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Chapter 2: Modeling Distributions of DataSection 2.1Describing Location in a Distribution

The Practice of Statistics, 4th edition - For AP*STARNES, YATES, MOORE

+Chapter 2Modeling Distributions of Data

2.1 Describing Location in a Distribution

2.2 Normal Distributions

+Section 2.1Describing Location in a Distribution

After this section, you should be able to…

MEASURE position using percentiles

INTERPRET cumulative relative frequency graphs

MEASURE position using z-scores

TRANSFORM data

DEFINE and DESCRIBE density curves

Learning Objectives

+Describing Location in a D

istribution 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.

Definition:

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?

Example, p. 85

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

+Exam

ple Example: Wins in Major League Baseball

The stemplot below shows the number of wins for each of the 30 Major League Baseball teams in 2009.

Calculate and interpret the percentiles for the Colorado Rockies who had 92 wins, the New York Yankees who had 103 wins, and the Cleveland Indians who had 65 wins.

Key: 5|9 represents a team with 59 wins.

+ Cumulative Relative Frequency

Cumulative Relative Frequency adds the counts in the frequency column for the current class and all previous classes.

Example: Let’s look at the frequency table for the ages of the 1st 44 US presidents when they were inaugurated.

Describing Location in a D

istribution

Age Frequency Relative Frequency

Cumulative Frequency

Cumulative Rel. Frequency

40 – 44 2 0.045 2 0.045

45 – 49 7 0.159 9 0.205

50 – 54 13 0.295 22 0.500

55 – 59 12 0.273 34 0.773

60 – 64 7 0.159 41 0.932

65 – 69 3 0.068 44 1.000

Total 44 1.00 or 100%

+ Cumulative Relative Frequency Graphs

A cumulative relative frequency graph displays the cumulative relative frequency of each class of a frequency distribution.

Describing Location in a D

istribution

Age of First 44 Presidents When They Were Inaugurated

Age Frequency 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%

+Describing Location in a D

istribution

Use the graph from page 88 to answer the following questions.

Was Barack Obama, who was inaugurated at age 47, unusually young?

Estimate and interpret the 65th percentile of the distribution

Interpreting Cumulative Relative Frequency Graphs

47

11

65

58

+Exam

ple Example: State Median Household Incomes

Here is a cumulative relative frequency graph showing the distribution of median household incomes for the 50 states and the District of Columbia.

0.0

0.2

0.4

0.6

0.8

1.0

35 40 45 50 55 60 65 70Median_Household_Income

The point (50, 0.49) means that 49% of the states had a median household income of less than $50,000.

a) California, with a median household income of $57,445, is at what percentile? Interpret this value.  b) What is the 25th percentile for this distribution? What is another name for this value?

+Describing Location in a D

istribution Measuring Position: z-Scores

A z-score is a value that tells ushow many standard deviations from the mean an observation falls, and in what direction.

Definition:

If x is an observation from a distribution that has known mean and standard deviation, the standardized value of x is:

A standardized value is often called a z-score.

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?

+Describing Location in a D

istribution Using z-scores for Comparison

We can use z-scores to compare the position of individuals in different distributions.

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, since he hit the most home runs in a single season. However, in a relative sense this may not be true. To make a fair comparison, we should see how these performances rate relative to others hitters during the same year. Calculate the standardized score for each player and compare.

+Exam

ple 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, since he hit the most home runs in a single season. However, in a relative sense this may not be true. To make a fair comparison, we should see how these performances rate relative to others hitters during the same year. Calculate the standardized score for each player and compare.

Ruth: 60 – 7.2 = 5.44 McGwire: 70 – 20.7 = 3.87

9.7 12.7

Maris: 61 – 18.8 = 3.16 Bonds: 73 – 21.4 = 3.91

13.4 13.2

Although all 4 performances were outstanding, Babe Ruth can still lay claim to being the single-season home run champ, relatively speaking.

+Describing Location in a D

istribution Transforming Data

Transforming converts the original observations from the original units of measurements to another scale. Transformations can affect the shape, center, and spread of a distribution.

Adding the same number a (either positive, zero, or negative) to each observation:

•adds a to measures of center and location (mean, median, quartiles, percentiles), but

•Does not change the shape of the distribution or measures of spread (range, IQR, standard deviation).

Effect of Adding (or Subracting) a Constant

+Exam

ple Example: Here is a graph and table of summary statistics

for a sample of 30 test scores. The maximum possible score on the test was a 50.

Suppose that the teacher is nice and wants to add 5 points to each test score. How would this affect the shape, center and spread of the data?

+Exam

ple The measures of center and position all increased by 5.

However, the shape and spread of the distribution did not change.

+Describing Location in a D

istribution Transforming Data

Multiplying (or dividing) each observation by the same number b (positive, negative, or zero):

•multiplies (divides) measures of center and location by b

•multiplies (divides) measures of spread by |b|, but

•does not change the shape of the distribution

Effect of Multiplying (or Dividing) by a Constant

Example: Suppose that the teacher wants to convert each test score to percents. Since the test was out of 50 points, he should just multiply each score by 2 to make them out of 100. How would this change the shape, center and spread of the data?

The measures of center, location and spread will all double. However, even though the distribution is more spread out, the shape will not change!!!

+Describing Location in a D

istribution Density Curves

In Chapter 1, we developed a kit of graphical and numerical tools for describing distributions. Now, we’ll add one more step to the strategy.

So far our strategy for exploring data is :

1. Graph the data to get an idea of the overall pattern

2. Calculate an appropriate numerical summary to describe the center and spread of the distribution.

Sometimes the overall pattern of a large number of observations is so regular, that we can describe it by a smooth curve, called a density curve.

+Describing Location in a D

istribution Density Curve

Definition:

A density curve is a curve that•is always on or above the horizontal axis, and•has area exactly 1 underneath it.

A density curve describes the overall pattern of a distribution. The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval.

The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars.

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Here’s the idea behind density curves:

+ A density curve has the following properties: The area under the density curve and above any range

of values is the proportion of all observations that fall in that range.

+Density C

urves NOTE: No real set of data is described by a density curve.

It is simply an approximation that is easy to use and accurate enough for practical use.

Describing Density Curves

The median of the density curve is the “equal-areas” point.

The mean of the density curve is the “balance” point.

+Section 2.1Describing Location in a Distribution

In this section, we learned that…

There are two ways of describing an individual’s location within a distribution – the percentile and z-score.

A cumulative relative frequency graph allows us to examine location within a distribution.

It is common to transform data, especially when changing units of measurement. Transforming data can affect the shape, center, and spread of a distribution.

We can sometimes describe the overall pattern of a distribution by a density curve (an idealized description of a distribution that smooths out the irregularities in the actual data).

Summary

+Looking Ahead…

We’ll learn about one particularly important class of density curves – the Normal Distributions

We’ll learn The 68-95-99.7 RuleThe Standard Normal DistributionNormal Distribution Calculations, andAssessing Normality

In the next Section…