Exploring Data Distributions: Histograms, Skewed and Symmetric Frequency Distribution

Average Main Dish Price (Dollars)
$10.12$	$9.29$	$8.29$	$9.78$	$10.69$
$9.68$	$12.09$	$8.94$	$10.81$	$8.62$
$11.39$	$12.62$	$8.71$	$10.74$	$10.52$
$10.77$	$10.15$	$9.18$	$8.45$	$9.52$
$11.89$	$9.77$	$9.44$	$13.24$	$11.01$
$10.62$	$9.38$	$12.15$	$9.68$	$9.60$
$10.32$	$11.31$	$11.41$	$8.62$	$9.27$
$10.96$	$9.18$	$10.28$	$10.71$	$10.02$

Average Main Dish Price (Dollars)

10.12

9.29

8.29

9.78

10.69

9.68

12.09

8.94

10.81

8.62

11.39

12.62

8.71

10.74

10.52

10.77

10.15

9.18

8.45

9.52

11.89

9.77

9.44

13.24

11.01

10.62

9.38

12.15

9.68

9.60

10.32

11.31

11.41

8.62

9.27

10.96

9.18

10.28

10.71

10.02

Game $1$	Game $2$
$32$	$21$	$27$	$46$	$114$	$87$	$96$	$92$
$9$	$16$	$19$	$19$	$101$	$111$	$80$	$106$
$40$	$28$	$42$	$36$	$85$	$112$	$117$	$94$
$11$	$38$	$23$	$28$	$62$	$43$	$106$	$66$
$8$	$18$	$26$	$59$	$104$	$51$	$76$	$91$
$62$	$40$			$111$	$78$

Game

1

Game

2

32

21

27

46

114

87

96

92

9

16

19

19

101

111

80

106

40

28

42

36

85

112

117

94

11

38

23

28

62

43

106

66

8

18

26

59

104

51

76

91

62

40

111

78

Cricket Runs Scored in Game $1$
Number of Runs Scored	Frequency
$0 - 9$	$2$
$10 - 19$	$5$
$20 - 29$	$6$
$30 - 39$	$3$
$40 - 49$	$4$
$50 - 59$	$1$
$60 - 69$	$1$

Cricket Runs Scored in Game

1

Number of Runs Scored

Frequency

0 - 9

2

10 - 19

5

20 - 29

6

30 - 39

3

40 - 49

4

50 - 59

1

60 - 69

1

Cricket Runs Scored in Game $2$
Number of Runs Scored	Frequency
$40 - 49$	$1$
$50 - 59$	$1$
$60 - 69$	$2$
$70 - 79$	$2$
$80 - 89$	$3$
$90 - 99$	$4$
$100 - 109$	$4$
$110 - 119$	$5$

Cricket Runs Scored in Game

2

Number of Runs Scored

Frequency

40 - 49

1

50 - 59

1

60 - 69

2

70 - 79

2

80 - 89

3

90 - 99

4

100 - 109

4

110 - 119

5

Ages of People Who Enter the Italian Restaurant on a Typical Day
$15$	$53$	$55$	$60$	$38$	$56$
$62$	$14$	$44$	$24$	$32$	$10$
$42$	$54$	$47$	$67$	$60$	$50$
$61$	$30$	$30$	$62$	$62$	$65$
$56$	$52$	$35$	$25$	$34$	$32$

Ages of People Who Enter the Italian Restaurant on a Typical Day

15

53

55

60

38

56

62

14

44

24

32

10

42

54

47

67

60

50

61

30

30

62

62

65

56

52

35

25

34

32

a To find the five-number summary of the data set, begin by ordering the data values from least to greatest.

101415242530303232343538424447505253545556566060616262626567

Notice that the minimum value is

10

and the maximum value is

67 .

Additionally, the number of data points is

30,

an even number. Therefore, the median of the data set will be given by the mean of the middle numbers

47

and

50 .

Median : \frac{4 7 + 5 0}{2} = 48.5

Finally, the first quartile, or the median of the lower half, is

32,

and the third quartile is

60 .

The following table summarizes this information. Each description is matched with its corresponding value.

Five Number Summary
Minimum Value	$10$
First Quartile	$32$
Median	$48.5$
Third Quartile	$60$
Maximum Value	$67$

b The five-number summary found in Part A can be used to draw the box plot of the data set. Start by drawing a number line that includes the minimum and maximum values of the data. Next, graph points above the number line for the five-number summary.

Now, draw a box from the $first$ quartile to the $third$ quartile. Then, draw a line through the median and the whiskers from the box to the minimum and maximum values.

Notice that this plot corresponds to option A.

c In the box-plot drawn in the previous part, notice that the left whisker is longer than the right and that the median is closer to the right whisker than it is to the left.

Comparison of the distance of the median from the whiskers.

This means that the data is skewed left. In a skewed distribution, the five-number summary best describes the center and spread of the data.

d In a box plot, each whisker represents

25 %

of the data and the box represents the middle

50 %

of the data. With this in mind, the following facts about the distribution of the data set can be determined.

$25 %$ of the people who enter the Italian restaurant in a regular day are between $10$ and $32$ years old.
$50 %$ of the people who enter the Italian restaurant in a regular day are between $32$ and $60$ years old.
$25 %$ of the people who enter the Italian restaurant in a regular day are between $60$ and $67$ years old.

Therefore, the option that states that $50 %$ of the people who enter the Italian restaurant in a regular day are between $32$ and $60$ years old is the right one.

Average Main Dish Price (Dollars)
$10.12$	$9.29$	$8.29$	$9.78$	$10.69$
$9.68$	$12.09$	$8.94$	$10.81$	$8.62$
$11.39$	$12.62$	$8.71$	$10.74$	$10.52$
$10.77$	$10.15$	$9.18$	$8.45$	$9.52$
$11.89$	$9.77$	$9.44$	$13.24$	$11.01$
$10.62$	$9.38$	$12.15$	$9.68$	$9.60$
$10.32$	$11.31$	$11.41$	$8.62$	$9.27$
$10.96$	$9.18$	$10.28$	$10.71$	$10.02$

Average Main Dish Price (Dollars)

10.12

9.29

8.29

9.78

10.69

9.68

12.09

8.94

10.81

8.62

11.39

12.62

8.71

10.74

10.52

10.77

10.15

9.18

8.45

9.52

11.89

9.77

9.44

13.24

11.01

10.62

9.38

12.15

9.68

9.60

10.32

11.31

11.41

8.62

9.27

10.96

9.18

10.28

10.71

10.02

Average Main Dish Price (Dollars)
Price Range	Frequency
$8.00 - 8.99$	$6$
$9.00 - 9.99$	$12$
$10.00 - 10.99$	$13$
$11.00 - 11.99$	$5$
$12.00 - 12.99$	$3$
$13.00 - 13.99$	$1$

Average Main Dish Price (Dollars)

Price Range

Frequency

8.00 - 8.99

6

9.00 - 9.99

12

10.00 - 10.99

13

11.00 - 11.99

5

12.00 - 12.99

3

13.00 - 13.99

1

Distribution	Measure of Center	Measure of Variation
Symmetric	Mean	Standard deviation
Skewed	Median	Five-number summary

Distribution

Measure of Center

Measure of Variation

Symmetric

Mean

Standard deviation

Skewed

Median

Five-number summary

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Catch-Up and Review

Challenge

Giving Meaning to a Data Set

Example

Finding the Distribution of Runs Scored in Cricket

Hint

Solution

Discussion

Uniform and Bimodal Distributions

Discussion

Box-and-Whisker Plots as Distributions

Example

Analyzing the Ages of Customers at an Italian Restaurant

Hint

Solution

Closure

Finding Insights and Drawing Conclusions From Histograms

Hint

Solution

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