1. Chapter 5 Z-Scores

2. Review ► We have finished the basic elements of descriptive statistics. ► Now we will begin to develop the concepts and skills that form the foundation for inferential statistics.

3. Review (cont.) ► Chapter 5: Present a method for describing the exact location of an individual score relative to the other scores in a distribution. ► Chapter 6: Determine probability values associated with different locations in a distribution of scores ► Chapter 7: Apply skills from Ch. 5 & 6 to sample means instead of individual scores.

4. Z-Scores ► Purpose of z-scores, or standard scores, is to identify and describe the exact location of every score in a distribution. ► To do this, we use the mean and standard deviation.

5. Z-Scores ► A statistical technique that uses the mean and the standard deviation to transform each score (X value) into a z-score or a standard score. ► Purpose of a z-score or a standard score is to identify the exact location of every score in a distribution.

6. Why are z-scores useful? ► If you got a 76 on a test, how did you do? ► You would need more information. ► You need to know the other scores in the distribution. ► What is the mean?

7. Why are z-scores useful? (cont.) ► Knowing the mean is not enough. ► You also need to know the standard deviation. ► The relative location within the distribution depends on the mean and the statistical deviation as well as your score.

8. Figure 5.1 Two distributions of exam scores Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning Score is in the extreme right hand tail – one of the highest in the distribution Score is only slightly above average.

9. Purpose for z-scores ► A score by itself does not necessarily provide much information about its position within a distribution  These are called raw scores ► To make raw scores more meaningful, they are often transformed into new values that contain more information. ► This transformation is one purpose for z- scores.

10. Purpose for z-scores (cont.) ► We can transform scores into z-scores to find out exactly where the original scores are located. ► A second purpose is to standardize an entire distribution.  IQ scores ► All are standardized with a mean of 100 and s.d. of 15 ► An IQ score of 95 is slightly below average and an IQ score of 145 is extremely high no matter what IQ test

11. To describe the exact location of the score within a distribution ► A z-score transforms an X score into a signed +/- number  + above the mean  - below the mean  The number tells the distance between the score and the mean in terms of the number of standard deviations.

12. Example ► In a distribution of standardized IQ scores with  and  and a score of X=130 ► The score of X=130 could be transformed into z= +2.00 ► z value indicates + (above the mean) ► by a distance of 2 standard deviations (30 points)

13. Definition ► A z-score  Specifies the precise location of each X value with a distribution  The sign of +/- signifies whether the score is above or below the mean  The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between the X and 

14. Z-scores ► Consist of two parts  +/-  Magnitude ► Both parts are necessary to describe completely where a raw score is located within a distribution.

15. Figure 5.2 The relationship between z-scores and locations in a distribution Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

16. ► a z-score of z = +1.00 corresponds to a position exactly 1 standard deviation above the mean. ► a z-score of z = +2.00 corresponds to a position exactly 2 standard deviations above the mean. ► The numerical value tells you the number of standard deviations from the mean.

17. Figure 5.1 Two distributions of exam scores Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning Now use a z-score to describe the position of X=76. Z= +2.00 The score is located above the mean by exactly 2 s.d. Z= +0.50 The score is located above the mean by 1/2 s.d.

18. Learning Check ► A negative z-score always indicates a location below the mean. ► What z-score value identifies each of the following locations in a distribution?  Above the mean by 2 s.d.  Below the mean by ½ s.d.  Above the mean by ¼ s.d.  Below the mean by 3 s.d.

19. Learning Check ► For a population with  = 50 and  = 10, find the z-score for each of the following scores:  X = 55  X = 40  X = 30 Z= +0.50 Z= - 1.00 Z = - 2.00

20. Figure 5.4 A z-score transformation Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning = 50 10 40 30

21. Learning Check ► For a population with  50 and  10, find the X value corresponding to each of the following z-scores:  z = + 1.00  z = - 0.50  z = + 2.00 X= 60 X = 45 X = 70

22. Figure 5.4 A z-score transformation Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning = 50 10 40 30 6070

23. Formula for transforming z-scores ► z = X –  

24. Example ► A distribution of scores has a mean of   and a standard deviation of  10. What z-score corresponds to a score of X=120 in this distribution? Z = X –  

25. Transforming z-scores into X values ► X =  +  z  = 60 + (-2.00)(5) = 60 + (-2.00)(5) = 60 + (-10.00) = 60 + (-10.00) = 50 = 50   Z = -2.00

26. Using z-scores to Standardize a Distribution ► When an entire population of scores is transformed into z-scores  The transformation does not change the shape of the population but;  The mean is transformed into a value of zero;  The s.d. is transformed into a value of 1.

27. Standardized Distribution ► A standardized distribution is composed of scores that have been transformed to create predetermined values for  and  ► Standardized distributions are used to make dissimilar distributions comparable.

28. Using z-scores to Make Comparisons ► Example: pg. 112-113 ► Psychology score = 60   50 and  10 ► Biology score = 56   48 and  4 ► z= X –    ► z= X –   