Box And Whisker

Box and Whisker Diagrams Averages use one number to describe a set of data, this can be misleading, depending upon which average you use. Box and whisker diagrams use the • median, • upper and lower quartiles and • range (Highest value – lowest value)

to draw a picture that can be used to interpret data Box plots can be drawn in two ways, from • cumulative frequency diagrams, or • the raw data 1

Box and Whisker Diagrams • Box plots from cumulative frequency diagrams We have already seen how to find the median, LQ and UQ from a cumulative frequency diagram 80

x x

Cumulative frequency

70

Minimum value

x

60

50

x

Median

40

Extend the the working out lines below the x axis

UQ

x

30

LQ 10

x 0

10

x 20

Maximum value

x

20

0

x

x 30

40

50

60

70

80

90

100

Weight (g)

Draw the box plot like this 2

Box and Whisker Diagrams Looking at the box plot on its own:

LQ

Median

UQ

Minimum value

whisker

Maximum value

whisker

3

Box and Whisker Diagrams •

Box plots from raw data

Ex 1) Fred keeps a record of his journey times to school each morning These are his time to the nearest minute for 25 days: 29, 21, 16, 25, 21, 19, 18, 30, 21, 21, 12, 26, 19, 21, 20, 19, 30, 29, 16, 21, 18, 18, 27, 18, 20

I.

First put them in order of size

12, 16, 16, 18, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21, 25, 26, 27, 29, 29, 30, 30 Median Maximum value Minimum value

Upper Quartile Lower Quartile number of pieces of data, n

Median position

n + 1 = 25 +1 = 13 2 2 Upper Quartile position 3(n + 1) = 3(25 +1) = 18.75 4 4 Lower Quartile position n + 1 = 25 +1 = 6.25 4 4

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Box and Whisker Diagrams 12

Minimum value Maximum value Median

30 21

Lower Quartile

18

Upper Quartile

25

Draw a sensible scale:

12 14

16

18

20 22 24 Time (minutes)

26

28 30

5

Box and Whisker Diagrams Now do this: Draw a box plot for this data: 18, 8, 19, 15, 27, 13, 10, 4, 8, 31, 26, 11, 29, 28, 23

Solution: Arrange the data in order of size: 4, 8, 8, 10, 11, 13, 15, 18, 19, 23, 26, 27, 28, 29, 31

Median position

n + 1 = 15 +1 = 8 2 2

LQ position

n + 1 = 15 +1 = 4 4 4

LQ = 10

UQ position

3(n + 1) = 3(15 +1) = 12 2 2

UQ = 27

Median = 18

0

4

8

10

14

16

20

24

28

32

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Box and Whisker Diagrams Interpreting Box and Whisker Diagrams

What can you say about the ages of the shoppers and what kind of shop they are? Dressnice has a much lower median age, so the shoppers are younger 75% of the Dressnice shoppers are younger than 75% of the Clotheswell shoppers

Clotheswell has broader appeal because the IQR is bigger Dressnice is a shop for younger people, perhaps more fashionable 7

Advantages of box plot: • From box plot, we can easily comment on the skewness of our data whether it is symmetrical, positively skewed or negatively skewed • We can easily identify the lowest and highest value Q1, Q2 & Q3 Disadvantage of box plot: • It does not show frequency

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