Download 09-18 lecture

Review
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X = {3, 5, 7, 9, 11}
Range?
Sum of squares?
Variance?
Standard deviation?
z-Scores
9/18
z-Scores
• How good (high, low, etc.) is a given value?
• How does it compare to other scores?
• Today's
z-scores
Solutionsanswer:
from before:
– Number
deviations
below) the mean
Compareoftostandard
mean, median,
min,above
max, (or
quartiles
1 the percentile
– Find
• Today's
answer:
z-scores
2 SDs below
mean
0.8
m = 3.5
Density
2 SDs above mean
Raw Score
Difference
mean
 z =from
+2 mean

zstandard
= -2
– Number
of
deviations
above
(or
below)
the
0.6
X  m s = .5
SDs from mean
z
s
0.4
0.2
2:30
0
1
2
3
2.5
4
Hours
5
4.5
6
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
– ShapeX unchanged
= [4, 8, 2, 5, 8, 5, 3]
– Just change mean to 0 and standard deviation to 1
1
0.6
X –0.6
m = [-1, 3, -3, 0, 3, 0, -2]
0.4
0.2
0
-10
mean = 03
m=3
0.8
0.4
s=1
s=2
X m
z  0.2
 [.5, 1.4,  1.4, 0, 1.4, 0,  .9]
s
Density
Density
0.8
1 s = 2.1
m = 5,
0
-0.2
-5
0
X
X z– m
5
10
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%
0
1
z
2
3
.003% .00003%
0.5
Density
0.4
0.3
0.2
0.1
0
-3
-2
-1
4
5
Comparison Between Distributions
• Different populations
– z-score gives value relative to the group
– Removes group differences, allows cross-group
comparison
• Swede – 6’1”
• Indonesian – 5’6”
(m = 5’11”, s = 2”)
(m = 5’2”, s = 2”)
z = +1
z = +2
• Different scales
– z-score removes indiosyncrasies of measurement
variable
– Puts everything on a common scale (cf. temperature)
• IQ = 115
• Digit span = 10
(m = 100, s = 15)
(m = 7, s = 2)
z = +1
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)
0.5
– e.g., blood doping and red blood cell count
0.50.4
Density
Density
0.40.3
0.30.2
0.2
0.1
0.1
0
-5
0
-3
-4
-2
-3
-1
-2
-1
0
z0
z
1
1
2
2
3
3
4
5