1. Chapter 2: Modeling Distributions of Data
Section 2.1
Describing Location in a Distribution
The Practice of Statistics, 4th edition - For AP*
STARNES, YATES, MOORE

2. Chapter 2 Modeling Distributions of Data
2.1 Describing Location in a Distribution
2.2 Normal Distributions

3. Section 2.1 Describing Location in a Distribution
Learning Objectives
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

4. Describing Location in a Distribution
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.
Describing Location in a Distribution
Definition:
The pth percentile of a distribution is the value with p percent of the observations less than it.
6 7
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
9 03

5. Describing Location in a Distribution
Measuring Position: Percentiles
Describing Location in a Distribution
5 9
10 3
The stemplot below shows the number of wins for each of the 30 Major League Baseball teams in 2009.
Alternate Example
Find the percentiles for the following teams:
(a) The Colorado Rockies, who won 92 games.
(b) The New York Yankees, who won 103 games.
(c) The Kansas City Royals and Cleveland Indians, who both won 65 games.
80th Percentile
97th Percentile
10th Percentile

6. Describing Location in a Distribution
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 Distribution
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%
2/44 =
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 41
41/44 = 93.2%
65-69 3
3/44 = 6.8% 44
44/44 = 100%

7. Describing Location in a Distribution
Interpreting Cumulative Relative Frequency Graphs
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
Describing Location in a Distribution 65 11 58 47

8. Describing Location in a Distribution
Alternate Example
Describing Location in a Distribution
Here is a table showing the distribution of median household incomes for the 50 states and the District of Columbia.
Median Income
$1000s
Freq
Relative frequency
Cumulative frequency
Cumulative relative frequency
35 to <40 1
1/51=0.020 or 2.0%
40 to <45 10
10/51=0.196 or 19.6% 11
11/51=0.216 or 21.6%
45 to <50 14
14/51=0.275 or 27.5% 25
25/51=0.490 or 49.0%
50 to <55 12
12/51=0.236 or 23.6% 37
37/51=0.725 or 72.5%
55 to <60 5
5/51=0.098 or 9.8% 42
42/51=0.824 or 82.4%
60 to <65 6
6/51=0.118 or 11.8% 48
48/51=0.941 or 94.1%
65 to <70 3
3/51=0.059 or 5.9% 51
51/51=1.000 or 100%

9. Describing Location in a Distribution
Interpreting Cumulative Relative Frequency Graphs
Describing Location in a Distribution
Use the graph from the alternate example to answer the following questions.
At what percentile is California, with a median income of $57,445?
Estimate and interpret the first quartile of the distribution.

10. Describing Location in a Distribution
Measuring Position: z-Scores
A z-score tells us how many standard deviations from the mean an observation falls, and in what direction.
Describing Location in a Distribution
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 What is her standardized score?

11. Describing Location in a Distribution
Using z-scores for Comparison
Describing Location in a Distribution
We can use z-scores to compare the position of individuals in different distributions.
Example, p. 91
Jenny earned a score of 86 on her statistics test. The class mean was 80 and the standard deviation was She earned a score of 82 on her chemistry test. The chemistry scores had a fairly symmetric distribution with a mean 76 and standard deviation of 4. On which test did Jenny perform better relative to the rest of her class?

12. Describing Location in a Distribution
Using z-scores for Comparison
Alternate Example
Describing Location in a Distribution
The single-season home run record for MLB has been set just three times since Babe Ruth hit 60 home runs in In an absolute sense, Barry Bonds had the best performance of the four players, since 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. To make a fair comparison, we should see how these performances rate relative to those of other hitters during the same year.
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
Compute the standardized scores for each performance. Which player had the most outstanding performance relative to his peers?
Babe Ruth – 5.44
Roger Maris – 3.16
Mark McGwire – 3.87
Barry Bonds – 3.91

13. Describing Location in a Distribution
Transforming Data
Describing Location in a Distribution
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.
Effect of Adding (or Subracting) a Constant
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).
Example, p. 93 n
Mean sx
Min Q1 M Q3
Max
IQR
Range
Guess(m) 44
16.02
7.14 8 11 15 17 40 6 32
Error (m)
3.02 -5 -2 2 4 27

14. Describing Location in a Distribution
Transforming Data
Describing Location in a Distribution
Effect of Multiplying (or Dividing) by a Constant
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 n
Mean sx
Min Q1 M Q3
Max
IQR
Range
Error(ft) 44
9.91
23.43
-16.4
-6.56
6.56
13.12
88.56
19.68
104.96
Error (m)
3.02
7.14 -5 -2 2 4 27 6 32
Example, p. 95

15. Describing Location in a Distribution
Transforming Data
Describing Location in a Distribution
Example, p. 96
During the winter months, the temperatures at the Starnes’ Colorado cabin can stay well below freezing for weeks at a time. To prevent the pipes from freezing, Mrs. Starnes sets thermostat at 50F and records the temperature each night at midnight, but in degrees Celsius.
Find the mean temp in degrees Fahrenheit. Does the thermostat seem accurate?
Calculate the standard deviation of the readings in degrees Fahrenheit. Interpret this value in context.
The 90th percentile of the readings was 11C. What is the 90th percentile temp in degrees Fahrenheit?

16. Describing Location in a Distribution
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.
Describing Location in a Distribution
Exploring Quantitative Data
Always plot your data: make a graph.
Look for the overall pattern (shape, center, and spread) and for striking departures such as outliers.
Calculate a numerical summary to briefly describe center and spread.
4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

17. Describing Location in a Distribution
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.
Describing Location in a Distribution
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.

18. Describing Location in a Distribution
Density Curve
Distinguishing the Median and Mean of a Density Curve
Describing Location in a Distribution
The median of a density curve is the equal-areas point, the point that divides the area under the curve in half.
The mean of a density curve is the balance point, at which the curve would balance if made of solid material.
The median and mean are the same for a symmetric density curve. The both lie in the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail.

19. Describing Location in a Distribution
Describing Density Curves
Our measures of center and spread apply to density curves as well as to actual sets of observations.
Describing Location in a Distribution
Distinguishing the Median and Mean of a Density Curve
The median of a density curve is the equal-areas point, the point that divides the area under the curve in half.
The mean of a density curve is the balance point, at which the curve would balance if made of solid material.
The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail.

20. Section 2.1 Describing Location in a Distribution
Summary
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).

21. Looking Ahead… In the next Section…
We’ll learn about one particularly important class of density curves – the Normal Distributions
We’ll learn
The Rule
The Standard Normal Distribution
Normal Distribution Calculations, and
Assessing Normality
In the next Section…