1. Some statistics questions answered:
"Well, am I significantly more popular than Nick Clegg?"

2. Requests: What's the difference between the SD and SEM?
What are "degrees of freedom"?
How to calculate t-tests.
When do you use an ANOVA?
How to use tables to assess statistical significance.

3. Standard deviation and standard error:
Standard deviation (SD): a measure of how much scores are spread out around their mean.
The bigger the SD, the less representative the mean is of the set of scores from which it was calculated.
1. Find the mean.
2. Find difference between each score and the mean.
3. Square the differences.
4. Add them up.
5. Divide by number of scores.
6. Take the square root of the result.
In practice - use the "standard deviation" function on your calculator!

4. Standard deviation and standard error:
Standard error of the mean (SE): a measure of how much sample means are spread out around the population mean.
The bigger the SE, the less confident we can be that the sample mean truly reflects the population mean.
1. Find the standard deviation.
2. Divide it by the square root of the number of scores.
means of different samples
actual population mean
means of different samples
actual population mean
successive attempts

5. What are "degrees of freedom"?:
Simple answer: the number of scores that are free to vary. 3 4 2 5 14 1 7 11 9 8
Top row total = 14: once you know the first three scores, fourth score is fixed - it's not free to vary.
Hence Chi-Square d.f. are (number of rows - 1) * (number of columns -1)
Here, d.f. = (4-1) * (5-1) = 12

6. Degrees of freedom for different tests:
Pearson's r: number of pairs of scores -2
Repeated measures t-test : number of participants - 1
Independent measures t-test:
(no. of participants in group A - 1) + (no. of participants in group B - 1)
Kruskal-Wallis: d.f. = number of groups - 1
Friedman's: d.f. = number of conditions - 1
Mann-Whitney: N1 and N2 = number of participants in each group
Wilcoxon: N = number of participants.

7. Degrees of freedom for different tests:
One-way independent measures ANOVA:
Total variation
d.f. = no. scores - 1
Between groups variation
d.f. = no. groups -1
Within-groups variation
d.f. = no. scores in group A -1 plus no. scores in group B -1, plus no. scores in group C - 1, etc.
e.g. ANOVA with 5 groups and 20 participants in each:
Source SS df MS F
Between groups
4 (= no. of groups - 1)
Within groups
95 (= 5 x 19)
Total
99 (= total no. scores – 1)
The value of F for (4,95) degrees of freedom is what would be reported from this.

8. One-way repeated measures ANOVA:
Total variation
d.f. = no. scores - 1
Between subjects variation
d.f. = number of subjects -1
Within-subjects variation
d.f. = no. subjects
Within-subjects variation due to experimental manipulations
d.f. = number of conditions - 1
Within-subjects variation due to random factors ("error")
d.f. = total within-subjects d.f. minus experimental d.f.
Source SS
d.f.
Mean Squares
F-ratio
Between subjects 5
Within subjects 12
Experimental 2
Error 10
Total 17

9. Degrees of freedom for different tests:
e.g. a one-way repeated measures ANOVA, with 3 conditions and 6 participants:
Source SS
d.f. MS F
Between subjects
5 (= number of participants -1)
Within subjects
12 (here = 6 * (3-1) = 6*2 )
Experimental
2 (= no. conditions -1)
Error
10 (here = (6-1) *(3-1) = 5*2 )
Total
17 (= total no. scores – 1)
The value of F for (2,10) degrees of freedom is what would be reported from this.

10. Tests whose output you should be able to understand:
Tests you need to be able to calculate by hand (for section 2 of the exam):
Chi-Square test of association , Chi-Square goodness of fit
Pearson's r , Spearman's rho
Wilcoxon , Mann-Whitney
Friedman's , Kruskal-Wallis
Repeated-measures t-test , Independent-measures t-test
Tests whose output you should be able to understand:
One-way independent-measures ANOVA
One-way repeated-measures ANOVA

11. Independent-measures t-test:
The t value represents the size of the difference between two means (essentially a z-score for small sample sizes).
The bigger the value of t, the more confident we can be that the difference between the means is "real", i.e. that it has not occurred just by chance.
t is the difference between two means, divided by an estimate of how much this difference is likely to vary from occasion to occasion (estimated standard error of the difference between means).
Repeated-measures t-test:
Same logic, except that we can capitalise on the fact that the same people did both conditions (and hence random variation in performance is likely to be less).

12. Independent-measures t-test:
t is the difference between two means, divided by an estimate of how much this difference is likely to vary from occasion to occasion (estimated standard error of the difference between means).
mean A
mean B
male height
female height
probability of occurrence
value of t
©McMaster University 12

13. Effects of food additives on children's activity levels:
Group A: eat tartrazine-containing nosh.
Group B: same nosh without tartrazine.
DV: time spent running around.
A: additive
B: no additive 11 5 15 6 4 14 7 18
mean:
14.00
5.57
SD:
2.68
0.98
Σ X1 = 84
Σ X2 =39
Data OK for a parametric test?
1. Normally distributed?
Here, too few scores to tell!
2. Interval or ratio?
Ratio (time).
3. Homogeneity of variance?
Yes - SD is roughly similar proportion of the mean (Levene's test p = .07).

14. Effects of food additives on children's activity levels:
Group A: tartrazine: mean =
Group B: no tartrazine: mean = 5.57.
top line of equation:

15. Effects of food additives on children's activity levels:
Group A: tartrazine. Group B: no tartrazine. DV: time spent running around.
bottom line of equation:
Quicker ways of calculating these...
= 36.00
=5.714

16. Effects of food additives
on children's activity levels:
Group A: tartrazine.
Group B: no tartrazine.
Two-tailed test (just predicting "an effect" of additives):
t = 7.79 with 11 d.f.,
p < .0001

17. My obtained value of t (9) = -3
My obtained value of t (9) = -3.2, which is smaller than the critical value of 2.26 (because -3.2 is smaller than zero). Does this mean it is not significant?
upper
t-critical value
2.26
lower
-2.26
0.025
Ignore the sign of t when using the table: whether t is positive or negative merely depends on which group you called X, and which you called Y.

18. When do you use a t-test, and when do you use a correlation?
Depends on whether you have an experimental or correlational design:
t-test:
Two groups or conditions (two levels of one IV)
Looking for differences between them on a single DV.
Does alcohol affect driving performance?
IV "alcohol dosage" (sober versus inebriated).
DV number of crashes.
Correlation:
Two different DVs (alcohol consumption and number of crashes).
Looking for a relationship between them - as alcohol consumption increases, so too should number of crashes.

19. When do you use Analysis of Variance?
One-way independent measures ANOVA:
Three or more groups of participants (three or more levels of one IV)
Looking for differences between the means for these conditions on a single DV.
Does alcohol affect driving performance?
IV "alcohol dosage" (sober versus two pints versus four pints).
DV number of crashes.
Three or more conditions, all done by the same participants (three or more levels of one IV)
DV number of crashes: each person produces three scores.