Unit 16 Section 4 : Box and Whisker Plots

A box and whisker plot is based on the minimum and maximum values, the upper and lower quartiles and the median. This type of plot provides a good way to compare two or more samples.

Note: Box and whisker plots must always be drawn accurately to scale.

Example 1

Given the information below, draw a box and whisker plot.

Minimum 82
Lower quartile 94
Median 95
Upper quartile 102
Maximum 110

The box and whisker plot is shown below.

Example 2

Draw a box and whisker plot for this sample:

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First list the sample in order, to determine the median and the quartiles.

The box and whisker plot is shown below:

Example 3

A gardener collected data on two types of tomato. The box and whisker plot below shows data for the masses in grams of the tomatoes in the two samples.
Compare and contrast the two types and advise the gardener which type of tomato he should grow in future.

Type AType B
Median52 grams52 grams
Lower Quartile49 grams51 grams
Upper Quartile57 grams54 grams
Range14 grams8 grams
Interquartile Range8 grams3 grams

From this table we can see that both types of tomato have the same average mass because their medians are the same.

Comparing the medians and interquartile ranges shows that there is far more variation in the masses of the type A tomatoes, which means that the masses of type B are more consistent than those of type A.

However, comparing the two box and whisker plots, and the upper quartiles, shows that type A tomatoes will generally have a larger mass than those of type B.

Nevertheless, there will be some type A tomatoes that are lighter than any of type B.

Taking all this together, the gardener would be best advised to plant type A tomatoes in future as he is likely to get a better yield from them than from type B.

Exercises

Question 1

Choose one of the box and whisker plots for a sample that has:

Minimum 10
Lower quartile 14
Median 16
Upper quartile 20
Maximum 29
Question 2

Choose one of the box and whisker plots for the following sample:

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Question 3

A sample has:

Minimum 3
Range 21
Semi-interquartile range 4
Median 17
Upper quartile 20

Choose a box and whisker plot for the sample.

Question 4

For the sample illustrated in the following box and whisker plot, determine:

(a)

the range,

(b)

the semi-interquartile range.

Question 5

What are the median and the semi-interquartile range of the following sample:

Median =
Semi-interquartile range =
Question 6

The two box and whisker plots show the data collected by the manufacturers on the life-span of light bulbs.

From this data, which manufacturer produces the better light bulb?

Manufacturer AManufacturer B
Median4000 hours5000 hours
Lower Quartile2500 hours3500 hours
Upper Quartile4500 hours6000 hours
Range4000 hours6500 hours
Interquartile Range2000 hours2500 hours

From this table we can see that, on average, the light bulbs produced by manufacturer B have the longer life-span.
Although there is far more variation in the life-spans of type B, the lower quartile for type B almost the same as the median for type A, which gives further support to manufacturer B producing the better light bulb.

Question 7

A maths test is given to two classes. The results are illustrated below.
Compare the results.

Class 9PClass 9Q
Median
Lower Quartile
Upper Quartile
Range
Interquartile Range

In general, tended to score better marks.

From the table we can see that both classes had the same average mark. There is far more variation in the scores for class 9P. Some students in 9Q performed better than their counterparts in 9P. In general, though, class 9P tended to score better marks than class 9Q.

Question 8

A builder can choose between two different types of brick that are coloured red or yellow. The box and whisker plots below illustrate the results of tests on the strength of the bricks.

From the data illustrated in the box and whisker plots:

(a)

give one reason why the builder might prefer to use red bricks. Complete the sentence.

The builder might prefer to use the red bricks because that type has the higher , i.e. they tend, on average, to be stronger than the yellow bricks.

(b)

give one reason why the builder might prefer to use yellow bricks. Complete the sentence.

The builder might prefer to use the yellow bricks because that type has the smaller , and the larger , which means that he is less likely to get a weak brick.

Also, if the red bricks tend, on average, to be stronger than the yellow bricks, then they are likely to be more expensive, so he might prefer the yellow bricks if he wishes to save money at the expense of structural stability.

Question 9

A class took an English test and a Maths test. Both tests had a maximum possible mark of 25. The results are illustrated below.

Compare the results.

EnglishMathematics
Median
Lower Quartile
Upper Quartile
Range
Interquartile Range

In general, the class tended to score better marks on the test.

From this table we can see that the class had the same average mark in both subjects. There is far more variation in the scores for Mathematics. Few pupils in the class scored high marks in the English test. Nobody scored full marks in either test. In general, though, the class tended to score better marks on the Mathematics test.

Question 10

A cinema is showing 3 films, A, B and C. The ages of people watching the films are illustrated in the following box and whisker plots:

Answer the following questions, giving reasons to support your answers.

(a)

Which film do you think you would not be allowed to watch?

minimum age to view is 18
(b)

Which film would you probably enjoy most?

50% of the audience was between 14 and 30
(c)

Which film would your parents probably enjoy most?

the median age of the audience was 38 years, and only 25% of the audience was under 34 years old.