Questions: Frequency Distributions and Contingency Tables

Question 1

In a survey, 40 out of 200 total respondents are women who prefer Brand A. There are 80 women total. What is the conditional frequency of preferring Brand A, given the respondent is a woman?

• A. 0.20 (40 out of 200 total respondents)
• B. 0.50 (40 out of 80 women)
• C. 0.40 (80 out of 200 total respondents)
• D. 0.33 (40 out of 120 non-women)

Answer: (B)

Conditional frequency restricts the denominator to the group you're conditioning on. Given the respondent is a woman, the relevant pool is the 80 women — not all 200 respondents. So the conditional frequency is 40/80 = 0.50. Option A (0.20) is the joint frequency: the proportion of the entire sample that is both female AND prefers Brand A. Option C (0.40) is the marginal frequency of women. Mixing up joint and conditional frequencies is the most common error in contingency table analysis.

Question 2

School A reports 60 students prefer online learning; School B reports 90 students prefer online learning. A student concludes School B has stronger preference for online learning. What critical information is missing?

• A. The specific online platforms used at each school
• B. The total number of students surveyed at each school, needed to compute relative frequencies
• C. Whether both surveys were conducted during the same semester
• D. The grade levels of the students surveyed

Answer: (B)

Raw frequency counts are uninterpretable without the total sample size. If School A surveyed 80 students (75% prefer online) and School B surveyed 900 (10% prefer online), School A actually has the stronger preference. Relative frequency — count divided by total — is the correct metric for comparison across samples of different sizes. This is one of the core lessons of frequency distributions: always normalize before comparing.

Question 3

In a contingency table, the marginal frequencies are the individual cell counts for each combination of categories.

• T. True
• F. False

Answer: False

Marginal frequencies are the row totals and column totals that appear at the 'margins' of the table. They show the distribution of each variable on its own, ignoring the other. The individual cell counts are called joint frequencies — they show how often each specific combination of both variables occurs. Marginal frequencies are obtained by summing across rows or columns; joint frequencies are the raw cells.

Question 4

A frequency distribution can be constructed for discrete quantitative data as well as categorical data.

• T. True
• F. False

Answer: True

Frequency distributions work for any type of data where you can enumerate values and count occurrences. For discrete quantitative data (like the number of siblings, or a 1-10 rating scale), you list each possible value and count how often it appears — exactly as with categorical data. Continuous quantitative data requires grouping into bins (class intervals) first, but the resulting grouped frequency distribution is equally valid.

Question 5

What is the difference between a joint frequency and a conditional frequency in a contingency table, and why does this distinction matter for detecting association between variables?

Model answer: A joint frequency is the count (or proportion) for a specific combination of both variables — e.g., the proportion of all respondents who are female AND prefer Brand A. A conditional frequency restricts the denominator to one category — e.g., among women only, the proportion who prefer Brand A. The distinction matters because association is detected by comparing conditional frequencies across groups: if the conditional distribution of Brand preference looks the same for men and women, the variables are independent. If it differs, there is an association. Joint frequencies alone can't reveal this because they're confounded by the marginal distributions.

The chi-square test formalizes exactly this comparison of conditional frequencies. If you only report joint frequencies, you cannot tell whether an observed pattern reflects a real association or merely the fact that one group is larger. Conditional frequencies control for group size and expose the genuine relationship (or lack thereof) between the two variables.