Download Section 2.1 - Effingham County Schools

+
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
+
Chapter 2
Modeling Distributions of Data
 2.1
Describing Location in a Distribution
 2.2
Normal Distributions
+ 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
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.
Example, p. 85
Jenny earned a score of 86 on her test. How did she perform
relative to the rest of the class?
6 7
7 2334
7 5777899
8 00123334
8 569
9 03
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.
Describing Location in a Distribution

Position: Percentiles
+
 Measuring
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.
Example
Key: 5|9 represents a
team with 59 wins.
+
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.
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%
Describing Location in a Distribution
 Cumulative
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
Frequency
Relative
frequency
Cumulative
frequency
Cumulative
relative
frequency
4044
2
2/44 =
4.5%
2
2/44 =
4.5%
4549
7
7/44 =
15.9%
9
9/44 =
20.5%
5054
13
13/44 =
29.5%
22
22/44 =
50.0%
5559
12
12/44 =
34%
34
34/44 =
77.3%
6064
7
7/44 =
15.9%
41
41/44 =
93.2%
6569
3
3/44 =
6.8%
44
44/44 =
100%
+
Relative Frequency Graphs
Describing Location in a Distribution
 Cumulative
Interpreting Cumulative Relative Frequency Graphs

Was Barack Obama, who was inaugurated at age 47,
unusually young?

Estimate and interpret the 65th percentile of the distribution
65
11
47
58
Describing Location in a Distribution
Use the graph from page 88 to answer the following questions.
+


Here is a cumulative relative frequency graph showing the
distribution of median household incomes for the 50 states
and the District of Columbia.
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?
+
Example: State Median Household Incomes
Example

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:
x - mean
z=
standard deviation
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?
x - mean
86 - 80
z=
=
= 0.99
standard deviation
6.07
Describing Location in a Distribution

Position: z-Scores
+
 Measuring
+
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.
86 - 80
zstats =
6.07
82 - 76
=
4
zchem
zstats = 0.99
zchem = 1.5
Describing Location in a Distribution
 Using

Ruth: 60 – 7.2 = 5.44
McGwire: 70 – 20.7 = 3.87
9.7

Maris: 61 – 18.8 = 3.16
13.4
12.7
Bonds: 73 – 21.4 = 3.91
13.2
Although all 4 performances were outstanding, Babe Ruth
can still lay claim to being the single-season home run
champ, relatively speaking.
+
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.
Example

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).
+
Data
Describing Location in a Distribution
 Transforming

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?
+
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.
Example

+
Example

The measures of center and position all increased by 5.
However, the shape and spread of the distribution did not
change.
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
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!!!
+
Data
Describing Location in a Distribution
 Transforming

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.
+
Curves
Describing Location in a Distribution
 Density
Curve
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.
Describing Location in a Distribution
Definition:
+
 Density
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.
+

+
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.
Density Curves
NOTE:
+
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).
+
Looking Ahead…
In the next Section…
We’ll learn about one particularly important class of
density curves – the Normal Distributions
We’ll learn
The 68-95-99.7 Rule
The Standard Normal Distribution
Normal Distribution Calculations, and
Assessing Normality