The Practice of Statistics, 5th Edition

Starnes, Tabor, Yates, Moore

Bedford Freeman Worth Publishers

CHAPTER 2 Modeling Distributions of Data 2.1 Describing Location in a Distribution

Learning Objectives

After this section, you should be able to:

The Practice of Statistics, 5th Edition 2

Percentiles: find and interpret the percentile of an individual value

within a distribution of data.

Cumulative relative frequency graph: estimate percentiles and

individual values using a cumulative relative frequency graph.

Z-score: find and interpret the standardized score (z-score) of an

individual value within a distribution of data.

Effect of adding, subtracting, multiplying by, or dividing by a

constant on the shape, center, and spread of a distribution of data.

Describing Location in a Distribution

The Practice of Statistics, 5th Edition 3

Measuring Position: Percentiles

One way to describe the location of a value in a distribution is to tell

what percent of observations are less than it.

The pth percentile of a distribution is the value with p percent of the

observations less than it.

6 7

7 2334

7 5777899

8 00123334

8 569

9 03

Jenny earned a score of 86 on her test. How did she perform

relative to the rest of the class? Michael got a 73 on his test.

How did he perform? And whose score is more unusual?

Example

Her score was greater than 21 of the 25

observations. Since 21 of the 25, or 84%, of the

scores are below hers, Jenny is at the 84th

percentile in the class’s test score distribution.

6 7

7 2334

7 5777899

8 00123334

8 569

9 03

The Practice of Statistics, 5th Edition 4

Practice: Wins in major League baseball

• The stemplot below shows the number of wins for each of the 30

Major League Baseball teams in 2012.

Problem: Find the percentiles for the following teams:

(a) The Minnesota Twins, who won 66 games.

(b) The Washington Nationals, who won 98 games.

(c) The Texas Rangers and Baltimore Orioles, who both won 93

games.

The Practice of Statistics, 5th Edition 5

Cumulative Relative Frequency Graphs

Interesting graphs with percentile: one being a cumulative relative

frequency graph displays the cumulative relative frequency of each

class of a frequency distribution.

Age of First 44 Presidents When They Were

Inaugurated

Age Frequenc

y

Relative

frequency

Cumulative

frequency

Cumulative

relative

frequency

40-44 2 2/44 =

4.5%

2 2/44 =

4.5%

45-49 7 7/44 =

15.9%

9 9/44 =

20.5%

50-54 13 13/44 =

29.5%

22 22/44 =

50.0%

55-59 12 12/44 =

34%

34 34/44 =

77.3%

60-64 7 7/44 =

15.9%

41 41/44 =

93.2%

65-69 3 3/44 =

6.8%

44 44/44 =

100%

0

20

40

60

80

100

40 45 50 55 60 65 70

Cu

mu

lati

ve r

ela

tive

fre

qu

en

cy (

%)

Age at inauguration

Relative

Freq Cumul.

freq

Cumulative

Relative

Frequency

The Practice of Statistics, 5th Edition 6

Practice: State median household incomes

• The table and cumulative relative frequency graph below show the

distribution of median household incomes for the 50 states and the

District of Columbia in a recent year.

• Problem: Use the cumulative relative frequency graph for the state

income data to answer each question.

• (a) At what percentile is California, with a median household income

of $57,445?

• (b) Estimate and interpret the first quartile of this distribution.

Median

income

($1000s)

Frequency Relative

frequency

Cumulative

frequency

Cumulative

relative

frequency

35 to < 40 1 1/51 = 0.020 1 1/51 = 0.020

40 to < 45 10 10/51 = 0.196 11 11/51 = 0.216

45 to < 50 14 14/51 = 0.275 25 25/51 = 0.490

50 to < 55 12 12/51 = 0.236 37 37/51 = 0.725

55 to < 60 5 5/51 = 0.098 42 42/51 = 0.824

60 to < 65 6 6/51 = 0.118 48 48/51 = 0.941

65 to < 70 3 3/51 = 0.059 51 51/51 = 1.000

The Practice of Statistics, 5th Edition 7

Measuring Position: z-Scores

A z-score tells us how many standard deviations from the mean an

observation falls, and in what direction.

If x is an observation from a distribution that has known mean and

standard deviation, the standardized score of x is:

A standardized score is often called a z-score.

 

z =x - mean

standard deviation

Jenny earned a score of 86 on her test. The class mean is 80 and

the standard deviation is 6.07. What is her standardized score?

 

z =x - mean

standard deviation=

86 - 80

6.07= 0.99

Example

The Practice of Statistics, 5th Edition 8

Year Player HR Mean SD

1927 Babe Ruth 60 7.2 9.7

1961 Roger

Maris 61 18.8 13.4

1998 Mark

McGwire 70 20.7 12.7

2001 Barry

Bonds 73 21.4 13.2

PRACTICE: The single-season home run record for Major League

Baseball has been set just three times since Babe Ruth hit 60 home runs

in 1927. Roger Maris hit 61 in 1961, Mark McGwire hit 70 in 1998,and

Barry Bonds hit 73 in 2001. In an absolute sense, Barry Bonds had the

best performance of these four players, because he hit the most home

runs in a single season. However, in a relative sense, this may not be true.

Baseball historians suggest that hitting a home run has been easier in

some eras than others. This is due to many factors, including quality of

batters, quality of pitchers, hardness of the baseball, dimensions of

ballparks, and possible use of performance-enhancing drugs. To make a

fair comparison, we should see how these performances rate relative to

those of other hitters during the same year.

Problem: Compute the standardized scores for each performance using

the information in the table. Which player had the most outstanding

performance relative to his peers?

The Practice of Statistics, 5th Edition 9

Homework

• Page100 #1-18 odd

• Extra Credit Project: Due October 3rd

– Recommended websites: www.censusatschool.com &

www.gapminder.com

– Individual work

– Extra credit: will replace your lowest POP Quiz grade

– Must make a poster and present the results to the class

• Chapter 2 Quiz: October 15th