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Section 2.2, Part 1
Standard Normal
Calculations
AP Statistics
Berkley High School/CASA
Comparing data sets
How do we compare results when they are
measured on two completely different
scales?
 One solution might be to look at
percentiles
 What might you say about a woman that is
in the 50th percentile and a man in the 15th
percentile?

AP Statistics, Section 2.2, Part 1
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Another way of comparing
Another way of comparing: Look at
whether the data point is above or below
the mean, and by how much.
 Example: A man is 64 inches tall. The
heights of men are normally distributed
with a mean of 69 inches and standard
deviation of 2.5 inches.

AP Statistics, Section 2.2, Part 1
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Another way of comparing
Example: A man is 64 inches tall. The
heights of men are normally distributed
with a mean of 69 inches and standard
deviation of 2.5 inches.
 We can see that the man is below the
mean, but by how much?

AP Statistics, Section 2.2, Part 1
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Another way of comparing

Example: A man is 64
inches tall. The
heights of men are
normally distributed
with a mean of 69
inches and standard
deviation of 2.5
inches.
z
x

64  69
z
2.5
z  2
AP Statistics, Section 2.2, Part 1
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z-scores


The z-score is a way
of looking at every
data set, because
each data set has a
mean and standard
deviation
We call the z-score
the “standardized”
score.
z
x

64  69
z
2.5
z  2
AP Statistics, Section 2.2, Part 1
6
z-scores



Positive z-scores mean
the data point is above
the mean.
Negative z-scores mean
the data point is below
the mean.
The larger the absolute
value of the z-score, the
more unusual it is.
z
x

64  69
z
2.5
z  2
AP Statistics, Section 2.2, Part 1
7
Using the z-table


We can use the ztable to find out the
percentile of the
observation.
A z-score of -2.0 is at
the 2.28 percentile.
z
x

64  69
z
2.5
z  2
AP Statistics, Section 2.2, Part 1
8
Cautions
The z-table only gives the amount of data
found below the z-score.
 If you want to find the portion found above
the z-score, subtract the probability found
on the table from 1.

AP Statistics, Section 2.2, Part 1
9
Standardized Normal Distribution
We should only use the z-table when the
distributions are normal, and data has
been standardized
 N(μ,σ) is a normal distribution
 N(0,1) is the standard normal distribution
 “Standardizing” is the process of doing a
linear translation from N(μ,σ) into N(0,1)

AP Statistics, Section 2.2, Part 1
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Example
Men’s heights are N(69,2.5).
 What percent of men are taller than 68
inches?

AP Statistics, Section 2.2, Part 1
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Working with intervals

What proportion of men are between 68 and
70 inches tall?
AP Statistics, Section 2.2, Part 1
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Working backwards

How tall must a man be in order to be in
the 90th percentile?
AP Statistics, Section 2.2, Part 1
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Working backwards

How tall must a woman be in order to be in
the top 15% of all women?
AP Statistics, Section 2.2, Part 1
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Working backwards

What range of values make up the middle
50% of men’s heights?
AP Statistics, Section 2.2, Part 1
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Assignment

Exercises 2.19 – 2.25, The Practice of
Statistics.
AP Statistics, Section 2.2, Part 1
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AP Statistics, Section 2.2, Part 1
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AP Statistics, Section 2.2, Part 1
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