Topics
9. Statistics
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United States of America
Grade 7

Review: Measures of center and variability in data displays

Lesson
Practice

Interpreting data displays

We've previously learned how to interpret data in dot plots, histograms, and box plots. Let's look at a few worked examples to review how to find measures of center (mean, median and mode) and measures of variability (range and interquartile range or IQR) from a data display.

Examples

Example 1

A group of adults is asked: "How old were you when you passed your driving test?". The responses were: 22,\,17,\,17,\,17,\,19,\,21,\,17,\,22,\,21,\,18,\,18,\,17,\,18,\,22,\,18

The dot plot represents the responses.

A dot plot titled Responses. Ranging from 17 to 22 in steps of 1. Ask your teacher for more information.
a

What is the range of this data set?

Worked Solution
Create a strategy

To find the range, use the formula: \text{Range}=\text{Greatest value}-\text{Least value}

Apply the idea
\displaystyle \text{Range}\displaystyle =\displaystyle 22-17Find the difference
\displaystyle =\displaystyle 5Evaluate the subtraction
b

What is the mode of this data set?

Worked Solution
Create a strategy

Choose the value which occurs most often.

Apply the idea

\text{Mode}=17

c

What is the median of this data set?

Worked Solution
Create a strategy

Order the values and find the middle value.

Apply the idea

Arrange the values in order:17,\,17,\,17,\,17,\,17,\,18,\,18,\,18,\,18,\,19,\,21,\,21,\,22,\,22,\,22

Remove the smallest and largest values until you are left with one remaining value.

\text{Median}=18

d

How many people passed their driving test on or after their 19th birthday?

Worked Solution
Create a strategy

Count the number of dots that are greater or equal to 19.

Apply the idea
\displaystyle \text{No. of people}\displaystyle =\displaystyle 6Count the values

Example 2

A government agency records how long people wait on hold to speak to their representatives. The results are displayed in the histogram below:

A histogram titled Time on hold with frequency on the y axis, Length of hold on x axis. Ask your teacher for more information.
a

Complete the corresponding frequency table:

Length of hold (minutes)Frequency
1
2
3
4
5
Worked Solution
Create a strategy

List the corresponding frequency of each length of hold (minutes).

Apply the idea
Length of hold (minutes)Frequency
111
212
311
42
54
b

How many phone calls were made?

Worked Solution
Create a strategy

Add all the frequencies from part (a).

Apply the idea
\displaystyle \text{No. of calls}\displaystyle =\displaystyle 11+12+11+2+4Find the sum of all frequencies
\displaystyle =\displaystyle 40Evaluate
c

How long in total did these people wait on the hold?

Worked Solution
Create a strategy

Multiply each hold time by its frequency, and add the results together.

Apply the idea
\displaystyle \text{Total time}\displaystyle =\displaystyle (1\times 11)+(2\times 12)+(3\times 11)+(4\times 2)+ (5\times 4)Multiply each time by the frequency
\displaystyle =\displaystyle 11+24+33+8+20Evaluate the multiplication
\displaystyle =\displaystyle 96\text{ minutes}Evaluate the addition
d

What was the mean wait time? Give your answer as a decimal.

Worked Solution
Create a strategy

To find the mean wait time, divide the total hold time by the total calls.

Apply the idea
\displaystyle \text{Mean}\displaystyle =\displaystyle \dfrac{96}{40}Divide the total hold time by the total calls
\displaystyle =\displaystyle 2.4\text{ minutes}Evaluate

Example 3

For the following box plot:

0
2
4
6
8
10
12
14
16
18
20
a

Find the lowest value.

Worked Solution
Create a strategy

The lowest value is at the end of the left whisker.

Apply the idea

\text{Lowest value}=3

b

Find the highest value.

Worked Solution
Create a strategy

The highest value is at the end of the right whisker.

Apply the idea

\text{Highest value}=18

c

Find the range.

Worked Solution
Create a strategy

The range is the difference between the highest value and the lowest value.

Apply the idea
\displaystyle \text{Range}\displaystyle =\displaystyle 18-3Find the difference of the values
\displaystyle =\displaystyle 15Evaluate the subtraction
d

Find the median.

Worked Solution
Create a strategy

The median is marked by a line between the lower and upper quartile.

Apply the idea

\text{Median}=10

e

Find the interquartile range (IQR).

Worked Solution
Create a strategy

The interquartile range (IQR) is the difference between the upper quartile and the lower quartile.

Apply the idea
\displaystyle \text{ Interquartile range (IQR) }\displaystyle =\displaystyle 15-8Find the difference between the quartiles
\displaystyle =\displaystyle 7Evaluate the subtraction
Idea summary

Summarizing data in a display often makes it easier to visualize and find measures of center and variability.

Mean

The sum of all the values divided by the total number of values

Median

The middle value in a data set

Mode

The value that appears the most in a data set

Range

The distance between the smallest and largest value in a data set

Interquartile range (IQR)

The distance between Quartile 1 and Quartile 3

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