Post on 12-Jul-2015
Statistics
Brian J. Piper, Ph.D.
ψ
Mark Twain (?)
• There are three types of lies, lies, damn lies, and statistics.
http://en.wikipedia.org/wiki/Lies,_damned_lies,_and_statistics
Goals
• Levels of measurement• Group comparisons (t)• Association (scatterplot/r)• Effect-size (d)
Levels of Measurement
• Nominal: categorical, example: sex• Ratio: quantitative, example: age• Ordinal: ranking, example: self-report
– Strongly disagree = 1– Disagree = 2– Neutral = 3– Agree = 4– Strongly agree = 5– S.D.-------------------------------------------------------------------------------------------S.A.
)
What do these countries have in common?
• Liberia• Burma• United States
Metric System
Unit Symbol Factor
tera T 1 x 1012
giga G 1 x 109
kilo K 1 x 103
--- -- 1
centi c 1 x 10-2
milli m 1 x 10-3
micro μ 1 x 10-6
DataSex Height
(m)Weight
(kg)
Sex Height (m)
Weight
(kg)
Female 2.0 60 Male 2.5 80
Female 1.9 58 Male 2.3 76
Female 1.8 56 Male 2.1 74
Female 1.7 54 Male 2.0 73
Female 1.6 52 Male 1.8 72
Female 1.5 50 Male 1.7 70
Female 1.4 48 Male 1.6 68
Female 1.3 46 Male 1.5 65
Female 1.6 57 Male 2 50
Sex Height (m)
Weight
(kg)
Sex Height (m)
Weight
(kg)
Female 2.0 60 Male 2.5 80
Female 1.9 58 Male 2.3 76
Female 1.8 56 Male 2.1 74
Female 1.7 54 Male 2.0 73
Female 1.6 52 Male 1.8 72
Female 1.5 50 Male 1.7 70
Female 1.4 48 Male 1.6 68
Female 1.3 46 Male 1.5 65
Female 1.6 57 Male 2 50Average 1.64 53.4 1.94 69.8
Mean (or average) = Sum (X) /N where N is the # of scores
Variability
• Variability: how much scores differ, on average, from mean– Variance = Sum (X – Mean)2 /N– Standard Deviation (SD) = √Variance– Standard Error of Mean (SEM) = SD / √ N
Group Comparisons I
• Are women lighter then men?– P = probability value
if p < .05 therefore statistically “significant”
– t test = (MeanMales - MeanFemales) / SEM
– t = 4.97, p = .0001
malefemale
SEX_
40
50
60
70
80
90
WE
IGH
T
0123456789Count
0 1 2 3 4 5 6 7 8 9Count
→
←←
►
►
Group Comparisons II
• Do men have a higher IQ then women?• T is the measure of variability (e.g. S.E.M.)
A. Sample Size = 40
Men (N = 20) Women (N=20)0
25
50
75
100
125
IQ
B. Sample Size = 4,000
Men (N = 2000) Women (N=2000)0
25
50
75
100
125
IQ
C. Sample Size = 4,000 ( * p < .05).
Men (N = 2000) Women (N=2000)90
95
100
105
*
IQ
A finding with a * refers to a “statistically significant” finding, e.g. men > women
→
Error Bars Example 2
Batterham et al. New England Journal of Medicine, 349, 941-948.
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0HEIGHT_F
44
48
52
56
60
64
WE
IGH
T_
F
Scatterplots
1.4 1.6 1.8 2.0 2.2 2.4 2.6HEIGHT_M
40
50
60
70
80
90
WE
IGH
T_M
ALEOutlier? ->
Outlier? ->
Positive Association
Study Hours
Score
3 80
5 90
2 75
6 80
7 90
1 50
2 65
7 85
1 40
7 100
Negative Association
Variable A
Variable B
Standardized “Z” Scores
• Z is a #• Z = 0 therefore average• Z > 0 therefore above average• Z < 0 therefore below average
• Z = (X – Mean) / SD• Z = (600 – 500) / 100
= 1.0
3.1
• r: quantifies relationship between two variables (e.g. x & y)• No association: r = 0.00 (C)• Positive association: r > 0.00 (A B)• Negative association: r < 0.00 (DE)• Strong association: A E, Weak association: B D
A B C
DE
r = Sum(Zx * Zy)/ N
3.6
Probability
Frequency Blue Brown
+ 2 10
- 999,998 999,990
Probability
Blue Brown
+ .000002 .000010
- .999998 .99999BrainCancer
Eyes Eyes
Risk
• Absolute Risk: Rate of condition/total population studied, e.g. .000010 or .000002
• Relative Risk: Rate of condition among group A divided by rate of condition among group B– .000010 / .000002 = 5.0
.0010% or .0002%
3.2
Effect-Size
• Procedure used to summarize the magnitude of group differences.– Cohen’s d = (MeanA – MeanB) / SD
• d = 0.20 small effect size• d = 0.50 medium effect size• d = 0.80 large effect size
Can be averaged for multiple studies (meta-analysis).
D.A.R.E.
• Founded in 1983 by Daryl Gates• Police officers give lectures to middle school• Found in 80% of U.S. school districts, 54
nations• Cohen’s d = (MeanD.A.R.E. – MeanControl )/ SDpop
d = 0.30 small, 0.50 medium, 0.70 large
http://www.dare.com/home/default.asp
Does D.A.R.E work?
Ennett S.T. (1994). American Journal of Public Health, 84, 1394-1401.
SummaryGoal Intuition Test
Difference in means
Bar Graphs with SEM
“t-test”
Relationship between variables (ratio x ratio)
Scatterplot Correlation “r”
Summarize many studies
Read papers Effect size “Cohen’s d”