1. Chapter Six z-Scores and the Normal Curve Model

2. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 2 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

3. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 3 Understanding z-Scores

4. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 4 Frequency Distribution of Attractiveness Scores

5. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 5 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

6. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 6 z-Score Formula The formula for computing a z-score for a raw score in a sample is

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

8. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 8 Interpreting z-Scores Using the z-Distribution

9. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 9 A z-Distribution A z-distribution is the distribution produced by transforming all raw scores in the data into z-scores.

10. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 10 z-Distribution of Attractiveness Scores

11. Copyright © Houghton Mifflin Company. All rights reserved.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 always equals 0 3. The standard deviation of any z-distribution always equals 1

12. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 12 Comparison of Two z-Distributions, Plotted on the Same Set of Axes

13. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 13 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.

14. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 14 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

15. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 15 Proportions of Total Area Under the Standard Normal Curve

16. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 16 Percentile The standard normal curve also can be used to determine a score’s percentile.

17. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 17 Proportions of the Standard Normal Curve at Approximately the 2nd Percentile [Insert new Figure 6.6 here.]

18. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 18 Using z-Scores to Describe Sample Means

19. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 19 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.

20. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 20 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.

21. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 21 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

22. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 22 z-Score Formula for a Sample Mean The formula for computing a z-score for a sample mean is

23. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 23 14 13151115 131012131413 14151714 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?

24. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 24 Assume that we know that and that Example z-Score

25. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 25 Again, assume we know that and that Example Raw Score from a z-Score

26. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 6 - 26 If = 13, N = 18,  = 12, and = 2.5, what is the z-score for this sample mean? Example z-Score for a Sample Mean