequencies to expected frequencies, helping to determine relationships or goodness of fit.

Core Principles

• Applies to categorical variables with frequency counts.
• Compares observed frequencies to expected frequencies under a null hypothesis.
• Two main versions: goodness of fit and test of independence.
• Assumes independence of observations.
• Requires expected frequencies to be sufficiently large (general rule: > 5).

Action Steps

• Identify categorical variables and frequency counts.
• Formulate the null and alternative hypotheses.
• Calculate expected frequencies for each cell.
• Compute the Chi-square statistic using the formula.
• Determine degrees of freedom (number of categories - 1 for goodness of fit; (rows-1)*(cols-1) for independence).
• Compare the calculated Chi-square statistic to a critical value or use p-value.
• Make a decision: reject or fail to reject the null hypothesis.
• Interpret the results in the context of the problem.

Formulas

• $E_{ij} = \frac{R_i C_j}{n}$
• $\chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}}$

Key Terms

• Categorical Variable: A variable that can take on one of a limited, and usually fixed, number of possible values, representing types or categories.
• Observed Frequency: The actual count of occurrences for a specific category or cell in the data.
• Expected Frequency: The count of occurrences that would be expected in a category or cell if the null hypothesis were true.
• Goodness of Fit Test: Tests if the observed frequencies for a single categorical variable match the expected frequencies.
• Test of Independence: Tests if two categorical variables are independent of each other.
• Degrees of Freedom (df): The number of independent values that can vary in the data. For Chi-square, it depends on the number of categories or the dimensions of the contingency table.

Timeline

• 1900: Karl Pearson introduces the Chi-square test.
• Early 20th Century: Development of the Chi-square test of independence.
• Mid-20th Century: Widespread adoption and application in various fields like biology, sociology, and psychology.
• Late 20th Century: Refinement of assumptions and computational methods.
• 21st Century: Continued use with advancements in statistical software and big data analysis.

People

• Karl Pearson: Statistician who developed the Chi-square test.

Quiz

• What type of data is typically analyzed using a Chi-square test?: Categorical data
• The Chi-square test of independence is used to determine:: If there is a relationship between two categorical variables
• What is a key assumption for the Chi-square test?: Observations must be independent