Review X = {3, 5, 7, 9, 11} Range? Sum of squares? Variance? Standard deviation?

z-Scores 9/13

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 z-Scores 2:30 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  = 3.5  = SDs below mean  z = -2 Raw Score Difference from mean SDs from mean SDs above mean  z = +2

Standardized Distributions Standardized distribution - the distribution of z-scores –Start with raw scores, X –Compute  –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]  = 5,  = 2.1 X –  = [-1, 3, -3, 0, 3, 0, -2]  = 3  = 2 3 mean = 0  = 1 X –  z

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”(  = 5’11”,  = 2”)z = +1 Indonesian – 5’6”(  = 5’2”,  = 2”)z = +2 Different scales –z-score removes indiosyncrasies of measurement variable –Puts everything on a common scale (cf. temperature) IQ = 115 (  = 100,  = 15)z = +1 Digit span = 10 (  = 7,  = 2)z = +1.5

Evaluating Effect Size How different are two populations? –z-score shows how important a difference is –Memory drug:  drug = 9,  pop = 7 –Important?  = 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