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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. Copyright © Houghton Mifflin Company. All rights reserved. Chapter 6 - 2 Understanding z-Scores Copyright © Houghton Mifflin Company. All rights reserved. Chapter 6 - 3 Frequency Distribution of Attractiveness Scores Copyright © Houghton Mifflin Company. All rights reserved. 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 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 6 - 5 z-Score Formula • The formula for computing a z-score for a raw score in a sample is z Copyright © Houghton Mifflin Company. All rights reserved. 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 ) Copyright © Houghton Mifflin Company. All rights reserved. Chapter 6 - 7 Interpreting z-Scores Using the z-Distribution Copyright © Houghton Mifflin Company. All rights reserved. Chapter 6 - 8 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 6 - 9 z-Distribution of Attractiveness Scores Copyright © Houghton Mifflin Company. All rights reserved. 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 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 6 - 11 Comparison of Two z-Distributions, Plotted on the Same Set of Axes Copyright © Houghton Mifflin Company. All rights reserved. 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. Copyright © Houghton Mifflin Company. All rights reserved. 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 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 6 - 14 Proportions of Total Area Under the Standard Normal Curve Copyright © Houghton Mifflin Company. All rights reserved. Chapter 6 - 15 Percentile The standard normal curve also can be used to determine a score’s percentile. Copyright © Houghton Mifflin Company. All rights reserved. Chapter 6 - 16 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 6 - 17 Using z-Scores to Describe Sample Means Copyright © Houghton Mifflin Company. All rights reserved. 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. Copyright © Houghton Mifflin Company. All rights reserved. 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. Copyright © Houghton Mifflin Company. All rights reserved. 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 Copyright © Houghton Mifflin Company. All rights reserved. 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 Copyright © Houghton Mifflin Company. All rights reserved. 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 Copyright © Houghton Mifflin Company. All rights reserved. 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 Copyright © Houghton Mifflin Company. All rights reserved. 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 Copyright © Houghton Mifflin Company. All rights reserved. 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 Copyright © Houghton Mifflin Company. All rights reserved. z X X 13 12 z 1.70 0.589 Chapter 6 - 26