1. Reasoning in Psychology Using Statistics
2017

2. Announcements Exam 2 in lecture and lab on Wednesday
Be prepared to do calculations (including square roots) on calculator
Announcements

3. Cautions with Correlations
Mathematical cautions
Different scales: convert to z-scores
Restriction of range (e.g., age & height)
Outliers (especially in small samples)
Interpretive caution
Causal claims
Cautions with Correlations

4. Pearson’s r, z transformation
Change all scores to z-scores
Both variables on same scale
Correlation stays the same
What happens to means? zy zx
-1.5
-1.0
-0.5
0.5 .5 1
1.5
Convert X and Y to z-scores
1.0 Y X 1 2 3 4 5 6
3.6
Pearson’s r, z transformation

5. Total data for positive correlation between SAT and GPA.
Get r > 0
What correlation between SAT and GPA in only those with admitted and studied (400 < SAT < 700)?
Get r = 0
Restriction of range

6. One extreme score can change correlation (especially in small sample).
On left, 5 observations, high X associated with high Y: good predictability.
On right, same 5 observations plus 1 other, high X associated with high or low Y: poor predictability.
Outliers

7. Causal claims We’d like to say: To be able to do this: X causes Y
The causal variable must come first
There must be co-variation between the two variables
Need to eliminate plausible alternative explanations
Causal claims

8. Causal claims We’d like to say: To be able to do this: X causes Y
The causal variable must come first
There must be co-variation between the two variables
Need to eliminate plausible alternative explanations
Directionality Problem (temporal precedence):
Happy people sleep well
- Or sleeping well makes you happy?
Causal claims

9. Causal claims We’d like to say: To be able to do this: X causes Y
The causal variable must come first
There must be co-variation between the two variables
Need to eliminate plausible alternative explanations
Third Variable Problem:
- Happy people sleep well
- Or does sleeping well make you happy?
OR something else makes people happy and sleep well!
Regular exercise
Minimal use of drugs & alcohol
Being a conscientious person
Being a good relationship
Other
Variable
Causal claims

10. Causal claims We’d like to say: To be able to do this: X causes Y
The causal variable must come first
There must be co-variation between the two variables
Need to eliminate plausible alternative explanations
Coincidence (random co-occurence)
r=0.52 correlation between the
number of republicans in US senate
and number of sunspots
From Fun with correlations
See also Spurious correlations
Causal claims
Correlation is not causation blog posts:
Internet’s favorite phrase
Why we keep saying it

11. Review for Exam 2: Descriptive statistics
Statistical procedures to help organize, summarize & simplify large sets of data
One variable (frequency distribution)
Display results in a frequency distribution table & histogram (or bar chart if categorical variable).
Make a deviations table to get measures of central tendency (mode, median, mean) & variability (range, standard deviation, variance).
Two variables (bivariate distribution)
Display results: Make a scatterplot.
Make a bivariate deviations or z-table table to get Pearson’s r.
Z-scores & normal distribution
Review for Exam 2: Descriptive statistics

12. Example Are hours sleeping related to GPA? You conduct a survey.
Your sample of 10 gives these results for average hours per night sleeping:
7, 6, 7, 8, 8, 7, 9, 5, 9, 6
You also have respondents give their overall GPA:
2.4, 3.9, 3.5, 2.8, 3.0, 2.1, 3.9, 2.9, 3.6, 2.7
We will focus on sleep results first and then both variables together.
What kind of scales are they?
To find standard deviation, will we use formula for population or sample?
Example

13. Step 1: Frequency distribution & histogram
Hrs. sleep
n=10
7,6,7,8,8
7,9,5,9,6 X f p % cf c% 9 8 7 6 5 ∑ 10
1.0
100
Step 1: Frequency distribution & histogram

14. Step 1: Frequency distribution & histogram
Hrs. sleep
n=10
7,6,7,8,8
7,9,5,9,6 X f p % cf c% 9 2 8 7 3 6 5 1 ∑ 10
1.0
100
Will enter first two columns as X and Y axes for frequency distribution
Step 1: Frequency distribution & histogram

15. Step 1: Frequency distribution & histogram
Hrs. sleep
n=10
p = f/n X f p % cf c% 9 2 .2 20 8 7 3 .3 30 6 5 1 .1 10 ∑ 10
1.0
100
Step 1: Frequency distribution & histogram

16. Step 1: Frequency distribution & histogramX f p % cf c% 9 2 .2 20 8 7 3 .3 30 6 5 1 .1 10 ∑ 10
1.0
100
Step 1: Frequency distribution & histogram

17. Step 1: Frequency distribution & histogramX f p % cf c% 9 2 .2 20 8 7 3 .3 30 6 5 1 .1 10 ∑ 10
1.0
100
Step 1: Frequency distribution & histogram

18. Step 1: Frequency distribution & histogramX f p % cf c% 9 2 .2 20 8 7 3 .3 30 6 60 5 1 .1 10 ∑ 10
1.0
100
Step 1: Frequency distribution & histogram

19. Step 1: Frequency distribution & histogramX f p % cf c% 9 2 .2 20 8 80 7 3 .3 30 6 60 5 1 .1 10 ∑ 10
1.0
100
Step 1: Frequency distribution & histogram

20. Step 1: Frequency distribution & histogramX f p % cf c% 9 2 .2 20 10
100 8 80 7 3 .3 30 6 60 5 1 .1 ∑ 10
1.0
100
Step 1: Frequency distribution & histogram

21. Step 1: Frequency distribution & histogram
Hrs. sleep F R E Q U N C Y 6 5 4 3 2 1 7 8 9 X f 9 2 8 7 3 6 5 1
SCORE
Step 1: Frequency distribution & histogram

22. A weighted mean Suppose that you combine two groups together.
How do you compute the new group mean?
Group 1
Group 2
New Group
110
110
110
140
110
110
140
140
110
110
A weighted mean

23. A weighted mean Suppose that you combine two groups together. X f 9 2
How do you compute the new group mean?
Be careful computing the mean of this distribution, remember there are groups here
Group 1
Group 2
New Group X f 9 2 8 7 3 6 5 1 9 8 7 6 5
110
110
110
140
110
110
140
140
110
110
A weighted mean

24. Characteristics of a mean & standard deviation
The mean
Change/add/delete a given score, then the mean will change.
Add/subtract a constant to each score, then the mean will change by adding(subtracting) that constant.
Multiply (or divide) each score by a constant, then the mean will change by being multiplied by that constant.
The standard deviation
Change/add/delete a given score, then the mean will change.
Add/subtract a constant to each score, then the standard deviation will NOT change.
Multiply (or divide) each score by a constant, then the standard deviation will change by being multiplied by that constant.
Characteristics of a mean & standard deviation

25. Step 2: Deviations tableX
Hrs. sleep
n = 10
Create table, sorted in descending order 9 8 7 6 5
Step 2: Deviations table

26. Step 2: Deviations tableX
Hrs. sleep
n = 10 9 8 7 6 5
Mode = 7 (filled in)
Median = 7 (arrow)
Mean = (∑X)/n = 72/10 = 7.2
Range = 5 to 9 ∑
Step 2: Deviations table

27. Step 2: Deviations tableX
Hrs. sleep
n = 10 9
1.8 8 .8 7
-.2 6
-1.2 5
-2.2
= 9-7.2
Mode = 7
Median = 7
Mean = (∑X)/n = 72/10 = 7.2
Range = 5 to 9 ∑
7.2
Step 2: Deviations table

28. Step 2: Deviations tableX
Hrs. sleep
n = 10 9
1.8
3.24 8 .8
.64 7
-.2
.04 6
-1.2
1.44 5
-2.2
4.84
= 1.82
Mode = 7
Median = 7
Mean = ∑X/n = 72/10 = 7.2
Range = 5 to 9
SD for sample
= √15.6/9 = √1.73 = 1.32 ∑
7.2
15.6 = SS
Step 2: Deviations table

29. Step 3: Scatterplot Person Hrs. GPA GPA A 7 2.4 B 6 3.9 C 7 3.5
D 8 2.8
E 8 3.0
F 7 2.1
G 9 3.9
H 5 2.9
I 9 3.6
J 6 2.7
4.0
3.5
3.0
2.5
2.0
1.5
1.0 5 6 7 8 9
GPA
Hours of sleep
Step 3: Scatterplot

30. Step 3: Scatterplot Person Hrs. GPA GPA A 7 2.4 B 6 3.9 C 7 3.5
D 8 2.8
E 8 3.0
F 7 2.1
G 9 3.9
H 5 2.9
I 9 3.6
J 6 2.7
4.0 B G
3.5 C I
3.0 H DE
2.5 J A
2.0 F
1.5
1.0 5 6 7 8 9
GPA
What does shape of envelope indicate about correlation?
low positive correlation
Hours of sleep
Step 3: Scatterplot

31. Step 3: Scatterplot, Effect of outlier
Person Hrs. GPA
A 7 2.4
B 6 3.9
C 7 3.5
D 8 2.8
E 8 3.0
F 7 2.1
G 9 3.9
H 5 2.9
I 9 3.6
J 6 2.7
K 5 1.0
4.0 B G
3.5 C I
3.0 H DE
2.5 J A
2.0 F
1.5
1.0 K 5 6 7 8 9
GPA
What does shape of envelope indicate about correlation?
moderate positive correlation
Hours of sleep
Step 3: Scatterplot, Effect of outlier

32. Step 3: Scatterplot, Effect of outlier
Person Hrs. GPA
A 7 2.4
B 6 3.9
C 7 3.5
D 8 2.8
E 8 3.0
F 7 2.1
G 9 3.9
H 5 2.9
I 9 3.6
J 6 2.7
K 9 1.0
4.0 B G
3.5 C I
3.0 H DE
2.5 J A
2.0 F
1.5
1.0 K 5 6 7 8 9
GPA
What does shape of envelope indicate about correlation?
low negative correlation
Hours of sleep
Step 3: Scatterplot, Effect of outlier

33. Step 4: Bivariate Deviations TableX Y 9
1.8
3.24
3.9
0.82
0.67
1.476
3.6
0.52
0.27
0.936 8
0.8
0.64
3.0
-0.08
0.01
-0.064
2.8
-0.28
0.86
-0.224 7
-0.2
0.40
3.5
0.42
0.18
-0.084
0.04
2.4
-0.68
0.46
0.136
2.1
-0.98
0.96
0.196 6
-1.2
1.44
-0.984
2.7
-0.38
0.14
0.456 5
-2.2
4.84
2.9
-0.18
0.03
0.396 72
0.0
15.6
30.8
3.47
2.24
7.2
SSX
3.08
SSY SP
n=10
Note signs!
Sum
Mean
+r or – r?
Step 4: Bivariate Deviations Table

34. Pearson’s r & summary statistics
XY co-deviations
___2.24___
√ 15.6 * 3.47
= _2.24_
√54.132
= _2.24_ = .304
7.357 =
X deviations, Y deviations
Pearson’s r & summary statistics

35. An example SRA (Scientific Reasoning Assessment) (fictional)
Based on normative data: Normal, μ = 50.0, σ = 10.0
Preparing for your analyses
Write down what you know
Make a sketch of the distribution (make a note: population or sample)
Determine the shape
What is best measure of center?
What is best measure of variability?
Mark the mean (center) and standard deviation on your sketch μ 40 60
An example

36. z-scores & Normal Distribution
SRA (Scientific Reasoning Assessment) (fictional)
Based on normative data: Normal distr., μ = 50.0, σ = 10.0
Question 1
If George got a 35 on the SRA, what is his percentile rank? m
Unit Normal Table
0.0668
Since a normal distribution, can use Unit Normal Table to infer percentile. 40 60
-1.0
1.0
That’s 6.68% at or below this score (definition of percentile)
z-scores & Normal Distribution

37. z-scores & Normal Distribution
SRA (Scientific Reasoning Assessment) (fictional)
Based on normative data: Normal distr., μ = 50.0, σ = 10.0
Question 2
Unit Normal Table
What proportion of people get between a 40 and 60 on the SRA? m
0.1587
0.1587 40 40 60 60
-1.0
1.0
That’s about 32% outside these two scores
Since a normal distribution, can use Unit Normal Table to infer percentile.
That leaves 68% between these two scores
z-scores & Normal Distribution

38. z-scores & Normal Distribution
SRA (Scientific Reasoning Assessment) (fictional)
Based on normative data: Normal distr., μ = 50.0, σ = 10.0
Question 3a
Suppose that Chandra took a different reasoning assessment (the RSE: Based on normative data, Normal distr., μ= 100, σ = 15). She received a 130 on the RSE. Assuming that they are highly positively correlated, what is the equivalent score on the SRA?
transformation
z-scores & Normal Distribution

39. z-scores & Normal Distribution
SRA (Scientific Reasoning Assessment) (fictional)
Based on normative data: Normal distr., μ = 50.0, σ = 10.0
Question 3a
(for RSE)
Suppose that Chandra took a different reasoning assessment (the RSE: Based on normative data, Normal distr., μ= 100, σ = 15). She received a 130 on the RSE. Assuming that they are highly positively correlated, what is the equivalent score on the SRA?
(for SRA)
transformation
z-scores & Normal Distribution

40. In lab: continue to review, including SPSS
Questions?
Wrap up