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Year 10AC9M10ST04

Two-Way Tables

Construct two-way tables and discuss possible relationship between categorical variables.

A two-way table sorts the same group of people by two things at once. Build it correctly and every probability question about that group can be read straight off it.

Method

The corner total is everyone
T=grand totalT = \text{grand total}
The bottom-right cell is the whole group. Every total must lead back to it.
Rows and columns both add to it
Each row total and each column total must add to the grand total. This is your check.
Four inner cells, four totals
With two categories you have 4 inner cells, 2 row totals, 2 column totals and 1 grand total. Nine numbers.
Any missing cell can be found
If you know three numbers in a row, you know the fourth. Work outwards from whatever you are given.

Worked example

e.g. Of 100 students, 60 are in Year 10. 45 students catch the bus, of whom 34 are in Year 10.\text{Of 100 students, 60 are in Year 10. 45 students catch the bus, of whom 34 are in Year 10.}
  1. Draw the frame. Two categories across, two down, with a Total row and column.BusWalkTotalYear 10Year 9Total\begin{array}{l|cc|c} & \text{Bus} & \text{Walk} & \text{Total} \\ \hline \text{Year 10} & & & \\ \text{Year 9} & & & \\ \hline \text{Total} & & & \end{array}
  2. Fill in everything you are told. Do not calculate yet.BusWalkTotalYear 103460Year 9Total45100\begin{array}{l|cc|c} & \text{Bus} & \text{Walk} & \text{Total} \\ \hline \text{Year 10} & 34 & & 60 \\ \text{Year 9} & & & \\ \hline \text{Total} & 45 & & 100 \end{array}
  3. Complete any row or column that is missing one number.60−34=26and45−34=1160 - 34 = 26 \quad \text{and} \quad 45 - 34 = 11
  4. Fill those in, then finish the rest the same way.BusWalkTotalYear 10342660Year 9112940Total4555100\begin{array}{l|cc|c} & \text{Bus} & \text{Walk} & \text{Total} \\ \hline \text{Year 10} & 34 & 26 & 60 \\ \text{Year 9} & 11 & 29 & 40 \\ \hline \text{Total} & 45 & 55 & 100 \end{array}
  5. Check both ways before using it. Rows across, columns down.34+26=6011+29=4034+11=4526+29=5534+26=60 \quad 11+29=40 \quad 34+11=45 \quad 26+29=55
  6. Now read probabilities straight off it. The denominator is whichever total you are inside.P(bus)=45100=0,45P(\text{bus}) = \frac{45}{100} = 0{,}45

Practice

Complete each table, then answer the question.

1
80 people. 50 own a dog. 30 own a cat. 18 own both. How many own neither?\text{80 people. 50 own a dog. 30 own a cat. 18 own both. How many own neither?}
Answer18 both, 32 dog only, 12 cat only, so 80−62=18 own neither18 \text{ both},\ 32 \text{ dog only},\ 12 \text{ cat only},\ \text{so } 80-62 = 18 \text{ own neither}
2
A row reads: 27, ?, 45. Find the missing cell.\text{A row reads: } 27,\ ?,\ 45. \text{ Find the missing cell.}
Answer45−27=1845 - 27 = 18
3
200 students: 120 study music, 90 study art, 40 study both. Find P(music only)\text{200 students: 120 study music, 90 study art, 40 study both. Find } P(\text{music only})
Answer120−40=80∴P=80200=0,4120 - 40 = 80 \quad\therefore\quad P = \frac{80}{200} = 0{,}4
4
Column totals are 45 and 55. Row totals are 60 and 45. What is wrong?\text{Column totals are } 45 \text{ and } 55. \text{ Row totals are } 60 \text{ and } 45. \text{ What is wrong?}
Answer45+55=100 but 60+45=105. The totals disagree, so a cell is wrong.45+55 = 100 \text{ but } 60+45 = 105. \text{ The totals disagree, so a cell is wrong.}
5
Of 60 people, 25 are left-handed and 12 of those wear glasses. 20 people wear glasses in total. Find the four inner cells.\text{Of 60 people, 25 are left-handed and 12 of those wear glasses. 20 people wear glasses in total. Find the four inner cells.}
Answer12, 13, 8, 2712,\ 13,\ 8,\ 27
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