Two questions at once
Tonight's idea
| Job | No job | Total | |
|---|---|---|---|
| Grade 9 | 12 | 108 | 120 |
| Grade 12 | 32 | 48 | 80 |
| Total | 44 | 156 | 200 |
Ask 200 students at a made-up school two questions: which grade are you in, and do you have a part-time job? A two-way table counts every combination.
A percentage needs an "out of". A row percentage divides by the row total. Of the 80 Grade 12 students, 32 have a job: 40%. For Grade 9 it is 12 out of 120: 10%.
A column percentage divides by the column total. Of the 44 students with a job, 32 are in Grade 12: about 73%.
Both 40% and 73% come from the same cell, 32. They answer different questions. The words after "of" name the total to divide by.
Linked or not? Compare the row percentages. Here 10% against 40%: grade and jobs are linked. If they were unrelated, every row would show roughly the whole-school figure, 44 out of 200, or 22%.
Watch one: does the bus make you late?
| Late | Never late | Total | |
|---|---|---|---|
| Bus | 18 | 42 | 60 |
| No bus | 27 | 63 | 90 |
| Total | 45 | 105 | 150 |
Of 150 students, 60 take the bus, and 18 of those were late at least once this month. Of the other 90, 27 were late. Subtraction fills the rest of the table.
Row percentages: \(18 \div 60 = 30\%\) of bus riders were late, and \(27 \div 90 = 30\%\) of the others. The rows match, so there is no link.
A column percentage: of the 45 late students, \(18 \div 45 = 40\%\) took the bus. That sounds like news, but 40% of all the students take the bus.
Your turn
Eighty people were asked about pets. (a) Fill in the five missing numbers. (b) What percentage of the cat owners have a dog? (c) What percentage of the dog owners have a cat?Dog No dog Total Cat 6 20 No cat Total 30 80 - Ninety students sat a test. Fifty went to a revision class, and 45 of those passed. Of the 40 who did not go, 28 passed. (a) Make the two-way table. (b) Find the two row percentages for passing. (c) Does the table prove that the class causes passes? Give one other explanation.
- In a made-up town, 900 cyclists wear helmets and 100 do not. Last year 27 helmet wearers and 18 of the others were hurt. A headline says "Most injured cyclists wore helmets". (a) Is the headline true? Which kind of percentage is it using? (b) Which percentages should you compare to judge helmets? Find them.
- A school has 400 students and 25% are in the choir. Of the choir members, 60% play an instrument. Of the rest, 20% do. (a) Build the table of counts. (b) What percentage of the instrument players are in the choir? (c) Why is that not 60%?
Stretch: the table with no link
A table has row totals 120 and 80, column totals 50 and 150, and 200 people in all. Fill in the four cells so that the rows and columns are not linked at all. Then find a rule that gives each cell from its row total, its column total and 200.
Answers
- (a) Cat row: 6, 14, 20. No cat row: 24, 36, 60. Totals: 30, 50, 80. (b) A row percentage: \(6 \div 20 = 30\%\). (c) A column percentage: \(6 \div 30 = 20\%\).
- (a) Went: 45 passed, 5 failed, total 50. Did not go: 28 passed, 12 failed, total 40. Totals: 73, 17, 90. (b) \(45 \div 50 = 90\%\) and \(28 \div 40 = 70\%\). (c) No. It shows a link, not a cause. Students who choose to go may be the ones who work harder anyway, for example.
- (a) True: 45 were hurt and \(27 \div 45 = 60\%\) wore helmets. It is a column percentage, and it is high only because nine in ten cyclists wear helmets. (b) The row percentages. Hurt among helmet wearers: \(27 \div 900 = 3\%\). Among the others: \(18 \div 100 = 18\%\). Riders without helmets were hurt six times as often.
- (a) Choir: 60 play, 40 do not, total 100. Not choir: 60 play, 240 do not, total 300. Totals: 120, 280, 400. (b) \(60 \div 120 = 50\%\). (c) 60% is a row percentage, out of the 100 choir members. The question asks for a column percentage, out of the 120 players. The 300 other students are a much bigger group, so even 20% of them adds as many players as the choir does.
Stretch. No link means each row has the same split as the totals: 50 out of 200, or 25%, in the first column. So the rows are 30, 90 and 20, 60. The rule: cell = row total × column total ÷ 200. Check: \(120 \times 50 \div 200 = 30\).