Conditional relative frequencies

When you reach for it

A two-way table counts one group of people or things by two questions at once. A made-up museum asked 80 visitors on their way out whether they took a guided tour and whether they bought something at the shop.

The four inner counts are the joint frequencies and the totals round the edge are the marginal frequencies (Joint and marginal frequencies). Now ask: “of the visitors who took a tour, what share bought something?” The answer is not 24 out of 80. The question’s first words narrow the whole down to the 32 tour visitors, and the answer is a share of them.

You reach for a conditional relative frequency whenever a question names a group and asks what share of that group has some other property, and whenever groups of different sizes have to be compared.

The procedure

Part 1: a cell over its row total or its column total

A conditional relative frequency is the count in one cell of a two-way table divided by the total of the row, or of the column, that the question picks out. That row or column is the condition: the group the question has narrowed down to.

This is the same part over whole as any share (Finding what percent one number is of another); what is new is only that the whole is a row or column total rather than the grand total. For the tour visitors, the group is the “took a tour” row, total 32, and those who also bought something are the cell 24:

So 0.75, or 75\%, of the tour visitors bought something.

Which total: read it from the wording. A conditional question always names its group, and the group’s total is what you divide by. Words such as “of”, “among”, “out of” and “what share of the” usually come just before the group.

• “Of the visitors who took a tour, what share bought something?” The group is the tour visitors, so divide by 32.
• “What share of the visitors who bought something took a tour?” The group is the buyers, so divide by 30.
• “Of those who play chess, what share are in the choir?” divides by the chess players, while “of those in the choir, what share play chess?” divides by the choir.

The cell is always the members of the group who also have the property asked about. When a question names no group, as in “what share of all the visitors took a tour and bought something?”, the whole is the grand total: \displaystyle \frac{24}{80} = 0.3. That is a share of the whole table, not a conditional one.

Shares may be written as fractions, decimals or percents. When a share does not come out exactly, this lesson rounds it to three decimal places and writes \approx.

Row-relative tables. Dividing every cell by its own row total gives a row-relative table, in which each row describes one group on its own scale. The tour row is \displaystyle \frac{24}{32} = 0.75 and \displaystyle \frac{8}{32} = 0.25; the no-tour row is \displaystyle \frac{6}{48} = 0.125 and \displaystyle \frac{42}{48} = 0.875; the Total row is \displaystyle \frac{30}{80} = 0.375 and \displaystyle \frac{50}{80} = 0.625.

Every row adds to 1, because the cells of a row add up to the row total. If a row holds counts a and b with total r = a + b, then

and the same works with three or more cells. In words: every tour visitor either bought something or did not. The columns of a row-relative table need not add to 1: 0.75 + 0.125 adds shares of two different groups.

Column-relative tables. Dividing every cell by its column total instead makes each column a group: the buyers and the non-buyers. The buyers’ column is \displaystyle \frac{24}{30} = 0.8 and \displaystyle \frac{6}{30} = 0.2; the non-buyers’ column is \displaystyle \frac{8}{50} = 0.16 and \displaystyle \frac{42}{50} = 0.84; the Total column is \displaystyle \frac{32}{80} = 0.4 and \displaystyle \frac{48}{80} = 0.6.

Now each column adds to 1, for the same reason. The entry 0.8 says that of the visitors who bought something, 80\% took a tour.

Part 2: two directions, fair comparisons, and working back

The two directions give different numbers. The cell “took a tour and bought something” holds 24 visitors. Over its row total it gives the share of tour visitors who bought something, \displaystyle \frac{24}{32} = 0.75. Over its column total it gives the share of buyers who took a tour, \displaystyle \frac{24}{30} = 0.8. The same 24 people, two different wholes, and two answers to two different questions.

The two agree only when the row total equals the column total, and they can be far apart. If a museum had only 4 tour visitors, all of whom bought something, and 100 buyers in all, then every tour visitor bought something, \displaystyle \frac{4}{4} = 1, while only \displaystyle \frac{4}{100} = 0.04 of the buyers took a tour. So a sentence has to keep its group. “75\% of tour visitors bought something” and “75\% of buyers took a tour” are different claims, and for Table 1 only the first is true.

Comparing groups of different sizes. A count mixes how common something is inside a group with how big the group is. In Table 1, 24 tour visitors and 6 others bought something, but out of wholes of 32 and 48. The shares 0.75 and 0.125 each answer “what share of this group bought something?”, so they can be set side by side fairly. The groups being compared decide the direction: to compare tour visitors with the others, divide by their row totals. Whether shares that differ from group to group mean the two questions are linked is the subject of Association in a two way table and segmented bar graphs.

Working back from shares. Since a share is the cell divided by the group’s total, multiplying both sides by that total gives the cell back:

For instance 0.75 \times 32 = 24. So from the shares and the group totals the counts can be filled in, and from the counts every other share follows, including the other direction. The totals are needed: doubling every count leaves every share the same, so shares alone fix the counts only up to scale.

Worked examples

Examples 1 to 4 go with Part 1, and Examples 5 to 7 with Part 2.

Boundaries

• The group must have members. A row or column whose total is 0 has no shares at all: “of the visitors who arrived by boat, what share bought something?” has no answer when nobody did; the answer is not 0.
• The cell must sit inside the group. Dividing a cell by the total of a row or column it is not in gives a number with no meaning.
• A row of a row-relative table adds to exactly 1 because each member of the group falls in exactly one cell of that row. If a survey let one person be counted in two columns, the shares of a row could add to more than 1. Rounded shares may add to 0.999 or 1.001; the exact fractions always add to 1.
• Shares alone do not give counts, and a row-relative table alone does not give column shares. Both need the group totals.

Where this goes next

The next step sets conditional shares side by side across a whole table. If the share of children wearing a helmet is very different from the share of adults, age group and helmet look connected; if the shares are about the same from row to row, they do not. That is Association in a two way table and segmented bar graphs, where each row of a row-relative table becomes one bar of a segmented bar graph.

Later the same division becomes Conditional probability: the chance of one event given that another happened. The two directions differ there for the same reason they differ here: one cell measured against two different wholes.

Exercises

Part 1: one share at a time

Table 2 is a made-up survey of 120 visitors at a second museum.

1. Table 2: Of the visitors who took a tour, what share bought something?
2. Table 2: Of the visitors who bought nothing, what share took no tour?

Part 1: row and column tables

Table 3 is a made-up cinema’s Saturday matinee, 150 tickets in all.

3. Make the row-relative table for Table 3, and check that each row adds to 1.
4. Example 1 found two shares from the park count. Complete the row-relative table for that count: the child, teen, adult and Total rows.

Part 1: which total?

For each question, name the total you divide by, then find the share.

5. Table 3: Among the musical tickets, what share were child tickets?
6. Table 3: What share of the animation tickets were adult tickets?
7. Table 2: Faolan writes, “70\% of the visitors who bought something took a tour.” Which total is he dividing by, and is he right?

Part 2: two directions

8. Table 2: Huxley writes, “One in five tour visitors bought nothing, so one in five visitors who bought nothing took a tour.” Check each half of his sentence.
9. Table 3: Find the share of adult tickets that were for a musical, and the share of musical tickets that were adult tickets. Which is larger, and why?

Part 2: comparing groups fairly

10. On a made-up Saturday, 42 of the 56 tour visitors at a museum bought something. On Sunday, 50 of the 75 tour visitors did. Which day had more tour visitors who bought something, and which had the larger share?
11. Table 3: (a) Is a child ticket or an adult ticket more likely to be for animation? (b) Is an animation ticket or a musical ticket more likely to be a child ticket? Compare shares.
12. On a second made-up day in the park, 28 of the 35 children and 54 of the 90 adults wore a helmet. Ines-Ruth says more adults than children wore helmets, so the adults were more careful. Compare the two groups fairly.

Part 2: filling a table back

13. A made-up museum had 200 visitors, and 80 of them took a tour. Of the tour visitors, 90\% bought something; of the others, 10\% did. Fill in the two-way table, then find the share of the buyers who took a tour.
14. A made-up cinema sold 400 tickets, 35\% of them child tickets. Half of the child tickets and 15\% of the adult tickets were for animation. What share of the animation tickets were child tickets?
15. In a made-up park count, 90 cyclists wore a helmet and 60 did not. Of the helmet wearers, 40\% were children; of those without a helmet, 25\% were children. What share of the children wore a helmet?

Reasoning

16. Kalliope adds the helmet column of the row-relative table from Exercise 4 and gets 0.75 + 0.45 + 0.62 = 1.82. She thinks something is wrong because it is not 1. Explain why nothing is wrong, and say which numbers in that table do have to add to 1.
17. A cell holds a people, with a > 0. Its row total is r and its column total is c. Show that the share of the row in that cell equals the share of the column in that cell exactly when r = c.
18. Two made-up museums have the same row-relative table as Table 1: 0.75 and 0.25 for the tour visitors, 0.125 and 0.875 for the others. The first is Table 1 itself. The second has 32 tour visitors and 16 other visitors. Find the share of the buyers who took a tour at each, and say what this shows about reading column shares from a row-relative table.
19. Odalys says, “Of the chess players, 40\% are in the choir, so 40\% of the choir play chess.” Without a table, explain why the second half does not follow from the first, and say what extra counts would settle it.
20. Explain why a table of shares alone, with no totals, can never be filled back into a table of counts.