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Chapter Six
z-Scores and the
Normal Curve Model
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
New Statistical Notation
• The absolute value of a number is the size
of that number, regardless of its sign.
• For example, the absolute value of +2 is 2
and the absolute value of -2 is 2.
• The symbol  means “plus or minus.”
Therefore,  1 means +1 and/or -1.
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 2
Understanding z-Scores
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 3
Frequency Distribution of
Attractiveness Scores
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 4
z-Scores
• A z-score is the distance a raw score is
from the mean when measured in
standard deviations.
• A z-score is a location on the distribution.
A z-score also communicates the raw
score’s distance from the mean.
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 5
z-Score Formula
• The formula for computing a z-score for a
raw score in a sample is
z
X 
X
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 6
Computing a Raw Score
• When a z-score and the associated  X
and  are known, this information can be
used to calculate the original raw score.
The formula for this is
X  ( z )( X )  
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 7
Interpreting z-Scores
Using The z-Distribution
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 8
A z-Distribution
A z-distribution is the distribution produced
by transforming all raw scores in the data
into z-scores.
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 9
z-Distribution of Attractiveness
Scores
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 10
Characteristics of the
z-Distribution
1. A z-distribution always has the same
shape as the raw score distribution
2. The mean of any z-distribution is 0
3. The standard deviation of any
z-distribution is1
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 11
Comparison of Two z-Distributions,
Plotted on the Same Set of Axes
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 12
Relative Frequency
• Relative frequency can be computed using
the proportion of the total area under the
curve
• The relative frequency at particular
z-scores will be the same on all normal
z-distributions
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 13
The Standard Normal Curve
The standard normal curve is a perfect
normal z-distribution that serves as our
model of any approximately normal raw
score distribution
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 14
Proportions of Total Area Under the
Standard Normal Curve
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 15
Relative Frequency
• For any approximately normal distribution,
transform the raw scores to z-scores and
use the standard normal curve to find the
relative frequency of the scores
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 16
Percentile
The standard normal curve also can be used
to determine a score’s percentile.
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 17
Proportions of the Standard Normal
Curve at Approximately the 2nd
Percentile
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 18
Using z-Scores to Describe
Sample Means
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 19
Sampling Distribution of Means
The frequency distribution of all possible
sample means that occur when an infinite
number of samples of the same size N are
randomly selected from one raw score
population is called the sampling
distribution of means.
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 20
Central Limit Theorem
The central limit theorem tells us the
sampling distribution of means
1. forms an approximately normal distribution,
2. has a  equal to the  of the underlying raw
score population, and
3. has a standard deviation that is
mathematically related to the standard
deviation of the raw score population.
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 21
Standard Error of the Mean
The standard deviation of the sampling
distribution of means is called the standard
error of the mean. The formula for the true
standard error of the mean is
X 
X
N
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 22
z-Score Formula for
a Sample Mean
The formula for computing a z-score for a
sample mean is
z
X 
X
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 23
Example
• Using the following data set, what is the
z-score for a raw score of 13? What is the
raw score for a z-score of -2?
14
14
13
15
11
15
13
10
12
13
14
13
14
15
17
14
14
15
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 24
Example z-Score
z
X 
X
Assume we know   13.67 and  X  1.56
13  13.67
z
  0.43
1.56
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 25
Example
Raw Score from a z-Score
X  ( z )( X )  
Again, assume we know   13.67 and
 X  1.56
X  (2)(1.56)  13.67  10.55
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 26
Example
z-Score for a Sample Mean
If X = 13 , N = 18,  = 12, and  X = 2.5,
what is the z-score for this sample
mean?
X 
X
N
2.5
X 
 0.589
18
z
X 
X
13  12
z
 1.70
0.589
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 27
Key Terms
•
•
•
•
•
•
•
•
central limit theorem
relative standing
sampling distribution of means
standard error of the mean
standard normal curve
standard score
z-distribution
z-score
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 6 - 28
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