Chapter 5

DESCRIBING DATA WITH Z-SCORES AND THE NORMAL CURVE

Going ForwardYour goals in this chapter are to learn:• What a z-score is• What a z-distribution is and how it indicates a

score’s relative standing• How the standard normal curve is used with z-

scores to determine relative frequency, simple frequency, and percentile

• What the sampling distribution of means is and what the standard error of the mean is

• How to compute z-scores for sample means and then determine their relative frequency

Understanding z-Scores

A Frequency Distribution

z-Scores

• A z-score is a location on the distribution • A z-score automatically communicates its

distance from the mean• A z-score indicates how far a raw score is

above or below the mean when measured in standard deviations

z-Score Formula

The formula for transforming a raw score into a z-score is

XS

XXz

Computing a Raw Score

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

XS X

XSzX X ))((

Using the z-Distribution toInterpret Scores

A z-Distribution

A z-distribution is the distribution produced by

transforming all raw scores in the data into z-

scores.

z-Distribution ofAttractiveness Scores

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 is 1

Using the z-Distribution toCompare Different Variables

Comparison of Distributions

Using the z-Distribution toCompute Relative Frequency

Relative Frequency

• The relative frequency of a particular z-score will be the same on all normal z-distributions

• Relative frequency can be computed using the proportion of the total area under the curve

The Standard Normal Curve

The standard normal curve is a perfect normal z-distribution that serves as our model of any approximately normal z-distribution.

Area Under the Standard Normal Curve

Using the z-Table

Using z-Scores to Describe Sample Means

Sampling Distribution of Means

A frequency distribution of all possible sample means occurring when an infinite number of samples of the same size N are selected from one raw score distribution is called the sampling distribution of means.

Central Limit Theorem

The central limit theorem tells us1.A sampling distribution is always an approximately normal distribution2.The mean of the sampling distribution equals the mean of the underlying raw score population used to create the sampling distribution3.The standard deviation of the sampling distribution is mathematically related to the standard deviation of the raw score population

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

NX

X

z-Score Formula for a Sample Mean

The formula for computing a z-score for a sample mean is

X

Xz

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

Example z-Score

XS

XXz

We know (from prior chapters) and for these data.

67.13X56.1XS

43.056.1

67.1313

z

ExampleRaw Score from a z-Score

XSzX X ))((

55.1067.13)56.1)(2( X

We know (from prior chapters) and for these data.

67.13X56.1XS

Examplez-Score for a Sample Mean

We know . . If m = 12 and = 2.5, what is the z-score for this sample mean?

67.13X

NX

X

X

Xz

589.018

5.2X 84.2

589.0

1267.13

z

X