The four kinds of number in a two-way table
Every two-way table classifies the same group of individuals by two variables at once. The survey below records the after-school activity of $300$ students.
| Music | Sports | Neither | Total | |
|---|---|---|---|---|
| Grade 11 | 48 | 72 | 30 | 150 |
| Grade 12 | 36 | 84 | 30 | 150 |
| Total | 84 | 156 | 60 | 300 |
- A cell such as $72$ counts students who are in grade 11 and chose sports.
- A row total such as $150$ counts everyone in grade 11, regardless of activity.
- A column total such as $156$ counts everyone who chose sports, regardless of grade.
- The grand total $300$ counts everyone.
Every question is a fraction. The numerator is almost always a cell; the whole exercise is deciding which of the four kinds of total belongs underneath it.
What percent of grade-11 students chose sports?
“Of grade-11 students” fixes the denominator to the grade-11 row total, $150$: $\dfrac{72}{150}=0.48=\boxed{48\%}$.
Of the students who chose sports, what percent are in grade 12?
Now the group named first is “students who chose sports”, so the denominator is the sports column total, $156$: $\dfrac{84}{156}\approx 0.538=\boxed{53.8\%}$.
What percent of all the students surveyed are grade-11 students who chose music?
No subgroup is named, so the denominator is the grand total: $\dfrac{48}{300}=0.16=\boxed{16\%}$.
Ba ví dụ trên dùng chung một tử số kiểu ô, chỉ khác mẫu số — và ra ba đáp án khác nhau. Kỹ thuật đọc đề: tìm cụm từ đứng ngay sau chữ “of” hoặc “among”, vì cụm đó chính là mẫu số. “Of grade-11 students” $\to$ tổng hàng. “Among those who chose sports” $\to$ tổng cột. Không có cụm nào $\to$ tổng chung. Đọc xong câu hỏi mà chưa khoanh được mẫu số thì đừng bấm máy tính vội.
Lấy tổng chung ($300$) làm mẫu số cho mọi câu là lỗi phổ biến nhất của chủ đề này, và nó không bao giờ ra một con số kỳ quặc để mà nghi ngờ — $\tfrac{72}{300}=24\%$ trông hoàn toàn bình thường, nên người ra đề thường để sẵn nó trong danh sách phương án.
Row percent, column percent, table percent
The same cell generates three different percentages. Name them before computing.
Cell $=72$ (grade 11 and sports). Row percent $\tfrac{72}{150}=48\%$ of grade-11 students. Column percent $\tfrac{72}{156}\approx 46.2\%$ of sports students. Table percent $\tfrac{72}{300}=24\%$ of all students.
Row percents across one row add to $100\%$. Column percents down one column add to $100\%$. Table percents over the whole table add to $100\%$. If a set of percentages you have computed does not add to $100\%$ in any of these three ways, one of them used the wrong denominator.
Completing a table from partial information
Questions often supply only a few entries and expect the rest to be reconstructed. Fill whatever a single subtraction gives, then repeat — the table unlocks itself.
A company surveyed $250$ customers about whether they use its mobile app. Complete the table.
| Uses app | Does not | Total | |
|---|---|---|---|
| Member | 160 | ||
| Nonmember | 36 | ||
| Total | 132 | 250 |
The “Uses app” column has one missing entry: members who use the app $=132-36=96$. Then the member row gives $160-96=64$ members who do not. The grand total gives $250-160=90$ nonmembers, so nonmembers who do not use the app $=90-36=54$. Finally the “Does not” column totals $64+54=118$, and $132+118=250$ checks out.
| Uses app | Does not | Total | |
|---|---|---|---|
| Member | 96 | 64 | 160 |
| Nonmember | 36 | 54 | 90 |
| Total | 132 | 118 | 250 |
Using the completed table: what percent of nonmembers use the app, and what percent of members use the app?
Nonmembers: $\dfrac{36}{90}=\boxed{40\%}$. Members: $\dfrac{96}{160}=\boxed{60\%}$.
Members are the more frequent users as a rate, even though the two groups differ enormously in size.
Using the same table: of the customers who use the app, what percent are members?
Denominator is now the app-user column: $\dfrac{96}{132}\approx\boxed{72.7\%}$.
Compare with the previous example. “Of members, what percent use the app” is $60\%$; “of app users, what percent are members” is $72.7\%$. Same cell, different questions, different answers.
Hai câu hỏi ở ví dụ vừa rồi là cặp hay bị đánh tráo nhất trong toàn bộ phần thống kê: “bao nhiêu phần trăm hội viên dùng app” và “bao nhiêu phần trăm người dùng app là hội viên”. Trong tiếng Việt hai câu này nghe cũng na ná, nên cách an toàn là đọc theo cấu trúc tiếng Anh: cụm ngay sau “of” là nhóm được lấy làm mốc, cụm sau “what percent are/do” là tính chất được đếm. Đổi chỗ hai cụm đó là đổi hẳn mẫu số.
Selecting at random from a subgroup
“A person is selected at random from those who …” is a two-way-table question wearing probability clothing. The phrase after “from” names the pool; that pool is the denominator, and the count of the pool members having the stated property is the numerator.
One of the $300$ students in the first table is selected at random from those who did not choose music. What is the probability that the student is in grade 12?
Students who did not choose music: $300-84=216$. Grade-12 students who did not choose music: $84+30=114$. Probability $=\dfrac{114}{216}=\dfrac{19}{36}\approx\boxed{0.528}$.
What a table supports, and what it does not
A table reports what happened in one group of individuals. Whether the finding extends beyond that group depends entirely on how the individuals were obtained.
| Reports poor sleep | Does not | Total | |
|---|---|---|---|
| Drinks coffee daily | 180 | 120 | 300 |
| Does not drink coffee | 60 | 140 | 200 |
| Total | 240 | 260 | 500 |
Among daily coffee drinkers, $\tfrac{180}{300}=60\%$ report poor sleep; among the others, $\tfrac{60}{200}=30\%$. The rates genuinely differ.
- Random sample from a population $\rightarrow$ results may be generalised to that population.
- Random assignment of subjects to groups $\rightarrow$ a difference between the groups may be attributed to the treatment: a causal claim is permitted.
- Observational study — subjects chose their own group, or were simply observed $\rightarrow$ only an association may be reported. No causal claim.
The two conditions are independent. Random sampling without random assignment gives a generalisable association but no cause. Random assignment without random sampling gives a cause within the study group but no licence to generalise.
The $500$ adults above answered a questionnaire about their own habits. Which conclusion is appropriate?
The adults chose their own coffee consumption, so this is observational. The defensible statement is: among these adults, daily coffee drinking is associated with a higher rate of reported poor sleep. Saying that coffee causes poor sleep is not supported, and neither is the reverse direction — people who sleep badly may drink coffee because of it. If, in addition, the $500$ adults were not a random sample, the association cannot even be extended beyond this group.
Ba chữ khác nhau, ba mức kết luận khác nhau, và đề SAT hỏi đúng chỗ này: chọn mẫu ngẫu nhiên cho phép nói về tổng thể; phân nhóm ngẫu nhiên cho phép nói nhân quả; quan sát chỉ cho phép nói có liên hệ. Khi bốn phương án chỉ khác nhau ở một động từ — “causes” hay “is associated with” hay “all” — thì câu đó đang kiểm tra đúng ba chữ này chứ không kiểm tra phép tính nào cả.
So số đếm thô thay vì so tỉ lệ. Một dây chuyền có $33$ sản phẩm lỗi và một dây chuyền có $21$ sản phẩm lỗi — chưa nói lên gì cả nếu dây thứ nhất sản xuất nhiều gấp rưỡi. Câu hỏi dùng chữ “rate”, “proportion”, “more likely” là bắt buộc phải chia cho tổng của hàng tương ứng.
Practice
Questions 1–7 refer to the table below. A survey recorded how many days per week each of $480$ adults exercises.
| 0–2 days | 3–4 days | 5+ days | Total | |
|---|---|---|---|---|
| Under 40 | 54 | 96 | 110 | 260 |
| 40 or older | 88 | 72 | 60 | 220 |
| Total | 142 | 168 | 170 | 480 |
Questions 8–10 refer to the table below. A factory inspected $900$ units produced on two lines.
| Defective | Not defective | Total | |
|---|---|---|---|
| Line A | |||
| Line B | 21 | 350 | |
| Total | 54 | 900 |
Questions 11–14 refer to the table below. Six hundred patients with the same condition each received one of two treatments.
| Improved | Did not improve | Total | |
|---|---|---|---|
| Treatment X | 198 | 102 | 300 |
| Treatment Y | 162 | 138 | 300 |
| Total | 360 | 240 | 600 |