This document provides an overview of key statistical concepts including levels of measurement, group comparisons using t-tests, assessing associations with scatterplots and correlation coefficients, and calculating effect sizes. Examples are given of nominal, ratio and ordinal levels of measurement. Group comparisons are demonstrated using t-tests to compare means and determine statistical significance. Scatterplots and correlation coefficients r are discussed as ways to assess the association between two variables. Effect sizes such as Cohen's d are introduced as a method for summarizing differences between groups across multiple studies.

1. ψ
Statistics
Brian J. Piper, Ph.D.

2. Mark Twain (?)
• There are three types of lies, lies, damn lies,
and statistics.
http://en.wikipedia.org/wiki/Lies,_damned_lies,_and_statistics

3. Goals
• Levels of measurement
• Group comparisons (t)
• Association (scatterplot/r)
• Effect-size (d)

4. )
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.

5. What do these countries have in
common?
• Liberia
• Burma
• United States

6. 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

7. Data
Sex Height Weight Sex Height Weight
(m) (kg) (m) (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

8. Sex Height Weight Sex Height Weight
(m) (kg) (m) (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
Average 1.64 53.4 1.94 69.8
Mean (or average) = Sum (X) /N where N is the # of scores

9. 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

10. 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
90
80 ►
←
70
WEIGHT
←
►
60
→
50 SEX_
female
40 male
9876543210123456789
Count Count

11. 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 B. Sample Size = 4,000 C. Sample Size = 4,000 ( * p < .05).
→
125 125 105
100 100
100
75 75 *
IQ
IQ
IQ
50 50
95
25 25
0 0 90
Men (N = 20) Women (N=20) Men (N = 2000) Women (N=2000) Men (N = 2000) Women (N=2000)
A finding with a * refers to a “statistically significant” finding, e.g. men > women

12. Error Bars Example 2
Batterham et al. New England Journal of Medicine, 349, 941-948.

13. Scatterplots
64 90
60 80
WEIGHT_MALE
Outlier? ->
WEIGHT_F
56 70
52 60
48 50 Outlier? ->
44 40
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 1.4 1.6 1.8 2.0 2.2 2.4 2.6
HEIGHT_F HEIGHT_M

14. Study Score Positive
Hours Association
3 80
5 90
2 75
6 80
7 90
1 50
2 65
7 85
1 40
7 100

15. Negative Association
Variable B
Variable A

17. 3.1
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

18. B C
A
3.6
r = Sum(Zx * Zy)/ N
D
E
• 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

19. Probability
Eyes Eyes
Probability
Frequency Blue Brown Blue Brown
.000002 .000010
+ 2 10 +
Brain
Cancer .999998 .99999
- 999,998 999,990 -

20. 3.2
Risk
• Absolute Risk: Rate of condition/total
population studied, e.g. .000010 or .000002
.0010% or .0002%
• Relative Risk: Rate of condition among group
A divided by rate of condition among group B
– .000010 / .000002 = 5.0

21. 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).

22. 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 = (Mean – Mean )/ SD D.A.R.E. Control pop
d = 0.30 small, 0.50 medium, 0.70 large
http://www.dare.com/home/default.asp

23. Does D.A.R.E work?
Ennett S.T. (1994). American Journal of Public Health, 84, 1394-1401.

24. Summary
Goal Intuition Test
Difference in Bar Graphs with “t-test”
means SEM
Relationship between
variables (ratio x ratio)
Scatterplot Correlation “r”
Summarize Read papers Effect size
many studies “Cohen’s d”