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

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

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

z-Scores 9/19

z-Scores 2:30 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 m = 3.5 2 SDs below mean  z = -2 2 SDs above mean  z = +2 Raw Score Difference from mean SDs from mean s = .5 2:30 2.5 4.5

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 mean = 0 3 m = 3 X – m = [-1, 3, -3, 0, 3, 0, -2] s = 1 s = 2 z X – m

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

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%

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

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

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

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

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