© 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

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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

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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

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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.

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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