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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 68-95-99.7 Rule for
a Normal Distribution
Copyright © 2011 Pearson Education, Inc.
6-C
Slide 6-6
6-C
Applying the 68-95-99.7 Rule
The resting heart rates for a sample of people are
normally distributed with a mean of 70 and a
standard deviation of 15.
 Find the percentage of heart rates that are
 Less than 55
 Less than 40
 Less than 100
 Between 55 and 85
 Greater than 115
Copyright © 2011 Pearson Education, Inc.
Slide 6-7
Applying the 68-95-99.7 Rule for
the Sample of Resting Heart Rates
6-C
34%
47.5%
25
40
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55
70
85
100
115
Slide 6-8
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
What is the z-score for the mean?
Data Value
above the mean
below the mean
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Standard Score
positive
→
negative
→
Slide 6-9
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 6-10
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 6-11
6-C
Standard Scores and Percentiles
Copyright © 2011 Pearson Education, Inc.
Slide 6-12
6-C
Application of Standard Scores and Percentiles
Scores on a chemistry exam were normally
distributed with a mean of 67 and a standard
deviation of 8



What is the standard score for an exam score of 67?
 Find the percentile for that exam score.
What is the standard score for an exam score of 88?
 Find the percentile for that exam score.
What is the standard score for an exam score of 59?
 Find the percentile for that exam score.
Copyright © 2011 Pearson Education, Inc.
Slide 6-13