Chapter Six z-Scores and the Normal Curve Model

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 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. New Statistical Notation

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Understanding z-Scores

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Frequency Distribution of Attractiveness Scores

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter z-Score Formula The formula for computing a z-score for a raw score in a sample is

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 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 Computing a Raw Score

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Interpreting z-Scores Using the z-Distribution

Copyright © Houghton Mifflin Company. All rights reserved.Chapter A z-Distribution A z-distribution is the distribution produced by transforming all raw scores in the data into z-scores.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter z-Distribution of Attractiveness Scores

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Comparison of Two z-Distributions, Plotted on the Same Set of Axes

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Proportions of Total Area Under the Standard Normal Curve

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Percentile The standard normal curve also can be used to determine a score’s percentile.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Proportions of the Standard Normal Curve at Approximately the 2nd Percentile [Insert new Figure 6.6 here.]

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Using z-Scores to Describe Sample Means

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter z-Score Formula for a Sample Mean The formula for computing a z-score for a sample mean is

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 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?

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Assume that we know that and that Example z-Score

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Again, assume we know that and that Example Raw Score from a z-Score

Copyright © Houghton Mifflin Company. All rights reserved.Chapter If = 13, N = 18,  = 12, and = 2.5, what is the z-score for this sample mean? Example z-Score for a Sample Mean