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Putting Statistics to Work
Copyright © 2011 Pearson Education, Inc.
Unit 6C
The Normal Distribution
Copyright © 2011 Pearson Education, Inc.
Slide 6-3
6-C
The Normal Distribution
The normal distribution is a symmetric, bellshaped distribution with a single peak. Its peak
corresponds to the mean, median, and mode of the
distribution.
Copyright © 2011 Pearson Education, Inc.
Slide 6-4
6-C
Conditions for a Normal Distribution
A data set satisfying the following criteria is likely to
have a nearly normal distribution.
1. Most data values are clustered near the mean,
giving the distribution a well-defined single peak.
2. Data values are spread evenly around the mean,
making the distribution symmetric.
3. Larger deviations from the mean are increasingly
rare, producing the tapering tails of the
distribution.
4. Individual data values result from a combination
of many different factors.
Copyright © 2011 Pearson Education, Inc.
Slide 6-5
The Empirical Rule (The 68-95-99.7 Rule)
for a Normal Distribution
Copyright © 2011 Pearson Education, Inc.
6-C
Slide 6-6
6-C
Standard Scores
The number of standard deviations that a data
value lies above or below the mean is called its
standard score (or z-score), defined by
data value  mean
z  standard score 
standard deviation
Data Value
above the mean
below the mean
Copyright © 2011 Pearson Education, Inc.
Standard Score
positive
→
negative
→
Slide 6-7
6-C
Standard Scores
Example: If the mean were 21 with a standard
deviation of 4.7 for scores on a nationwide test, find
the z-score for a 30. What does this mean?
data value  mean
z
standard deviation
30  21

 1.91
4.7
This means that a test score of 30 would be about
1.91 standard deviations above the mean of 21.
Copyright © 2011 Pearson Education, Inc.
Slide 6-8
6-C
Using z-scores
The mean height of males 20 years or older is 69.1 inches
with a standard deviation of 2.8 inches. The mean height
of females 20 years or older is 63.7 inches with a standard
deviation of 2.7 inches. Data based on information
obtained from National Health and Examination Survey.
Who is relatively taller?
Kevin Garnett whose height is 83 inches
or
Candace Parker whose height is 76 inches
Copyright © 2011 Pearson Education, Inc.
Slide 6-9
6-C
Using z-scores (cont.)
Kevin Garnett
Candace Parker
83  69.1
zkg 
2.8
 4.96
76  63.7
zcp 
2.7
 4.56
Kevin Garnett’s height is 4.96 standard deviations above the
mean. Candace Parker’s height is 4.56 standard deviations
above the mean. Kevin Garnett is relatively taller.
Copyright © 2011 Pearson Education, Inc.
Slide 6-10
6-C
You try it!
The mean commute time in the U.S. is 24.4
minutes with a standard deviation of 6.5
minutes. Find the z-score that corresponds
to a commute time of 15 minutes.
A. 1.45
B. –1.45
C. 11.25
D. –9.4
Copyright © 2011 Pearson Education, Inc.
Slide 6-11
6-C
You try it! (Answer)
The mean commute time in the U.S. is 24.4
minutes with a standard deviation of 6.5
minutes. Find the z-score that corresponds
to a commute time of 15 minutes.
A. 1.45
B. –1.45
C. 11.25
D. –9.4
Copyright © 2011 Pearson Education, Inc.
Slide 6-12
6-C
Standard Scores and Percentiles

The nth percentile of a data set is the smallest
value in the set with the property that n% of the
data are less than or equal to it.

A data value that lies between two percentiles is
said to lie in the lower percentile.
Copyright © 2011 Pearson Education, Inc.
Slide 6-13
6-C
Standard Scores and Percentiles
Copyright © 2011 Pearson Education, Inc.
Slide 6-14
6-C
Assignment

P. 398 – 400 5-18, 20 – 28, 30, 37
Copyright © 2011 Pearson Education, Inc.
Slide 6-15