Conditional Relative Frequency Examples in Statistics
Start with the recap, study the fully worked examples, then use the practice problems to
check your understanding of Conditional Relative Frequency.
This page combines explanation, solved examples, and follow-up practice so you can move
from recognition to confident problem-solving in Statistics.
Concept Recap
Conditional relative frequency is the proportion of cases in one group that also belong to another category, measured within a chosen row or column total of a two-way table. Joint and marginal relative frequencies describe the cell shares and row or column totals that support this calculation.
A two-way table becomes much more informative once you stop reading raw counts and start reading percentages within the relevant group.
Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:Conditional Relative Frequency asks whether the same cases connect two variables or groups in a pattern that can be described carefully.
Common stuck point:Students often know a procedure related to conditional relative frequency but skip the recognition step: Am I studying a relationship between variables, and have I separated association from causation? That leads to a calculation or graph that looks reasonable but answers a different question.
Sense of Study hint:Ask: Am I studying a relationship between variables, and have I separated association from causation?
Common Mistakes to Watch For
Before you work through the examples, skim the mistake guide so you know which shortcuts and
sign errors to avoid.
Using the same table (Teen: 45/5; Adult: 80/20), find the conditional relative frequency of 'Yes' among adults and compare.Conditional relative frequency of phone ownership among adults
Answer
0.8(less than 0.9)
First step
1
Adult row total: 80+20=100.
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Setup·Key insight·Why it works·Common pitfall·Connection
A two-way table has 200 cases total. Row 'A' has 80 cases, with 20 in the 'Yes' column. Find both the joint and conditional (within row A) relative frequencies of 'A and Yes'.
Example 3
medium
A 2x2 table has row totals 40 and 60. The 'Yes' column has 14 in row 1 and 48 in row 2. Find the within-row 'Yes' conditional frequencies.Find the within-row Yes conditional frequencies for each row
Example 4
hard
In a 2x3 table: Row A is 10,20,30; row B is 20,30,50. Find the row-conditional frequencies for each row.Find row-conditional frequencies for each row
Example 5
hard
In a 2x2 table the conditional rate of 'Yes' is 0.6 in row 1 and 0.6 in row 2. What does this say about an association?
Example 6
hard
Table: Treatment (T1/T2/T3) by Outcome (Cured/Not). T1: 36 cured of 60. T2: 54 cured of 90. T3: 30 cured of 50. Which treatment has the highest within-row cure rate?Which treatment has the highest within-row cure rate?
Example 7
hard
A diagnostic test has P(positive∣disease)=0.95 and P(positive∣no disease)=0.10. In a sample of 1000 with disease prevalence 5%, what is the within-column conditional rate of disease among positives?
Example 8
challenge
Two-way table by Age (Young/Old) and Outcome: Young row total 100, Old row total 300. Within-row 'Yes' rates are 0.40 and 0.20. What is the marginal proportion of 'Yes'?What is the marginal proportion of Yes across both age groups?
Example 9
challenge
In a 2x2 table, all four cell counts are a,b,c,d with row totals a+b and c+d. Show that 'independence' means a+ba=c+dc.
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
Table: Studied (Yes/No) by Passed (Yes/No). Studied-Yes: 18 passed, 2 failed. What fraction of students who studied passed?Find the fraction of students who studied that passed
Example 2
easy
In a row with cells 30 and 10 (row total 40), what is the conditional relative frequency of the first cell within the row?Conditional relative frequency of the first cell within the row
Example 3
easy
Column 'Yes' has cells 15 (top) and 5 (bottom), column total 20. What is the conditional relative frequency of the top cell within the column?Conditional relative frequency of the top cell within column Yes
Example 4
easy
Among 50 dog owners, 20 also own a cat. What is the conditional relative frequency of cat ownership among dog owners?
Example 5
easy
A row has cells 6, 9, 5 (row total 20). What is the conditional relative frequency of the middle cell within the row?Conditional relative frequency of the middle cell within the row
Example 6
easy
Within a row, the conditional relative frequencies are 0.3, 0.5, and the rest. What is the last one?
Example 7
easy
A column total is 25 and one cell in it is 10. What is that cell's conditional relative frequency within the column?Cell 10 within a column of total 25
Example 8
easy
Why is a conditional relative frequency different from a joint relative frequency for the same cell?
Example 9
medium
Table: Treatment (A/B) by Outcome (Cured/Not). A: 36 cured, 12 not. B: 20 cured, 12 not. Which treatment has the higher cure rate among its own patients?Which treatment has the higher cure rate among its own patients?
Example 10
medium
Table: Gender (M/F) by Choice (X/Y). M: 12 X, 18 Y. F: 21 X, 9 Y. What proportion of those who chose X are female?What proportion of those who chose X are female?
Example 11
medium
Same table (M: 12 X, 18 Y; F: 21 X, 9 Y). What proportion of females chose Y?What proportion of females chose Y?
Example 12
medium
Table: Exercise (Yes/No) by Healthy (Yes/No). Exercise-Yes: 45 healthy, 5 not. Exercise-No: 20 healthy, 30 not. Compare the conditional healthy rate for exercisers vs non-exercisers.Compare the conditional healthy rate for exercisers vs non-exercisers
Example 13
medium
A two-way table shows the conditional relative frequency of success is 0.8 within a row whose total is 35. How many successes are in that row?
Example 14
medium
Table: Region (N/S) by Preference (Tea/Coffee). N: 24 tea, 16 coffee. S: 9 tea, 51 coffee. In which region is tea preference more common (as a within-region rate)?In which region is tea preference more common?
Example 15
medium
Why might the conditional relative frequency of 'female given chose X' differ from 'chose X given female'?
Example 16
medium
In a row of totals 50, the success conditional rate is reported as 0.46. Explain why the success count must be a whole number and give it.
Example 17
medium
Table: Age (Young/Old) by Drink (Tea/Coffee). Young: 12 tea, 28 coffee. Old: 27 tea, 13 coffee. What is the tea rate among the Young?What is the tea rate among the Young?
Example 18
challenge
Table: Group (A/B) by Result (Win/Lose). A: win rate 0.7 over 40; B: win rate 0.4 over 60. Pooling both groups, what is the conditional win rate given Group B, and what is the overall win rate? Explain why they differ.Group B conditional win rate vs overall win rate
Example 19
challenge
A 2×2 table has row totals r and 100−r. The conditional success rate is 0.8 in row 1 and 0.5 in row 2, and the overall success rate is 0.65. Find r.
Example 20
challenge
Two variables are 'independent' in a table when every conditional row distribution equals the marginal column distribution. A table has column marginals Tea =0.4, Coffee =0.6. If row N has 50 people and is independent, how many in row N prefer tea?
Example 21
easy
In a row with cells 12 and 8 (row total 20), what is the conditional relative frequency of the first cell within the row?Conditional relative frequency of the first cell within the row
Example 22
easy
Out of 80 shoppers, 20 bought milk. What is the conditional relative frequency of buying milk among the 80 shoppers?Conditional relative frequency of buying milk among the 80 shoppers
Example 23
easy
A row reads: 5, 10, 15. What is the conditional relative frequency of the largest cell within the row?Conditional relative frequency of the largest cell within the row
Example 24
easy
In a 2x2 table, the cell for 'Yes/Yes' is 24 and the column 'Yes' total is 40. What is the conditional relative frequency within column 'Yes'?Conditional relative frequency within column Yes
Example 25
easy
A column has cells 9, 3, 8 (column total 20). What is the conditional relative frequency of the first cell within the column?Conditional relative frequency of the first cell within the column
Example 26
medium
Table: Has Phone (Yes/No) by Age (Teen/Adult). Teen: 45 yes, 5 no. Adult: 80 yes, 20 no. What is the conditional relative frequency of 'Yes' among teens?Conditional relative frequency of phone ownership among teens
Example 27
medium
Table: Pet (Cat/Dog) by Likes Walks (Yes/No). Cat: 5 yes, 25 no. Dog: 35 yes, 5 no. What fraction of those who like walks are dog owners?What fraction of those who like walks are dog owners?
Example 28
medium
Survey of 100: 40 adults, 60 teens. Of adults, 30 own a car; of teens, 6 own a car. Conditional probability of car ownership among adults?Conditional probability of car ownership among adults
Example 29
medium
Table: Sport (Run/Swim) by Injured (Yes/No). Run: 15 yes, 35 no. Swim: 4 yes, 46 no. Compare injury rates within each sport.Compare injury rates within each sport
Example 30
medium
Why might raw counts in two-way tables be misleading for comparing groups of different sizes?
Example 31
hard
A study finds the within-row positive rate for women is 0.40, and for men is 0.20. Without more data, can you conclude women outnumber men in positives?
Example 32
hard
Conditional rates differ by row: 0.30,0.50,0.70. What does this pattern suggest about the row variable's relationship to the 'Yes' column?
Example 33
hard
In a 2x2 table, the marginal proportion of 'Yes' is 0.50. Within row 1 the conditional 'Yes' rate is 0.70. What must the within-row 'Yes' rate in row 2 be, given equal row totals?Find the missing within-row Yes rate for Row 2 given equal row totals and marginal rate 0.50
Example 34
hard
A 2x2 table has joint relative frequencies summing to 1 across all four cells. Why might the conditional frequencies within a row sum to 1 but the conditional frequencies across all four cells NOT sum to 1?
Example 35
challenge
A college reports its conditional admission rate for women is 0.40 and for men 0.50. Across two departments separately, women's rates exceed men's. What paradox is this?