UNIT 2Two-way tablesLesson 1 of 11 Two-Way Tables
Two-Way Tables
A two-way table counts answers to two questions at once. Percents of a row or a column show if the answers are linked.
| Sibling | No sibling | Total | |
|---|---|---|---|
| Pet | 14 | ? | 25 |
| No pet | 20 | ? | ? |
| Total | ? | ? | 60 |
What you'll be able to do
- Read and fill in a two-way table.
- Find row percents and column percents.
- Pick the right total to divide by.
- Decide if two answers are linked.
The big idea
Two questions at once. Rows show the answer to one question. Columns show the answer to the other.
Totals. Each row adds to its row total. Each column adds to its column total. Both add to the grand total.
Fill a blank. Find a row or column with only one blank. Subtract what you know from its total.
Row percent. Divide a cell by its row total. It answers 'of the grade 8 students, how many …?'
Column percent. Divide a cell by its column total. It answers 'of the walkers, how many …?'
Pick the total. The words after 'of the' tell you which total to use.
Is there a link? Compare row percents. If they are far apart, the answers are linked. If they are close, they are not.
Counts can fool you. A bigger group has bigger counts. Compare percents, not counts.
Words you need
- Two-way table
- A table that sorts answers to two questions into rows and columns.
- Row total
- The sum of a row.
- Column total
- The sum of a column.
- Relative frequency
- A count out of a total, often as a percent.
- Row percent
- A cell divided by its row total.
- Column percent
- A cell divided by its column total.
- Association
- A link between two answers: one group answers differently from the other.
Which total do I use?
| The question says | Divide by |
|---|---|
| Of the grade 8 students | The grade 8 row total |
| Of the walkers | The walk column total |
| Of all students | The grand total |
Worked examples
Example 1
60 students: 25 have a pet. 14 of the pet owners have a sibling. 20 students with no pet have a sibling. Make the table.
- Pet row: 25 − 14 = 11 have no sibling.
- No pet row: 60 − 25 = 35 students. 35 − 20 = 15 have no sibling.
- Columns: 14 + 20 = 34 have a sibling. 11 + 15 = 26 don't.
Answer: Pet, no sibling: 11. No pet, no sibling: 15. No pet total: 35. Sibling total: 34. No sibling total: 26
| Sibling | No sibling | Total | |
|---|---|---|---|
| Pet | 14 | ? | 25 |
| No pet | 20 | ? | ? |
| Total | ? | ? | 60 |
Example 2
In that table, what percent of pet owners have a sibling? What percent of students with no pet do?
- Pet owners: 14 ÷ 25 = 0.56, so 56%.
- No pet: 20 ÷ 35 ≈ 0.571, so about 57%.
Answer: 56% of pet owners. About 57% of students with no pet
| Sibling | No sibling | |
|---|---|---|
| Pet | 14 | 11 |
| No pet | 20 | 15 |
Example 3
Do the pet table's row percents show an association?
- The row percents are 56% and 57%.
- They are almost the same.
- So having a pet and having a sibling show no association here.
Answer: No. The row percents are the same or close
| Sibling | No sibling | |
|---|---|---|
| Pet | 14 | 11 |
| No pet | 20 | 15 |
Common mistakes
Dividing by the grand total. 'Of the grade 8 students' means divide by the grade 8 total.
Mixing up row and column. 'Of the walkers' uses the walk column total.
Comparing counts. Groups can differ in size. Compare percents.
Reading 'most' as a rate. Most hurt riders can wear helmets just because most riders do.
Practice: 20 questions
20 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- The table shows 100 students. Sport: instrument 18, no instrument 42. No sport: instrument 22, no instrument 18. How many play an instrument? What fraction play both? Word problem
Instrument No instrument Total Sport 18 42 60 No sport 22 18 40 Total 40 60 100 - The table shows how 100 students get to school, with four cells blank. Grade 7: bus 34, total 55. Grade 8: walk 16. Bus total: 63. Fill in the blanks. Word problem
Bus Walk Total Grade 7 34 ? 55 Grade 8 ? 16 ? Total 63 ? 100 - The table shows how 100 students get to school. Grade 7: bus 34, walk 21, total 55. Grade 8: bus 29, walk 16, total 45. Totals: bus 63, walk 37. What percent of grade 8 students take the bus? Round to the nearest percent. Word problem
Bus Walk Total Grade 7 34 21 55 Grade 8 29 16 45 Total 63 37 100 - The table shows how 100 students get to school. Grade 7: bus 34, walk 21, total 55. Grade 8: bus 29, walk 16, total 45. Totals: bus 63, walk 37. What percent of walkers are in grade 7? Round to the nearest percent. Word problem
Bus Walk Total Grade 7 34 21 55 Grade 8 29 16 45 Total 63 37 100 - The table shows how 100 students get to school. Grade 7: bus 34, walk 21, total 55. Grade 8: bus 29, walk 16, total 45. Totals: bus 63, walk 37. Ezra says 16% of grade 8 students walk. His reason: 16 of all 100 students do. Find his mistake. Word problem
Bus Walk Total Grade 7 34 21 55 Grade 8 29 16 45 Total 63 37 100
Tier 2 Intermediate One more step to think through
- 80 campers: 35 can swim. 28 of the swimmers are over 12. 15 of the non-swimmers are over 12. Make a two-way table with totals. Word problem
- The table shows 100 students. Sport: instrument 18, no instrument 42. No sport: instrument 22, no instrument 18. Is there an association between sport and instrument? Word problem
Instrument No instrument Total Sport 18 42 60 No sport 22 18 40 Total 40 60 100 - The table shows art club members. Grade 7: in the club 12, not 48, total 60. Grade 8: in the club 18, not 22, total 40. What percent of grade 8 are in the club? What percent of the club are in grade 8? Word problem
In the club Not in it Total Grade 7 12 48 60 Grade 8 18 22 40 Total 30 70 100 - 40 of 80 grade 8 students like the new lunch. 30 of 50 grade 7 students like it. Bo says grade 8 likes it more, because 40 is more than 30. Find his mistake. Word problem
- 48 students eat breakfast. 12 of them feel tired by noon. 32 skip breakfast. 8 of them feel tired. Is there an association? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- A survey asks 200 people about jazz. 40% are adults. 75% of the adults like jazz. 30% of the kids like jazz. How many people like jazz? Word problem
- Grade 6: 30 bus, 20 walk, 10 car. Grade 7: 24 bus, 30 walk, 6 car. Grade 8: 18 bus, 44 walk, 18 car. Which grade walks most often? Use row percents. Word problem
Bus Walk Car Total Grade 6 30 20 10 60 Grade 7 24 30 6 60 Grade 8 18 44 18 80 Total 72 94 34 200 - Club A: 30 of 50 members who practice win, and 10 of 50 who don't. Club B: 26 of 50 who practice win, and 24 of 50 who don't. Which club shows the stronger association? Word problem
Club A Win Lose Practice 30 20 No practice 10 40 Club B Win Lose Practice 26 24 No practice 24 26 - 200 riders wore helmets, and 14 were hurt. 50 riders did not, and 11 were hurt. Cal says helmets don't help, because most hurt riders wore helmets. Find his mistake. Word problem
- Dog owners: 36 adults and 24 kids. Cat owners: 18 adults and 22 kids. Write each row as percents. Round to the nearest percent. Word problem
Adults Kids Total Dog owners 36 24 60 Cat owners 18 22 40 Total 54 46 100
Tier 4 Challenge Stretch problems, some above grade level
- A table has row totals 30 and 70 and column totals 40 and 60. The top-left cell is x. For which x is there no association? Word problem
Column 1 Column 2 Total Row 1 x ? 30 Row 2 ? ? 70 Total 40 60 100 - 60% of 40 boys like a song. 80% of 60 girls like it. Pia says 85% of all 100 students like it. Find her mistake. Word problem
- 50 people own a cat. 30 of them are kids. Two percents match: kids who own a cat, and cat owners who are kids. How many kids were asked? Word problem
- A table gives percents of all students. Music and art: 15%. Music, no art: 45%. No music, art: p%. Neither: 25%. Find p. What percent of art students also take music? Word problem
Art No art Music 15% 45% No music ? 25% Total ? ? - Of 200 emails, 30% are spam. A filter flags 90% of the spam and 5% of the other emails. What percent of flagged emails are spam? Round to the nearest percent. Word problem Above grade level
Answers
Tier 1 Foundations
- 40 play an instrument. 18/100 play both. Instrument: 40. 18 of the 100 in all.
- Grade 7, walk: 21. Grade 8, bus: 29. Grade 8 total: 45. Walk total: 37. Each row adds to its row total, and each column adds to its column total. Fill the cell that is the only blank in its row or column first.
- About 64%. Divide by the Grade 8 row total: 29 ÷ 45 ≈ 0.644, so about 64%.
- About 57%. Divide by the Walk column total: 21 ÷ 37 ≈ 0.568, so about 57%.
- About 36%. The question is about grade 8 only. Ezra divided by the grand total. Divide by the Grade 8 row total: 16 ÷ 45 ≈ 0.356, so about 36%.
Tier 2 Intermediate
- Swim, 12 or under: 7. No swim, 12 or under: 30. No swim total: 45. Over 12 total: 43. 12 or under total: 37. Each row adds to its row total, and each column adds to its column total. Fill the cell that is the only blank in its row or column first.
- Yes. No sport: 55% instrument, against Sport: 30%. Compare row percents, not counts. Sport: 18/60 = 30%. No sport: 22/40 = 55%.
- 45% of grade 8 are in the club. 60% of the club are in grade 8. Divide by the Grade 8 row total: 18 ÷ 40 gives 45%. Divide by the In the club column total: 18 ÷ 30 gives 60%.
- Grade 7 likes it more: 60% against 50%. The groups are different sizes, so Bo must compare percents. Compare row percents, not counts. Grade 7: 30/50 = 60%. Grade 8: 40/80 = 50%.
- No. The row percents are the same or close. Compare row percents, not counts. Breakfast: 12/48 = 25%. No breakfast: 8/32 = 25%.
Tier 3 Advanced
- 96. Adults: 40% of 200 = 80, and 75% of them = 60. Kids: 60% of 200 = 120, and 30% of them = 36. In all: 60 + 36 = 96.
- Grade 8: 55% walk. Compare row percents, not counts. Grade 6: 20/60 ≈ 33%. Grade 7: 30/60 = 50%. Grade 8: 44/80 = 55%.
- Club A. Club A: 60% against 20%, a 40-point gap. Club B: 52% against 48%, a 4-point gap. The bigger gap shows the stronger association.
- Helmets help: 7% of helmet riders were hurt, against 22% without. Most hurt riders wore helmets only because most riders wore helmets. Compare the rate in each row. Compare row percents, not counts. Helmet: 14/200 = 7%. No helmet: 11/50 = 22%.
- Dog owners: 60% adults, 40% kids. Cat owners: 45% adults, 55% kids. Divide each cell by its row total. Each row then adds to 100%.
Tier 4 Challenge
- x = 12. No association means both rows have the same share in column 1. That share is 40 of 100, so 40%. 40% of 30 = 12.
- 72%. 85% is more than both groups' percents. A total can't be higher than every group. Count the students who like it: 24 + 48 = 72 of 100. The total always falls between 60% and 80%.
- 50 kids. Cat owners who are kids: 30 of 50, so 60%. Kids who own a cat: 30 of x. These match only when x = 50.
- p = 15, and 50% of art students take music. The four cells add to 100%, so p = 100 − 15 − 45 − 25 = 15. Art is 15% + 15% = 30% of students. 15 ÷ 30 = 50%.
- About 89%. Spam: 30% of 200 = 60, and 90% of them = 54. Other: 70% of 200 = 140, and 5% of them = 7. 54 ÷ 61 ≈ 0.885, so about 89%.
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