1. Psych 231: Research Methods in Psychology
Statistics (cont.)
Psych 231: Research Methods in Psychology

2. Announcements Quiz 10 (chapter 7) is due on Nov. 13th at midnight
Journal Summary 2 assignment
Due in class NEXT week (Wednesday, Nov. 18th) <- moved due date
Group projects
Plan to have your analyses done before Thanksgiving break, GAs will be available during lab times to help
Poster sessions are last lab sections of the semester (last week of classes), so start thinking about your posters. I will lecture about poster presentations on the Monday before Thanksgiving break.
Announcements

3. Statistics 2 General kinds of Statistics 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?
Population
Inferential statistics used to generalize back
Sample
Statistics

4. Inferential Statistics
Purpose: To make claims about populations based on data collected from samples
What’s the big deal?
Population
Sample A
Treatment
X = 80%
Sample B
No Treatment
X = 76%
Example Experiment:
Group A - gets treatment to improve memory
Group B - gets no treatment (control)
After treatment period test both groups for memory
Results:
Group A’s average memory score is 80%
Group B’s is 76%
Is the 4% difference a “real” difference (statistically significant) or is it just sampling error?
Inferential Statistics

5. Testing Hypotheses Step 1: State your hypotheses
Null hypothesis (H0)
Alternative hypothesis(ses) (HA)
Step 2: Set your decision criteria
Step 3: Collect your data from your sample(s)
Step 4: Compute your test statistics
Step 5: Make a decision about your null hypothesis
Reject H0 “statistically significant differences”
Fail to reject H0 “not statistically significant differences”
Type I error (α): concluding that there is an effect (a difference between groups) when there really isn’t.
Type II error (β): concluding that there isn’t an effect, when there really is.
Testing Hypotheses

6. Summary to this point XA XB About populations Example Experiment:
Group A - gets treatment to improve memory
Group B - gets no treatment (control)
After treatment period test both groups for memory
Results:
Group A’s average memory score is 80%
Group B’s is 76%
H0: μA = μB
H0: there is no difference between Grp A and Grp B
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II error
Is the 4% difference a “real” difference (statistically
significant) or is it just sampling error?
Two sample
distributions XA XB
76%
80%
Summary to this point

7. Statistical significance
“Statistically significant differences”
When you “reject your null hypothesis”
Essentially this means that the observed difference is above what you’d expect by chance
“Chance” is determined by estimating how much sampling error there is
Factors affecting “chance”
Sample size
Population variability
Statistical significance

8. (Pop mean - sample mean)
Population mean
Population
Distribution x
Sampling error
(Pop mean - sample mean)
n = 1
Sampling error

9. (Pop mean - sample mean)
Population mean
Population
Distribution
Sample mean x x
Sampling error
(Pop mean - sample mean)
n = 2
Sampling error

10. (Pop mean - sample mean)
Generally, as the sample size increases, the sampling error decreases
Population mean
Population
Distribution
Sample mean x
Sampling error
(Pop mean - sample mean)
n = 10
Sampling error

11. Typically the narrower the population distribution, the narrower the range of possible samples, and the smaller the “chance”
Large population variability
Small population variability
Sampling error

12. Sampling error Population Distribution of sample means
These two factors combine to impact the distribution of sample means.
The distribution of sample means is a distribution of all possible sample means of a particular sample size that can be drawn from the population
Population
Distribution of sample means XC
Samples of size = n XA XD
Avg. Sampling error XB
“chance”
Sampling error
More info

13. “Generic” statistical test
Tests the question:
Are there differences between groups due to a treatment?
H0 is true (no treatment effect)
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II error
Two possibilities in the “real world”
One population
Two sample
distributions XA XB
76%
80%
“Generic” statistical test

14. “Generic” statistical test
Tests the question:
Are there differences between groups due to a treatment?
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II error
Two possibilities in the “real world”
H0 is true (no treatment effect)
H0 is false (is a treatment effect)
Two populations XA XB XB XA
76%
80%
76%
80%
People who get the treatment change,
they form a new population
(the “treatment population)
“Generic” statistical test

15. “Generic” statistical testXB XA
ER: Random sampling error
ID: Individual differences (if between subjects factor)
TR: The effect of a treatment
Why might the samples be different?
(What is the source of the variability between groups)?
“Generic” statistical test

16. “Generic” statistical testXB XA
ER: Random sampling error
ID: Individual differences (if between subjects factor)
TR: The effect of a treatment
The generic test statistic - is a ratio of sources of variability
Observed difference
TR + ID + ER
ID + ER
Computed
test statistic = =
Difference from chance
“Generic” statistical test

17. Sampling error Population Difference from chance
The distribution of sample means is a distribution of all possible sample means of a particular sample size that can be drawn from the population
Population
Distribution of sample means XC
Samples of size = n XA XD
Avg. Sampling error XB
Difference from chance
Sampling error

18. “Generic” statistical test
Things that affect the computed test statistic
Size of the treatment effect
The bigger the effect, the bigger the computed test statistic
Difference expected by chance (sample error)
Sample size
Variability in the population
TR + ID + ER
ID + ER
TR + ID + ER
ID + ER XB XA XB XA
TR + ID + ER
ID + ER
“Generic” statistical test

19. Some inferential statistical tests
1 factor with two groups
T-tests
Between groups: 2-independent samples
Within groups: Repeated measures samples (matched, related)
1 factor with more than two groups
Analysis of Variance (ANOVA) (either between groups or repeated measures)
Multi-factorial
Factorial ANOVA
Some inferential statistical tests

20. T-test Design Formula: Observed difference X1 - X2 T =
2 separate experimental conditions
Degrees of freedom
Based on the size of the sample and the kind of t-test
Formula:
Observed difference
T =
X X2
Diff by chance
Based on sample error
Computation differs for
between and within t-tests
T-test

21. T-test Reporting your results
The observed difference between conditions
Kind of t-test
Computed T-statistic
Degrees of freedom for the test
The “p-value” of the test
“The mean of the treatment group was 12 points higher than the control group. An independent samples t-test yielded a significant difference, t(24) = 5.67, p < 0.05.”
“The mean score of the post-test was 12 points higher than the pre-test. A repeated measures t-test demonstrated that this difference was significant significant, t(12) = 5.67, p < 0.05.”
T-test

22. Analysis of Variance XB XA XC Designs Test statistic is an F-ratio
More than two groups
1 Factor ANOVA, Factorial ANOVA
Both Within and Between Groups Factors
Test statistic is an F-ratio
Degrees of freedom
Several to keep track of
The number of them depends on the design
Analysis of Variance

23. Analysis of Variance More than two groups F-ratio = XB XA XC
Now we can’t just compute a simple difference score since there are more than one difference
So we use variance instead of simply the difference
Variance is essentially an average difference
Observed variance
Variance from chance
F-ratio =
Analysis of Variance

24. 1 factor ANOVA 1 Factor, with more than two levels XB XA XC
Now we can’t just compute a simple difference score since there are more than one difference
A - B, B - C, & A - C
1 factor ANOVA

25. 1 factor ANOVA The ANOVA tests this one!! XA = XB = XC XA ≠ XB ≠ XC
Null hypothesis:
H0: all the groups are equal
The ANOVA tests this one!!
XA = XB = XC
Do further tests to pick between these
Alternative hypotheses
HA: not all the groups are equal
XA ≠ XB ≠ XC
XA ≠ XB = XC
XA = XB ≠ XC
XA = XC ≠ XB
1 factor ANOVA

26. 1 factor ANOVA Planned contrasts and post-hoc tests:
- Further tests used to rule out the different Alternative hypotheses
XA ≠ XB ≠ XC
Test 1: A ≠ B
XA = XB ≠ XC
Test 2: A ≠ C
XA ≠ XB = XC
Test 3: B = C
XA = XC ≠ XB
1 factor ANOVA

27. 1 factor ANOVA Reporting your results The observed differences
Kind of test
Computed F-ratio
Degrees of freedom for the test
The “p-value” of the test
Any post-hoc or planned comparison results
“The mean score of Group A was 12, Group B was 25, and Group C was 27. A 1-way ANOVA was conducted and the results yielded a significant difference, F(2,25) = 5.67, p < Post hoc tests revealed that the differences between groups A and B and A and C were statistically reliable (respectively t(1) = 5.67, p < 0.05 & t(1) = 6.02, p < 0.05). Groups B and C did not differ significantly from one another”
1 factor ANOVA

28. We covered much of this in our experimental design lecture
More than one factor
Factors may be within or between
Overall design may be entirely within, entirely between, or mixed
Many F-ratios may be computed
An F-ratio is computed to test the main effect of each factor
An F-ratio is computed to test each of the potential interactions between the factors
Factorial ANOVAs

29. Factorial ANOVAs Reporting your results The observed differences
Because there may be a lot of these, may present them in a table instead of directly in the text
Kind of design
e.g. “2 x 2 completely between factorial design”
Computed F-ratios
May see separate paragraphs for each factor, and for interactions
Degrees of freedom for the test
Each F-ratio will have its own set of df’s
The “p-value” of the test
May want to just say “all tests were tested with an alpha level of 0.05”
Any post-hoc or planned comparison results
Typically only the theoretically interesting comparisons are presented
Factorial ANOVAs

30. Distribution of sample means
The following slides are available for a little more concrete review of distribution of sample means discussion.
Distribution of sample means

31. Distribution of sample means
Distribution of sample means is a “theoretical” distribution between the sample and population
Mean of a group of scores
Comparison distribution is distribution of means
Population
Distribution of sample means
Sample
Distribution of sample means

32. Distribution of sample means
A simple case
Population: 2 4 6 8
All possible samples of size n = 2
Assumption: sampling with replacement
Distribution of sample means

33. Distribution of sample means
A simple case
Population: 2 4 6 8
All possible samples of size n = 2
There are 16 of them
mean 2 2 2 6 4 5 6 4 7 8 4 6 8 2 2 4 3 8 4 6 2 4 6 4 8 2 8 2 5 4 2 3 4 4
Distribution of sample means

34. Distribution of sample means2 3 4 5 6 7 8 1
In long run, the random selection of tiles leads to a predictable pattern
mean
mean
mean 2 2 2 4 6 5 8 2 5 2 4 3 4 8 6 8 4 6 2 6 4 6 2 4 8 6 7 2 8 5 6 4 5 8 8 8 4 2 3 6 6 6 4 4 4 6 8 7
Distribution of sample means

35. Properties of the distribution of sample means
Shape
If population is Normal, then the dist of sample means will be Normal
If the sample size is large (n > 30), regardless of shape of the population
Distribution of sample means
Population
N > 30
Properties of the distribution of sample means

36. Properties of the distribution of sample means
Center
The mean of the dist of sample means is equal to the mean of the population
Population
Distribution of sample means
same numeric value
different conceptual values
Properties of the distribution of sample means

37. Properties of the distribution of sample means
Center
The mean of the dist of sample means is equal to the mean of the population
Consider our earlier example 2 4 6 8
Population
Distribution of sample means
means 2 3 4 5 6 7 8 1 4
μ =
= 5 16 =
= 5
Properties of the distribution of sample means

38. Properties of the distribution of sample means
Spread
Standard deviation of the population
Sample size
Putting them together we get the standard deviation of the distribution of sample means
Commonly called the standard error
Properties of the distribution of sample means

39. The standard error is the average amount that you’d expect a sample (of size n) to deviate from the population mean
In other words, it is an estimate of the error that you’d expect by chance (or by sampling)
Standard error

40. All three of these properties are combined to form the Central Limit Theorem
For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will approach a normal distribution with a mean of and a standard deviation of as n approaches infinity
(good approximation if n > 30).
Central Limit Theorem

41. Distribution of sample means
Keep your distributions straight by taking care with your notation
Population σ μ
Distribution of sample means
Sample s X
Distribution of sample means
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