What a Conditional Distribution by Row Describes
In Joint Relative Frequencies, you compared a cell count with the grand total to describe a share of the entire sample. A different question asks about a share within one particular group: for example, “What percent of seniors play a sport?” The relevant total is now the number of seniors, not the number of everyone surveyed.
When rows of a two-way table represent groups and columns represent possible outcomes or categories, a conditional distribution by row describes the column categories for one row group at a time. “Conditional” signals that the group named by the row is already specified. A row percentage is the percentage of individuals in that row who fall into a particular column category.
The denominator follows the wording of the question. In “What percent of seniors play a sport?”, “of seniors” identifies the group being considered. Find the count of seniors who play a sport, then divide by the total number of seniors. The resulting percentage describes seniors in the table, not all students and not all students who play a sport.
Calculate a Row Percentage
Consider a fictional survey of students who were each classified by grade and whether they play a school or community sport. The table contains counts. The row totals tell how many students are in each grade.
| Grade | Plays a sport | Does not play a sport | Row total |
|---|---|---|---|
| Grade 9 | 54 | 36 | 90 |
| Grade 10 | 63 | 42 | 105 |
| Grade 11 | 48 | 52 | 100 |
| Grade 12 | 84 | 36 | 120 |
| Column total | 249 | 166 | 415 |
There are 84 seniors who play a sport and 120 seniors altogether. Divide the cell count by the Grade 12 row total:
In this fictional survey, 70% of the Grade 12 students surveyed played a sport. The denominator is 120 because the question asks about the percentage of seniors. If we instead divided 84 by the grand total of 415, we would find the percentage of all surveyed students who were seniors who played a sport. That is a joint relative frequency and answers a different question.
To find a full conditional distribution by row, repeat the calculation for every column category in each row. For example, the Grade 12 percentage who did not play a sport is \(36/120 \times 100\%=30\%\). The two percentages in that row add to 100%: they account for all seniors in the table.
Compare Conditional Distributions Across Rows
Once row percentages are calculated, you can compare how the column categories are distributed across row groups. In the sports table, 48 of the 100 Grade 11 students played a sport, so the Grade 11 row percentage is \(48/100 \times 100\%=48\%\). For Grade 12, it is 70%. The difference is \(70\%-48\%=22\) percentage points. In this sample, the percentage who played a sport was 22 percentage points higher among Grade 12 students than among Grade 11 students.
When comparing percentages, say which groups and which outcome you mean. A difference between two percentages is usually stated in percentage points, not as a difference in percent. Here, 70% is 22 percentage points higher than 48%. This describes the sample; by itself, it does not establish why the groups differ or show that grade level causes sports participation.
Worked Examples: Conditional Distributions by Row
Worked Example: What Percent of Seniors Play a Sport?
Use the fictional grade-and-sports table above. Find the percentage of surveyed seniors who play a sport, then compare it with the percentage of surveyed Grade 11 students who play a sport.
Identify the groups and counts: The Grade 12 row contains 84 students who play a sport out of 120 seniors. The Grade 11 row contains 48 students who play a sport out of 100 Grade 11 students.
Calculate the row percentages:
Compare: \(70\%-48\%=22\) percentage points. In this survey, 70% of seniors played a sport, compared with 48% of Grade 11 students, a difference of 22 percentage points. Each percentage uses its own grade’s row total.
The denominator matters: using 415 for both calculations would describe each cell as a share of the entire sample, not the share within its grade. The question asks for a percentage among a specified grade, so each grade’s row total is appropriate.
Worked Example: Complete and Compare Row Distributions
A fictional survey asks 180 residents how they primarily access a community media service. The table classifies each resident by home type and response. Calculate the conditional distribution of access method for each home type, then compare the streaming percentages.
| Home type | Streaming | Downloading | Neither | Row total |
|---|---|---|---|---|
| Apartment | 45 | 15 | 30 | 90 |
| Detached house | 24 | 36 | 30 | 90 |
Calculate the apartment row: Each cell is divided by the apartment row total, 90.
The apartment percentages total \(50\%+16.7\%+33.3\%=100\%\) after rounding.
Calculate the detached-house row: Each cell is divided by that row’s total, also 90.
These percentages total \(26.7\%+40\%+33.3\%=100\%\) after rounding. In this sample, 50% of apartment residents primarily streamed the service, compared with about 26.7% of detached-house residents. The apartment percentage was about \(50\%-26.7\%=23.3\) percentage points higher. This comparison describes the residents surveyed; the table alone does not explain the difference.
Worked Example: Four Steps for a Row-Percentage Question
A fictional community survey records whether residents living near or far from a new walking path visited it during the past month. Find and compare the percentage who visited in each distance group.
| Distance from path | Visited | Did not visit | Row total |
|---|---|---|---|
| Near | 52 | 28 | 80 |
| Far | 45 | 75 | 120 |
We want to compare the percentage of residents who visited the path among those living near it and those living far from it.
For each distance group, divide the “Visited” cell count by that group’s row total and multiply by 100%. The row total represents all surveyed residents in that distance group.
Near: \(52/80 \times 100\%=65\%\). Far: \(45/120 \times 100\%=37.5\%\). The difference is \(65\%-37.5\%=27.5\) percentage points.
In this fictional survey, 65% of residents living near the path visited it during the past month, compared with 37.5% of residents living far from it. The near-group percentage was 27.5 percentage points higher.
As a check, the other row percentages are \(28/80 \times 100\%=35\%\) for residents living near the path who did not visit, and \(75/120 \times 100\%=62.5\%\) for those living far away who did not visit. Each row totals 100%. The comparison describes these surveyed residents; it does not show that distance alone caused the difference in visits.
Choose the Denominator and Read the Question Carefully
The phrase “among [row group]” or “of [row group]” is a useful clue that the question calls for a row percentage. Identify the group named in the question first. Then use the total for that row as the denominator. For example, “What percentage of detached-house residents primarily download?” uses the detached-house row total. The numerator is the count in the Downloading cell of that row.
Find the group described by words such as “of seniors,” “among apartment residents,” or “for residents living near the path.”
Locate the count in the column category named by the question within that row.
Calculate cell count ÷ row total and multiply by 100% if the question asks for a percentage.
Name the row group and column category in your answer. When comparing two percentages, state the difference in percentage points.
A row percentage is not the same as a joint relative frequency, introduced in the previous tutorial. A joint relative frequency uses the grand total and describes a share of the entire sample. A row percentage uses a row total and describes a share within one row group. The same cell count can therefore produce different percentages depending on the question and denominator.
For example, the 84 seniors who play a sport are \(84/415\), or about 20.2%, of all students in the sports table. They are \(84/120\), or 70%, of the seniors. Both calculations use the same cell, but they answer different questions because they use different denominators.
If you calculate all percentages for every row, each row should add to 100% because the column categories account for everyone in that row. A displayed total such as 99.9% or 100.1% may result from rounding. The percentages down a column generally do not need to add to 100%, because each row percentage uses a different row total.
Common Mistakes and AP Exam Tips
- Using the grand total instead of the row total. This gives a joint relative frequency, not a percentage within the row group. For “what percent of seniors,” divide by the total number of seniors.
- Dividing by the column total. A column total counts everyone in that outcome category across the table. It does not represent the row group named by a question such as “among seniors.”
- Leaving out the condition in the interpretation. “70% play a sport” is incomplete if the table includes several grades. A clear answer says, “70% of the Grade 12 students surveyed played a sport.”
- Comparing counts instead of percentages. Groups may have different row totals. Compare the within-row percentages when the question asks how common a category is in each group.
- Calling a percentage-point difference a percent difference. Subtract the percentages to get a percentage-point difference. Here, \(70\%-48\%=22\) percentage points.
- Assuming a difference explains its cause. A two-way table can describe how categories vary across groups, but a difference in row percentages alone does not show that group membership caused an outcome.
- Expecting every row percentage to be identical. Each percentage has its own numerator and row denominator. Calculate each row separately, and use the fact that a complete row should total 100% as a check.
For a strong written response, show the cell count divided by the correct row total, report the percentage with sensible rounding, and interpret it in context. If comparing groups, include both percentages and the difference in percentage points.
Check Your Understanding
Use the grade-and-sports table in this tutorial. Show each calculation and include the group named by the denominator in your interpretation.
- What percentage of Grade 9 students surveyed played a sport?
- What percentage of Grade 11 students surveyed did not play a sport?
- What is the conditional distribution by row for Grade 10 students? Check that its percentages add to 100%.
- Compare the percentage of Grade 12 students who played a sport with the percentage of Grade 11 students who played a sport. State the difference in percentage points.
- Explain why dividing the Grade 12 students who played a sport by 415 would not answer the question, “What percentage of seniors played a sport?”