CONTINGENCY TABLES
Learning outcome
Construct and interpret contingency tables.
A contingency table provides a way of displaying data that can facilitate
calculating probabilities. The table can be used to describe the sample
space of an experiment. Contingency tables allow us to break down a
sample pace when two variables are involved.
When reading a contingency table:
The left-side column lists all of the values for one of the variables. In
the table shown above, the left-side column shows the variable about
whether or not someone uses a cell phone while driving.
The top row lists all of the values for the other variable. In the table
shown above, the top row shows the variable about whether or not
someone had a speeding violation in the last year.
In the body of the table, the cells contain the number of outcomes that
fall into both of the categories corresponding to the intersecting row
and column. In the table shown above, the number of 25 at the
intersection of the “cell phone user” row and “speeding violation in the
last year” column tells us that there are 25 people who have both of
these characteristics.
The bottom row gives the totals in each column. In the table shown
above, the number 685 in the bottom of the “no speeding violation in
the last year” tells us that there are 685 people who did not have a
speeding violation in the last year.
The right-side column gives the totals in each row. In the table shown
above, the number 305 in the right side of the “cell phone user” row
tells us that there are 305 people who use cell phones while driving.
The number in the bottom right corner is the size of the sample space.
In the table shown above, the number in the bottom right corner is
755, which tells us that there 755 people in the sample space.
EXAMPLE
Suppose a study of speeding violations and drivers who use cell phones
while driving produced the following fictional data:
Speeding violation in No speeding violation Tota
the last year in the last year l
Cell phone
25 280 305
user
Not a cell
45 405 450
phone user
Total 70 685 755
Calculate the following probabilities:
1. What is the probability that a randomly selected person is a cell
phone user?
2. What is the probability that a randomly selected person had no
speeding violations in the last year?
3. What is the probability that a randomly selected person had a
speeding violation in the last year and does not use a cell phone?
4. What is the probability that a randomly selected person uses a cell
phone and had no speeding violations in the last year?
This table shows the number of athletes who stretch before exercising and
how many had injuries within the past year.
Injury in No injury in
Total
last year last year
Stretches 55 295 350
Does not
231 219 450
stretch
Total 286 514 800
1. What is the probability that a randomly selected athlete stretches
before exercising?
2. What is the probability that a randomly selected athlete had an injury
in the last year?
3. What is the probability that a randomly selected athlete does not
stretch before exercising and had no injuries in the last year?
4. What is the probability that a randomly selected athlete stretches
before exercising and had no injuries in the last year?
Solution
Example
The table below shows a random sample of 100 hikers broken down by
gender and the areas of hiking they prefer.
Gende The Near Lakes and On Mountain
Total
r Coastline Streams Peaks
Femal
18 16 45
e
Male 14 55
Gende The Near Lakes and On Mountain
Total
r Coastline Streams Peaks
Total 41
1. Fill in the missing values in the table
2. What is the probability that a randomly selected hiker is female?
3. What is the probability that a randomly selected hiker prefers to hike
on the coast?
4. What is the probability that a randomly selected hiker is male and
prefers to hike near lakes and streams?
5. What is the probability that a randomly selected hiker is female and
prefers to hike on mountains?
Example
The table below relates the weights and heights of a group of individuals
participating in an observational study.
Weight/ Tal Mediu Shor Total
Height l m t s
Obese 18 28 14
Normal 20 51 28
Underweight 12 25 9
Totals
1. Find the total for each row and column.
2. Find the probability that a randomly chosen individual from this group
is tall.
3. Find the probability that a randomly chosen individual from this group
is normal.
4. Find the probability that a randomly chosen individual from this group
is obese and short.
5. Find the probability that a randomly chosen individual from this group
is underweight and medium.
Example
