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Probability and Statistics
Chapter 3 – Modeling Distribution of Data
3.1 Measuring
Location in a
Distribution
Objective:
 MEASURE position using percentiles
 INTERPRET cumulative relative frequency graphs
 MEASURE position using z-scores
 TRANSFORM data
 DEFINE and DESCRIBE density curves
Measuring Position:
Percentiles
The pth percentile of a distribution is the value with p percent of the observations less than it.
Examples:
Here are the scores of all 25 students in Mr. Pryor’s statistics class on their first test:
79
77
81
83
80
86
77
90
73
79
83
85
74
83
93
89
78
84
80
82
75
77
67
72
73
Problem: Use the scores on Mr. Pryor’s test to find the percentiles for the for the following students
(how did they perform relative to their classmates):
a) Jenny, who earned an 86.
b) Norman, who earned a 72.
c) Katie, who earned a 93.
Example
Measuring Position:
z-Scores
d) the two students who earned scores of 80.
PSAT scores
In October 2007, about 1.4 million college-bound high school juniors took the PSAT. The mean score on
the Critical Reading test was 46.7 and the standard deviation was 11.3. Nationally, 6 percent of testtakers earned a score higher than 65 on the Critical Reading test’s 20 to 80 scale. 3tps3ech2
Scott was one of 50 junior boys to take the PSAT at his school. He scored 65 on the Critical
Reading test. This placed Scott at the 68th percentile within the group of boys. Looking at all 50 boys’
Critical Reading scores, the mean was 58.2 and the standard deviation was 9.4.
Write a sentence or two comparing Scott’s percentile among the national group of test takers and
among the 50 boys at his school.
A z-score tells us how many standard deviations from the mean an observation falls, and in what
direction.
To compare data from distributions with different means and standard deviations, we need to find a
common scale. We accomplish this by using standard deviation units (z-scores) as our scale. Changing
to these units is called standardizing. Standardizing data shifts the data by subtracting the mean and
rescales the values by dividing by their standard deviation.
z  score 
datavalue  mean
st .dev.
or
z
x

Standardizing does not change the shape of the distribution. It changes the center (shifts it to zero) and
the spread by making the standard deviation one.
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Example:
Example:
PSAT scores (continued)
Refer to the previous example. Calculate and compare Scott’s z-score among these same two groups of
test takers.
Assignment 3.1
Part 1, page 105
#3.1-3.6
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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.
Effect of Adding (or
Subtracting) 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).
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:
Remember: Exploring
Quantitative Data
To describe a distribution:
- Make a graph
- Look for overall patterns (shape, center, and spread) and outliers
- Calculate a numerical summary to describe the center (mean, median) and spread (minimum,
maximum, Q1, Q3, range, IQR, standard deviation)
 In addition to the above distributions sometimes the overall pattern of a large number of
observations is so regular that we can describe it by a smooth curve.
Density Curves
A density curve describes the
overall pattern of a distribution
o Is always on or above the
horizontal axis
o Has exactly 1 underneath it
o The area under the curve and
above any range of values is the
proportion of all observations
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.
Median and Mean of
a Density Curve
Median of a density curve is the equal areas point, the point that divides the are under the curve in half
Mean of a density curve is the balance point, at which the curve would balance if made of solid
material.
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Examples:
Use the figure shown to answer the following questions.
1. Explain why this is a legitimate density curve.
2. About what proportion of observations lie
between 7 and 8?
3. Mark the approximate location of the median.
4. Mark the approximate location of the mean.
Explain why the mean and median have the
relationship that they do in this case.
Examples:
Assignment 3.1
Part 2, page 110
#3.7-3.12
Assignement 3.1
Part 3, page 113
#3.13-3.20
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3.2 Normal
Distributions
Objectives:
 DESCRIBE and APPLY the 68-95-99.7 Rule
 DESCRIBE the standard Normal Distribution
 PERFORM Normal distribution calculations
 ASSESS Normality
Normal
Distributions
N(μ, σ)







The 68-95-99.7 Rule
All Normal curves have the same overall shape: symmetric, single-peaked, bell shaped.
A Normal distribution is described by a Normal density curve.
A Normal distribution can be fully described by two parameters, its mean μ and standard deviation σ
The mean, µ, of a Normal distribution is at the center of the symmetric Normal curve and is the same
as the median.
The standard deviation σ controls the spread of a Normal curve. Curves with larger standard
deviations are more spread out.
The standard deviation, σ, is the distance from the center to the change-of-curvature points on either
side.
A short-cut notation for the normal distribution in N(μ,σ).
All normal curves obey the 68-95-99.7% (Empirical) Rule.
This rule tells us that in a normal distribution approximately
68% of the data values fall within one standard
deviation (1σ) of the mean,
95% of the values fall within 2σ of the mean, and
99.7% (almost all) of the values fall
within 3σ of the mean.
Application of the
68-95-99.7 Rule
Distribution of the heights of young women aged 18 to 24
What is the mean μ?
What is the ?
What is the height range for 95% of young women?
What is the percentile for 64.5 in.?
What is the percentile for 59.5 in.?
What is the percentile for 67 in.?
What is the percentile for 72 in.?
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Example SAT performance
Students’ scores on the SAT Critical Reading test follow a Normal distribution with mean 500 and
standard deviation 100. What percent of students earn scores above 700?
Assignment 3.2
Part 1, page 121
#3.21-3.26
The Standard
Normal Distribution
The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1.
If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the
standardized variable
Z-Score Table
z
x

has the standard Normal distribution, N(0,1).
Because all Normal distributions are the same when we standardize, we can find areas under any Normal
curve from a single table.
Table A is a table of areas under the standard Normal curve. The table entry for each value z is the area
under the curve to the left of z.
Table A practice
Example Use Table A to find the proportion of observations from a standard Normal distribution that falls in each
of the following regions. In each case, sketch a standard Normal curve and shade the area representing
the region.
(a) z  2.25
(b) z  2.25
(c) z  1.77
(d) 2.25  z  1.77
Example
Finding z-scores from proportions
Use Table A to find the value z of a standard Normal variable that satisfies each of the following
conditions. In each case, sketch a standard Normal curve with your value of z marked on the axis.
(a) The point z with 70% of the observations falling below it.
(b) The point z with 85% of the observations falling above it.
Assignment 3.2
Part 2, page 127
#3.27-3.32
(c) Find the number z such that the proportion of observations less than z is 0.8.
(d) Find the number z such that 90% of all observations are greater than z.
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4-Step Process
How to Solve Problems Involving Normal Distributions
Step 1:
Step 2:
Step 3:
Step 4:
Normal calculations
Example: a.
Women’s heights are approximately normal with N(64.5, 2.5).
What proportion of all young women are less than 68 inches tall?
b.
On the driving range, Tiger Woods practices his swing with a particular club by hitting many,
many balls. When tiger hits his driver, the distance the balls travels follows a Normal
distribution with mean 304 yards and standard deviation 8 yards. What percent of Tiger’s
drives travel at least 290 yards?
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c.
What percent of Tiger’s drives travel between 305 and 325?
Using Table A in
Reverse
Example: d. What distance would a ball have to travel to be at the 80th percentile f Tiger’s drive lengths?
Assignment 3.2 Part
3 page 132 #3.333.38
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Summary
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