© 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 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.
Chapter 6 - 2
• 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.
New Statistical Notation
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Chapter 6 - 3
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 - 4
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 - 5
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 - 6
X
Xz
z-Score Formula
• The formula for computing a z-score for a raw score in a sample is
© 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
• When a z-score and the associated and are known, this information can be used to calculate the original raw score. The formula for this is
X
))(( XzX
Computing a Raw 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 - 8
Interpreting z-ScoresUsing The z-Distribution
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Chapter 6 - 9
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 - 10
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 - 11
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 - 12
Comparison of Two z-Distributions, Plotted on the Same Set of Axes
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Chapter 6 - 13
• 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
Relative Frequency
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Chapter 6 - 14
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 - 15
Proportions of Total Area Under the Standard Normal Curve
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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
Chapter 6 - 16
© 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
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 - 18
Proportions of the Standard Normal Curve at Approximately the 2nd
Percentile
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Chapter 6 - 19
Using z-Scores to Describe Sample Means
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Chapter 6 - 20
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 - 21
Central Limit Theorem
The central limit theorem tells us the
sampling distribution of means1. 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 - 22
NX
X
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
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Chapter 6 - 23
X
Xz
z-Score Formula for a Sample Mean
The formula for computing a z-score for a sample mean is
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Chapter 6 - 24
14 14 13 15 11 15
13 10 12 13 14 13
14 15 17 14 14 15
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?
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Chapter 6 - 25
X
Xz
Assume we know and67.13 56.1X
43.056.167.1313
z
Example z-Score
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Chapter 6 - 26
Again, assume we know and67.1356.1X
))(( XzX
55.1067.13)56.1)(2( X
ExampleRaw Score from a z-Score
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Chapter 6 - 27
If = 13 , N = 18, = 12, and = 2.5, what is the z-score for this sample mean?
X
NX
X
X
Xz
589.0185.2
X 70.1589.01213
z
X
Examplez-Score for a Sample 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.
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
Chapter 6 - 28
