1. z-Scores

2. 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

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

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

5. 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%

6. 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 = (m = 100, s = 15) z = +1
Digit span = 10 (m = 7, s = 2) z = +1.5

7. 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