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AP Statistics
Chapter 2 - The Normal Distribution
2.1 Density Curves
and the Normal
Distribution
What You should
Know From
Chapter 1:
Objective:
1) Know that the area under a density curve represent proportions of all observations and that the total
area under a density curve is 1.
2) Approximately locate the median (equal-area point) and mean (balance point) on a density curve.
3) Know that the mean and median both lie at the center of a symmetric density curve and that the
mean moves farther toward the long tail of a skewed curve.
4) Estimate μ and  from a normal curve
5) Use the 68-95-99.7 rule and symmetry to state what percent of the observations from a normal
distribution fall between two points
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.
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
80
85
81
75
83
80
67
89
77
73
84
73
77
82
83
83
77
74
86
72
93
90
78
79
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 89.
b) Norman, who earned a 72.
c) Katie, who earned a 93.
Density Curves
d) the two students who earned scores of 80.
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
Example 2.1 page 79
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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. See figures 2.5 and 2.6 page 81-82.
Density Curve
Mean (μ)
S.D. ()
Example:
Computed from actual observations
Mean ( x )
S.D. (s)
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.
Practice Problem 2.2 Uniform Density Curve
page 83 #2.2
a.
b.
c.
d.
e.
a.
Practice Problem
page 84 #2.3
b.
c.
d.
e.
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Normal Curve




All Normal curves have the same overall shape: symmetric, single-peaked, bell shaped.
A Normal distribution can be fully described by two parameters, its mean μ and standard deviation σ
The mean is located in the center of the symmetric 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.
Normal Distributions
N(μ,)



A Normal distribution is described by a Normal density curve.
The mean, µ, of a Normal distribution is at the center of the symmetric Normal curve.
The standard deviation, σ, is the deviation 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,).

The 68-95-99.7 Rule
All normal curves they 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 (1) of the mean,
95% of the values fall within 2 of the mean, and
99.7% (almost all) of the values fall
within 3 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 s.d. ?
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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Practice Problem
page 89 #2.6 and 2.7
 
Objective:
2.2 Standard Normal 6) Find and interpret the z score of an observation
Calculations
7) Use the Z table and calculator to calculate the proportion of values above, below, or between two
The Standard
Normal Distribution
Z-Score
stated numbers
8) Calculate the point having a stated proportion of all values above or below it.
9) Construct and interpret a Normal Probability Plot
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.
The standard normal distribution is the normal distribution N(0,1) with mean 0 and standard deviation 1.
A z-score tells us how many standard deviations the original observation falls away from the mean, and
in which direction. Observations larger than the mean are positive when standardized, and observations
smaller than the mean are negative.
The standard Normal
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Table
Normal Distribution
Calculations
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.
For example, lets say that we know a girl named Georgia who is 60 inches tall and a girl named
Deanna that is 68 inches tall. What are Georgia’s and Deanna’s standardized heights? Women’s
heights are approximately normal with N(64.5, 2.5).
a. For example, what proportion of all young women are less than 68 inches tall
what is the percentile for Deanna’s height)?
(in other words
b. In what percentile does Georgia fall?
How to Solve
Problems Involving
Normal Distributions
State: Express the problem in terms of the observed variable x.
Plan: Draw a picture of the distribution and shade the area of interest under the curve.
Do: Perform calculations.
 Standardize x to restate the problem in terms of a standard Normal variable x.
 Use Table A and the fact that the total area under the curve is 1 to find the required area under the
standard Normal curve.
Conclude: Write your conclusion in context of the problem
Normal calculations
Example: 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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More complicated
calculations
Example: What percent of Tiger’s drives travel between 305 and 325?
Using Table A in
High levels of cholesterol in the blood increase the risk of heart disease. For 14 year old boys, the
Reverse
distribution of blood cholesterol is approximately Normal with mean µ = 170 milligrams of cholesterol
Example:
per deciliter of blood (mg/dl) and standard deviation σ = 30 mg/dl. What is the first quartile off the
distribution of blood cholesterol?
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Normal Probability
Plots
Assessing Normality using the Calculator
To decide if a set of data is normal we can construct a normal probability plot.
See Technology Toolbox page 105-106
Normal Distributions
on the Calculator (See
Technology Box page
115-117)

Finding Areas with ShadeNorm

Finding areas with normalcdf

Finding z values with invNorm
Assignment 2.2 page
109 #2.28-2.33,2.35
Summary
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