1. Chapter 5: z-Scores

2. 12 82 x = 76 (a) X = 76 is slightly below average 12 70 x = 76 (b) X = 76 is slightly above average 3 70 x = 76 (c) X = 76 is far above average Normal curve distribution comparisons

3. Definition of z-score A z-score specifies the precise location of each x-value within a distribution. The sign of the z-score (+ or - ) signifies whether the score is above the mean (positive) or below the mean (negative). The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and µ.

4. Normal curve segmented by sd’s X z -20+1+2

5. Example 5.2 A distribution of exam scores has a mean (µ) of 50 and a standard deviation (σ) of 8.

6. Z formula

7. Normal curve with sd’s    = 4 60 6468 µ X 66

8. Example 5.5 A distribution has a mean of µ = 40 and a standard deviation of  = 6.

9. To get the raw score from the z-score:

10. If we transform every score in a distribution by assigning a z-score, new distribution: 1.Same shape as original distribution 2.Mean for the new distribution will be zero 3.The standard deviation will be equal to 1

11. Z-distribution X z -20+1+2 1001101209080

12. A small population 0, 6, 5, 2, 3, 2N=6 xx - µ(x - µ) 2 00 - 3 = -39 66 - 3 = +39 55 - 3 = +24 22 - 3 = -11 33 - 3 = 00 22 - 3 = -11

13. Frequency distributions with sd’s 2 1 0123456 µ  X frequency (a) 2 1 -1.5 -0.50+0.5+1.0+1.5 µ  z frequency (b)

14. Let’s transform every raw score into a z-score using: x = 0 x = 6 x = 5 x = 2 x = 3 x = 2 = -1.5 = +1.5 = +1.0 = -0.5 = 0 = -0.5

15. Z problem computation Mean of z-score distribution : Standard deviation: zz - µ z (z - µ z ) 2 -1.5-1.5 - 0 = -1.52.25 +1.5+1.5 - 0 = +1.52.25 +1.0+1.0 - 0 = +1.01.00 -0.5-0.5 - 0 = -0.50.25 00 - 0 = 00 -0.5-0.5 - 0 = -0.50.25

16. Psychology vs. Biology 10 50 µ X = 60 X Psychology 4 48 µ X = 56 X Biology 52

17. Converting Distributions of Scores

18. Original vs. standardized distribution 14 57 µ Joe X = 64 z = +0.50 X Original Distribution 10 50 µ X Standardized Distribution Maria X = 43 z = -1.00 Maria X = 40 z = -1.00 Joe X = 55 z = +0.50

19. Correlating Two Distributions of Scores with z-scores

20. Distributions of height and weight  = 4 µ = 68 Person A Height = 72 inches Distribution of adult heights (in inches)  = 16 µ = 140 Person B Weight = 156 pounds Distribution of adult weights (in pounds)