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AP Stats
2 Describing Location in a Distribution
2.2 Normal Distributions Day 1
Date _______
Objectives
 Identify the main properties of the Normal Curve as a particular density curve.
 List 3 reasons why Normal distributions are important in statistics.
 Explain the 68-95-99.7 rule.
 Explain the notation N(μ, σ)
 Define the standard Normal distribution.
 Use a table of values for the standard Normal curve (Table A) to compute the
proportion of observations that are less than or greater than a particular z-score or
between two give z-scores.
Normal Distributions
An important class of density curves are called Normal curves. They describe __________
distributions. Normal curves are symmetric, single-peaked (unimodal), and bell-shaped.
ALL Normal distributions have the _______ general shape! The exact density curve for a
particular Normal distribution is described by giving its mean μ and its standard deviation σ.
*** μ and σ completely determine the shape of a Normal curve.*** The mean is located
at the center of the SYMMETRIC curve, and is the same as the __________. The
standard deviation σ controls the __________ of a Normal curve. One Normal curve will be
more spread out if it has a larger ___ than another Normal curve.
As you move out in either direction from the center μ, the curve changes from falling
steeply to not as steep (concave down to concave up). The points at which this change of
curvature takes place are located _____ on either side of the mean μ. These are also
known as inflection points.
3 reasons why Normal distributions are important in statistics?
 They are good descriptions for some distributions of _______ _______.
o Distributions that are close to Normal
 Scores on tests (SAT and many psychological tests)
 Repeated careful measurements of the same quantity
 Characteristics of biological populations (yields of corn and lengths of
pregnancies)
 They are good approximations to the results of many kinds of _______ _________.
o Tossing a coin many times
 Many statistical inference procedures based on Normal distributions work well for
other roughly _________________ distributions. (We’ll see this later in the course.)
The 68-95-99.7 Rule
There are _______ Normal curves, however they all have __________ properties. All
Normal distributions obey the following rule:
In the Normal distribution with mean μ and standard deviation σ
 Approximately _____ of the observations fall within 1σ of the mean μ.
 Approximately _____ of the observations fall within 2σ of the mean μ.
 Approximately _____ of the observations fall within 3σ of the mean μ.
Remembering this rule allows us to think about Normal distributions without constantly
making detailed calculations when rough approximations are good enough.
The 68-95-99.7 rule gives us much more __________ information about how the
observations fall in a __________ distribution than Chebyshev’s inequality did.
How come???
Example 2.6 Young Women’s Heights
The distribution of heights of young women aged 18-24 is approximately Normal with
mean μ = 64.5 inches and standard deviation σ = 2.5 inches. So:
 Approximately 68% of women are between _____ and _____ inches tall.
 Approximately 95% of women are between _____ and _____ inches tall.
 Approximately 99.7% of women are between _____ and _____ inches tall.
For the visual learners out there see pg. 136 and copy the picture into your notes.
Check out www.whfreeman.com/tps3e “Normal curve applet” reinforces the 68-95-99.7 rule
The Standard Normal Distribution is the Normal distribution N(0, 1) with mean ___ and
standard deviation ___. If a variable x has any Normal distribution (μ, σ) with mean μ and
standard deviation σ, then the standardized variable
has the standard Normal distribution.
Standard Normal Calculations
An area under a density curve is a _______________ of the observations in a distribution.
Because all Normal distribution are the same when we _______________, we can find
areas under any Normal curve from a single table. Table A in the text is the standard
Normal table.
Table A is a table of _______ under the standard Normal curve. The table entry for each
value z is the area under the curve to the _______ of z.
See examples 2.7 & 2.8 (pgs. 140 & 141) to understand how to use the standard Normal
table.
***BE CAREFUL. If a questions asks for the are to the RIGHT of a z-value then you have
to do ________________________________.***
YOU TRY:
2.29 (pg. 142)
a)
b)
c)
d)
HW- pgs. 137-138 (2.23-2.26) and pg. 142 (2.30) Ch. 2 Test next Friday
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