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```Small-N designs
& Basic Statistical Concepts
Psych 231: Research
Methods in Psychology

What are they?


Historically, these were the typical kind of design
used until 1920’s when there was a shift to using
larger sample sizes
Even today, in some sub-areas, using small N
designs is common place
• (e.g., psychophysics, clinical settings, expertise, etc.)
Small N designs

One or a few participants



Data are not analyzed statistically; rather rely on
visual interpretation of the data
Observations begin in the absence of treatment
(BASELINE)
Then treatment is implemented and changes in
frequency, magnitude, or intensity of behavior are
recorded
Small N designs

Baseline experiments – the basic idea is to show:
1. When the IV occurs, you get the effect
2. When the IV doesn’t occur, you don’t get the effect
(reversibility)

Before introducing treatment (IV), baseline
needs to be stable
• Measure level and trend
• Level – how frequent (how intense) is behavior?
• Are all the data points high or low?
• Trend – does behavior seem to increase (or
decrease)
• Are data points “flat” or on a slope?
Small N designs

AB design (baseline, treatment)
– Can look for levels and trends before and after the
treatment
A
B
AB design

ABA design (baseline, treatment, baseline)
– The reversibility is necessary, otherwise
something else may have caused the effect
other than the IV (e.g., history, maturation, etc.)
A
B
A
ABA design

ABAB design (baseline, treatment, baseline, treatment)
– The second AB part essentially provides a
replication of the first (and you can see the reversal
effect)
– A useful control for ruling out potential threats to internal
validity (e.g., history)
A
B
A
ABAB design
B
Reversibility
| Transition2






Focus on individual performance, not fooled by group
averaging effects
Focus is on big effects (small effects typically can’t be seen
without using large groups)
Avoid some ethical problems – e.g., with non-treatments
Allows to look at unusual (and rare) types of subjects (e.g.,
case studies of amnesiacs, experts vs. novices)
Often used to supplement large N studies, with more
observations on fewer subjects
Small N designs



Effects may be small relative to variability of situation
Some effects are by definition between subjects
• Treatment leads to a lasting change, so you don’t get
reversals

Difficult to determine how generalizable the effects are
Small N designs

Mistrust of statistics?


It is all in how you use them
They are a critical tool in research
Statistics

Why do we use them?

Descriptive statistics
• Used to describe, simplify, & organize data sets
• Describing distributions of scores

Inferential statistics
• Used to test claims about the population, based on data
gathered from samples
• Takes sampling error into account, are the results above
and beyond what you’d expect by random chance
Statistics


Recall that a variable is a characteristic that can take
different values.
The distribution of a variable is a summary of all the
different values of a variable
 Both type (each value) and token (each instance)
How much do you like psy231?
5 values (1, 2, 3, 4, 5)
1-2-3-4-5
Hate it
Love it
1
5
5
Distribution
7 tokens (1,1,2,3,4,5,5)
4
1
3
2

Properties of a distribution

Shape
• Symmetric v. asymmetric (skew)
• Unimodal v. multimodal

Center
• Where most of the data in the distribution are
• Mean, Median, Mode

• How similar/dissimilar are the scores in the distribution?
• Standard deviation (variance), Range
Distribution

A picture of the distribution is usually helpful


Gives a good sense of the properties of the distribution
Many different ways to display distribution

Graphs
• Continuous variable:
• histogram, line graph (frequency polygons)
• Categorical variable:
• pie chart, bar chart

Table
• Frequency distribution table
Distribution

A frequency histogram
Example: Distribution of scores on an exam
Frequency

20
18
16
14
12
10
8
6
4
2
0
18
17
12
11
10
8
7
5
3
1
5054
5559
60- 6564 69
70- 7574 79
80- 8584 89
9094
95100
Exam scores
Graph for continuous variables

Bar chart

Pie chart
Cutting
Doe
Missing
Smith
Graphs for categorical variables
VAR00 003
Va lid
1.00
Fre quen cy
2
Percent
7.7
Va lid Perce nt
7.7
Cumu lati ve
Percent
7.7
2.00
3.00
4.00
3
3
5
11 .5
11 .5
19 .2
11 .5
11 .5
19 .2
19 .2
30 .8
50 .0
5.00
6.00
7.00
8.00
4
2
4
2
15 .4
7.7
15 .4
7.7
15 .4
7.7
15 .4
7.7
65 .4
73 .1
88 .5
96 .2
9.00
To tal
1
26
3.8
10 0.0
3.8
10 0.0
10 0.0
Values
Counts
Percentages
(types)
Frequency distribution table

Symmetric
Asymmetric (skewed)
Negative Skew
Positive Skew
tail
Properties of distributions: Shape
tail

Unimodal (one mode)
Multimodal
Minor
mode
Major
mode
Bimodal examples
Properties of distributions: Shape

There are three main measures of center

Mean (M): the arithmetic average
• Add up all of the scores and divide by the total number
• Most used measure of center

Median (Mdn): the middle score in terms of location
• The score that cuts off the top 50% of the from the bottom
50%
• Good for skewed distributions (e.g. net worth)

Mode: the most frequent score
• Good for nominal scales (e.g. eye color)
• A must for multi-modal distributions
Properties of distributions: Center


The most commonly used measure of center
The arithmetic average

Computing the mean
– The formula for the population
mean is (a parameter):
X

N
– The formula for the sample
mean is (a statistic):
X
X
n


The Mean
Divide by the
total number in
the population
the X’s
Divide by the
total number in
the sample

How similar are the scores?

Range: the maximum value - minimum value
• Only takes two scores from the distribution into account
• Influenced by extreme values (outliers)

Standard deviation (SD): (essentially) the average amount that
the scores in the distribution deviate from the mean
• Takes all of the scores into account
• Also influenced by extreme values (but not as much as the range)

Variance: standard deviation squared
Low variability
High variability
The scores are fairly similar
The scores are fairly dissimilar
mean
Variability
mean

The standard deviation is the most popular and most
important measure of variability.

The standard deviation measures how far off all of the
individuals in the distribution are from a standard, where that
standard is the mean of the distribution.
• Essentially, the average of the deviations.

Standard deviation
Our population
2, 4, 6, 8
 X 2  4  6  8 20


  5.0
N
4
4
1 2 3 4 5 6 7 8 9 10


An Example: Computing the Mean


Our population
2, 4, 6, 8
Step 1: To get a measure of the
deviation we need to subtract the
population mean from every
individual in our distribution.
 X 2  4  6  8 20


  5.0
N
4
4
X -  = deviation scores
2 - 5 = -3
-3
1 2 3 4 5 6 7 8 9 10

An Example: Computing Standard
Deviation (population)


Our population
2, 4, 6, 8
Step 1: To get a measure of the
deviation we need to subtract the
population mean from every
individual in our distribution.
 X 2  4  6  8 20


  5.0
N
4
4
X -  = deviation scores
2 - 5 = -3
4 - 5 = -1
-1
1 2 3 4 5 6 7 8 9 10

An Example: Computing Standard
Deviation (population)


Our population
2, 4, 6, 8
Step 1: To get a measure of the
deviation we need to subtract the
population mean from every
individual in our distribution.
 X 2  4  6  8 20


  5.0
N
4
4
X -  = deviation scores
2 - 5 = -3
4 - 5 = -1
6 - 5 = +1
1
1 2 3 4 5 6 7 8 9 10

An Example: Computing Standard
Deviation (population)


Our population
2, 4, 6, 8
Step 1: To get a measure of the
deviation we need to subtract the
population mean from every
individual in our distribution.
 X 2  4  6  8 20


  5.0
N
4
4
X -  = deviation scores
2 - 5 = -3
4 - 5 = -1
6 - 5 = +1
8 - 5 = +3
3
1 2 3 4 5 6 7 8 9 10

Notice that if you add up
all of the deviations they
must equal 0.
An Example: Computing Standard
Deviation (population)

Step 2: So what we have to do is get rid of the negative
signs. We do this by squaring the deviations and then
taking the square root of the sum of the squared
deviations (SS).
X -  = deviation scores
2 - 5 = -3
4 - 5 = -1
6 - 5 = +1
8 - 5 = +3
SS =  (X - )2
= (-3)2 + (-1)2 + (+1)2 + (+3)2
= 9 + 1 + 1 + 9 = 20
An Example: Computing Standard
Deviation (population)

Step 3: ComputeVariance (which is simply the average
of the squared deviations (SS))

So to get the mean, we need to divide by the number of
individuals in the population.
variance = 2 = SS/N
An Example: Computing Standard
Deviation (population)

Step 4: Compute Standard Deviation

To get this we need to take the square root of the population
variance.
X  
2
standard deviation =  =  
2
N

An Example: Computing Standard
Deviation (population)

To review:



Step 1: Compute deviation scores
Step 2: Compute the SS
Step 3: Determine the variance
• Take the average of the squared deviations
• Divide the SS by the N

Step 4: Determine the standard deviation
• Take the square root of the variance
An Example: Computing Standard
Deviation (population)

To review:



Step 1: Compute deviation scores
Step 2: Compute the SS
Step 3: Determine the variance
• Take the average of the squared deviations
• Divide the SS by the N-1

Step 4: Determine the standard deviation
• Take the square root of the variance
An Example: Computing Standard
Deviation (sample)
```
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