Download chapter Six - Bakersfield College

Chapter Six
z-Scores and the
Normal Curve Model
New Statistical Notation
• The absolute value of a number is the
size of that number, regardless of its
sign. That is, 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.
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Chapter 6 - 2
Understanding z-Scores
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Chapter 6 - 3
Frequency Distribution of Attractiveness
Scores
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Chapter 6 - 4
z-Scores
• Like any raw score, a z-score is a
location on the distribution. A z-score
also automatically communicates the
raw score’s distance from the mean
• A z-score describes a raw score’s
location in terms of how far above or
below the mean it is when measured in
standard deviations
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Chapter 6 - 5
z-Score Formula
• The formula for computing a z-score for
a raw score in a sample is
z
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X 
X
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 )  
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Chapter 6 - 7
Interpreting z-Scores
Using the z-Distribution
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Chapter 6 - 8
A z-Distribution
A z-distribution is the distribution
produced by transforming all raw scores
in the data into z-scores.
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Chapter 6 - 9
z-Distribution of Attractiveness Scores
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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 always
equals 0
3. The standard deviation of any
z-distribution always equals 1
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Chapter 6 - 11
Comparison of Two z-Distributions,
Plotted on the Same Set of Axes
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Chapter 6 - 12
Relative Frequency
• Relative frequency can be computed
using the proportion of the total area
under the curve.
• The relative frequency of a particular
z-score will be the same on all normal
z-distributions.
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Chapter 6 - 13
The Standard Normal Curve
The standard normal curve is a perfect
normal z-distribution that serves as our
model of the z-distribution that would
result from any approximately normal raw
score distribution
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Chapter 6 - 14
Proportions of Total Area Under the
Standard Normal Curve
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Chapter 6 - 15
Percentile
The standard normal curve also can be
used to determine a score’s percentile.
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Chapter 6 - 16
Proportions of the Standard Normal Curve at
Approximately the 2nd Percentile
[Insert new Figure 6.6 here.]
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Chapter 6 - 17
Using z-Scores to Describe
Sample Means
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Chapter 6 - 18
Sampling Distribution of Means
A distribution which shows 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.
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Chapter 6 - 19
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.
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Chapter 6 - 20
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 
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X
N
Chapter 6 - 21
z-Score Formula for
a Sample Mean
The formula for computing a z-score for a
sample mean is
z
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X 
X
Chapter 6 - 22
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
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Chapter 6 - 23
Example z-Score
z
X 
X
Assume that we know that   13.67
and that  X  1.56
13  13.67
z
  0.43
1.56
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Chapter 6 - 24
Example
Raw Score from a z-Score
X  ( z )( X )  
Again, assume we know that   13.67
and that  X  1.56
X  (2)(1.56)  13.67  10.55
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Chapter 6 - 25
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
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z
X 
X
13  12
z
 1.70
0.589
Chapter 6 - 26