1. Reasoning in Psychology Using Statistics
2018

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

3. 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

4. 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

5. 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

6. 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

7. Review for Exam 2: Descriptive statistics
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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

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

21. 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

22. 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

23. 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

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 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

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