1. Psychology 202a Advanced Psychological Statistics
August 29, 2017

2. Overview of today’s class
Start discussion of variables and distributions (first using R)
Working definition of a variable
Working definition of a distribution
Graphical and numerical methods for understanding distributions
Introduce SAS

3. What is a variable?
We are going to be using R and SAS as tools to help us understand the behavior of variables.
We need a working definition of “variable.”
For our purposes, a variable consists of
numbers
that convey information
about some well-defined entity.

4. Is this a variable?
Numbers 
that convey information 
about a well-defined entity 

5. Numbers that convey information…
Peabody Picture Vocabulary scores
The Peabody is intended to be a test of verbal ability.
Items consist of a set of pictures and words; the task is to choose the picture that matches the word.
Example from youtube.

6. …about some well-defined entity.
These scores are a sample from the “Child Health and Development Study.”
10-year-old children
Members of the Kaiser-Permanente health plan in Oakland, CA
Data were collected at the beginning of the 1970s.

7. Is this a variable?
numbers 
that convey information 
about a well-defined entity 
Yes, it’s a variable.

8. But it’s not easy to say much about the variable.
data are not organized
difficult to see structure

9. R can help us see the structure.
Peabody <- c(
69, 72, 94, 64, 80, 77, 96, 86, 89, 69,
92, 71, 81, 90, 84,100, 76, 57, 61, 84,
81, 65, 87, 92, 89, 79, 91, 65, 91, 81,
86, 85, 95, 93, 83, 76, 84, 90, 95, 67)
“Peabody gets cee of…”

10. How can R help us see the structure?
How many scores are there?
length(Peabody)
What’s a big score or a small score?
sort(Peabody)
Note that there are lots of scores in the 80s, not so many in the 70s and 90s, and very few in the 50s or 100s.

11. What is a distribution?
We’ve been looking at what values of Peabody occur, and how often they occur.
That’s what a distribution is:
the values that a variable takes on, together with…
…the frequencies (or relative frequencies) of those values.

12. Understanding distributions…
We could interpret that idea very literally.
There is one score of 57.
There is one score of 61.
There is one score of 64.
There are two scores of 65.
This would rapidly become tedious…
…and would not be very useful.

13. …by ignoring detail.
The problem with that approach is that there is too much information.
Simplify, ignore detail to see structure.
Ways to do that:
group the data
use pictures
use summary numbers (descriptive statistics)

14. Grouping data
Looking at our sorted data, we can see that there is (or are)
one number in the 50s
seven numbers in the 60s
six numbers in the 70s
fourteen numbers in the 80s
eleven numbers in the 90s
one number in the 100s.

15. Grouping data We’ve gone too far: that’s not enough information.
Here’s a general principle: try to group so that there are between seven and fifteen categories.
(as with any rule of thumb, there will be exceptions)

16. Peabody Distribution Values Frequency 55 – 59 1 60 – 64 2 65 – 69 5
70 – 74
75 – 79 4
80 – 84 8
85 – 89 6
90 – 94
95 – 99 3
100 – 104

17. Some details about grouping
continuous and discrete variables
real limits

18. Lower and upper real limits
What we say
What we mean
55 – 59
54.5 – 59.5
60 – 64
59.5 – 64.5
65 – 69
64.5 – 69.5
70 – 74
69.5 – 74.5
75 – 79
74.5 – 79.5
80 – 84
79.5 – 84.5
85 – 89
84.5 – 89.5
90 – 94
89.5 – 94.5
95 – 99
94.5 – 99.5
100 – 104
99.5 – 104.5

19. Relative frequency distribution
Peabody Values
Relative Frequency
55 – 59
.025
60 – 64
.050
65 – 69
.125
70 – 74
75 – 79
.100
80 – 84
.200
85 – 89
.150
90 – 94
95 – 99
.075
100 – 104

20. What can we say about the distribution?
There is variation in the scores.
Peabody scores are most frequent in the 80s and 90s.
Scores at the extremes of the distribution are much less frequent than scores at the center.
But it’s still a little hard to see all this.

21. A simple graphical technique
Stem-and-leaf plot
Divide the numbers into fine-grained and coarse-grained information.
coarse = “stem”
fine = “leaf”
Manual demonstration
stem(Peabody)

22. Descriptive Statistics
Numbers that convey information about aspects of the shape of the distribution
Example: What values are most typical? (Central tendency.)
Mean, median, mode
Descriptive statistics in SAS