Review

1. Review Class test scores have the following statistics: Minimum = 54 Maximum = 99 25th percentile = 61 75th percentile = 87 Median = 78 Mean = 76 What is the interquartile range? • 34 • 26 • 45 • 46 • 9

2. Review A population of 100 people has a sum of squares of 3600. What is the standard deviation? • 36 • 60 • 6 • 0.6 • Not enough information

3. Review You weigh 50 people and calculate a variance of 240. Then you realize the scale was off,and everyone’s weight needs to be increased by 5 lb. What happens to the variance? • Increase • Decrease • No change

4. z-Scores 9/19

5. m = 3.5 Raw Score Difference from mean s = .5 SDs from mean 2.5 4.5 z-Scores • How good (high, low, etc.) is a given value? • How does it compare to other scores? • Today's answer: z-scores • Number of standard deviations above (or below) the mean • How good (high, low, etc.) is a given value? • How does it compare to other scores? • Solutions from before: • Compare to mean, median, min, max, quartiles • Find the percentile • Today's answer: z-scores • Number of standard deviations above (or below) the mean 2 SDs above mean  z = +2 2 SDs below mean  z = -2 2:30

6. 3 Standardized Distributions • Standardized distribution - the distribution of z-scores • Start with raw scores, X • Compute m, s • Compute z for every subject • Now look at distribution of z • Relationship to original distribution • Shape unchanged • Just change mean to 0 and standard deviation to 1 X = [4, 8, 2, 5, 8, 5, 3] m = 5, s = 2.1 m = 3 mean = 0 X – m= [-1, 3, -3, 0, 3, 0, -2] s = 1 s = 2 z X – m

7. Uses for z-scores • Interpretation of individual scores • Comparison between distributions • Evaluating effect sizes

8. Interpretation of Individual Scores • z-score gives universal standard for interpreting variables • Relative to other members of population • How extreme; how likely • z-scores and the Normal distribution • If distribution is Normal, we know exactly how likely any z-score is • Other shapes give different answers, but Normal gives good rule of thumb p(Z  z): 50% 16% 2% .1% .003% .00003%

9. Comparison Between Distributions • Different populations • z-score gives value relative to the group • Removes group differences, allows cross-group comparison • Swede – 6’1” (m = 5’11”, s = 2”) z = +1 • Indonesian – 5’6” (m = 5’2”, s = 2”) z = +2 • Different scales • z-score removes indiosyncrasies of measurement variable • Puts everything on a common scale (cf. temperature) • IQ = 115 (m = 100, s = 15) z = +1 • Digit span = 10 (m = 7, s = 2) z = +1.5

10. Evaluating Effect Size • How different are two populations? • z-score shows how important a difference is • Memory drug: mdrug = 9, mpop = 7 • Important? s = 2  z = +1 • Is an individual likely a member of a population? • z-score tells chances of score being that high (or low) • e.g., blood doping and red blood cell count

11. Review Your z-score is 0.15. This implies you are • Above average • Below average • Exactly at the mean • Not enough information

12. Review What is the z-score for a score of 12, if µ = 50 and s = 5? • -7.6

13. Review What is the raw score corresponding to z = 4, if µ = 10 and s = 2? • -3 • 18 • 2 • 16 • 12