1. Inferential Statistics
Mann-Whitney U
Wilcoxon matched pairs signed ranks test
Probability values
Observed and critical values
Inferential Statistics

2. Can you recall each of these symbols from GCSE Maths?
Starter
Can you recall each of these symbols from GCSE Maths? = ≤
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> ≥

3. Learning Objective:
To develop an understanding of the process of statistical testing.
Success Criteria
Recall basic mathematical symbols.
Conduct two inferential statistical tests (Mann-Whitney U and Wilcoxon).
Challenge Look for critical values in a table.

4. Inferential Tests
Inferential tests are very important to psychologists.
They allow the psychologists to test whether a correlation or a difference between a set of results is significant.
In the exam you will not have to calculate the inferential tests. But you may be asked to justify why an inferential test is used, whether the result is significant, to sketch a graph of results, or identify co-variables, (to name just a few).
In today’s lesson we will look at the first two tests:
Mann-Whitney U
Wilcoxon matched pairs signed ranks test

5. EXAM TIP!
You will need to know how to choose inferential tests – but don’t panic…you do not need to actually work out the statistics!
The tests you will be soon be familiar with are:
Spearman’s rho
Chi-square (χ2)
Mann-Whitney U
Wilcoxon T
Sign test

6. Levels of Measurement
Before you can work out the significance of a set of data, let’s recap the types of data you have.
Over the next few slides you will look back at levels of measurement that you learnt in AS.
Can you recall the images you drew for each data type?

7. Levels of Measurement
Nominal data in categories, e.g grouping people in class into ‘short’ and ‘tall’, or ‘boys’ and ‘girls’.
Ordinal data that is ordered, e.g lining people up in height order.
Interval data measured in equal intervals, e.g. measuring someone’s height or weight.

8. Probability = number of particular outcomes
Scenario: You roll a dice...what is the chance of getting a 3?
1/6
This is basic probability.
Psychological research relies on probability too.
Probability is expressed as a number between 0 and 1 (where 0 means something definitely will not happen, and 1 means that is definitely will).
To calculate probability a psychologist uses this equation:
Probability = number of particular outcomes
number of possible outcomes
For example, if you were to flip a coin, what is the probability that it will be a head?
Sometimes you can convert probability to a percentage, what is the answer above as a %?

9. Statistical Significance
When we carry out a statistical test we want to know how accurately our sample reflects the general population (i.e. the extent to which we can generalise from our sample to the target population).
In order to carry out an inferential test you need a null hypothesis (H0) and an alternative (otherwise known as experimental) hypothesis (H1).
Inferential statistics allow psychologists to look at patterns of results to see if they have arisen by chance. However, if the results could not have arisen by chance then the pattern is said to be significant.

10. Statistical Significance
But what is a small enough probability? This is known as a level of significance.
In psychology we generally accept a 0.05 or 5% level of significance (written as p0.05 level of significance; 1 in 20 probability that the results are due to chance factors).
This means that anything less than 0.05 is significant and it is unlikely that the null hypothesis is true. Anything above 0.05 is not significant and we reject the null hypothesis.
If the researcher wants to be more certain about their results then they use a more stringent probability, such as p0.01 (1 in 100) or p0.001 (1 in 1000). Whatever level is chosen, the p is the significance level.

11. Mann-Whitney U

12. Mann-Whitney U
This test is used to predict a difference between two sets of data.
The two sets of data are from separate groups or participants (independent groups).
The data can be ordinal or interval.
Think of some data that could be tested with Mann-Whitney:
Testing two groups of participants to see if it is better to revise with or without music.

13. Mann-Whitney U
Step 1. Write a null hypothesis and an alternative hypothesis.
Step 2. Record the data in a table and allocate points.
Step 3. Find the observed value of U.
Step 4. Find the critical value of U.
Step 5. State the conclusion.

14. Mann-Whitney U
Step 1.
Alternative hypothesis = students can revise more effectively in a quiet room, and get a higher score on a test, than those participants listening to an iPod.
Null hypothesis = there will be no difference in the test scores of participants after revising in a quiet room or while listening to an iPod.

15. Test score with no music (?/10) Test score with iPod (?/10)
Mann-Whitney U
To allocate points you look at each score one at a time. Compare this one score with all other scores in the other group. Give one point for every score higher than this score. And give half a point to every equal score.
Step 2. Record the data in a table and allocate points.
Test score with no music (?/10)
Points
Test score with iPod (?/10) 7 5 8 6 3 9 10 2
N1= 9
N2 = 9
1.5
0.5 5
15.5 8 7 9
5.5 3
65.5
Data has been recorded in this table. Test marks were scored out of 10.

16. PAUSE!

17. Observed and critical values
The purpose of applying the test is to measure the observed value against the critical value to see if the null hypothesis can be accepted or rejected.
The observed value is based on the observations you have made.
The critical value is the value that needs to be reached in order for the null hypothesis to be rejected.
The statistical test, along with the level of significance, allows a researcher to estimate the extent to which results could have occurred by chance. The researcher will look at the table of critical values to see if the null hypothesis is to be rejected.
Each inferential test has its own table of critical values.
To find the appropriate critical value you need:
The degrees of freedom (df) which in most cases is the number of participants (N). In cases where an independent groups design is used there are two values for N, one for each group (N1 and N2)
One-tailed (directional hypothesis) or two-tailed (non-directional hypothesis)
Significance level (normally p0.05)

18. Mann-Whitney U Step 3. Find the observed value of U.
U is the lowest number of points
U = 15.5

19. R
Some tests are significant if the observed value is greater than the critical value, while some tests are the reverse.
You will also find this under each table.
But try to remember this:
If there is an R then the observed value should be gReateR than the critical value (e.g. Spearman’s and chi-square). If there is no R (e.g. Mann-Whitney and Wilcoxon) then the observed value should be less than the critical value.

20. Is the result significant?
Mann-Whitney U
Step 4. Find the critical value of U.
Is the result significant?

21. Mann-Whitney U Step 5. State the conclusion.
The observed value (15.5) is less than the critical value (21) so we must reject the null hypothesis (at p≤0.05) and conclude there is a difference in the test scores of participants after revising in a quiet room compared to listening to an iPod.

22. Wilcoxon T

23. Wilcoxon T
This test is used to predict a difference between two sets of data.
The two sets of data are from one person (or a matched pair) so the data is related.
The data can be ordinal or interval.
Think of some data that could be tested with Wilcoxon:
Testing one group of participants to see if it is better to revise with or without music.

24. Wilcoxon T
Step 1. Write a null hypothesis and an alternative hypothesis.
Step 2. Record the data, calculate the difference between scores and rank.
Step 3. Find the observed value of T.
Step 4. Find the critical value of T.
Step 5. State the conclusion.

25. Wilcoxon T
Step 1. Write a null hypothesis and an alternative hypothesis.
Alternative hypothesis = students can revise more effectively in a quiet room, and get a higher score on a test, than when they listen to an iPod when revising.
Null hypothesis = there will be no difference in the test scores of participants after revising in a quiet room or while listening to an iPod.

26. Wilcoxon T
You rank from low to high. You should ignore the signs. If there are two or more of the same score you should work out the mean of the ranks that would be given. If there is a difference of 0 omit this from ranking and reduce N accordingly.
Step 2. Record the data, calculate the difference between scores and rank.
Ppt
With iPod
No iPod
Difference
Rank 1 5 6 2 4 3 7 8
Data has been recorded in this table. Test marks were scored out of 10. -1
Omit 1
3.5
To calculate the difference subtract the third column from the second column.

27. Wilcoxon T Step 3. Find the observed value of T.
T is the sum of the ranks of the less frequent sign.
Here the less frequent sign is +, so T = 3.5
Ppt
With iPod
No iPod
Difference
Rank 1 5 6 -1
3.5 2 4 3
omit 7 8

28. Is the result significant?
Wilcoxon T
Step 4. Find the critical value of T.
N = 6 (one score omitted)
Is the hypothesis directional or non-
directional?
Step 5. State the conclusion.
The observed value (3.5) is greater than the critical value (2) so we must accept the null hypothesis (at p≤0.05) and conclude there is no difference in the test scores of participants after revising in a quiet room or while listening to an iPod.
Is the result significant?

29. Final Task: is this significant?
Participant number
New Mathematics Help Scheme
No Scheme implemented 1 11 14 2 12 13 3 9 4 8 16 5 15 6 7 10 17 19 18 20
Mann Whitney U
U = 8
Critical value table:
p0.05
one-tailed
N1 =10
N2=9
Critical value = ??
So is this significant?
O is less than C
So yes = significant
Reject null

30. Final Task: is it significant?
Participant number
Neutral Words
Emotionally threatening words 1 14 15 2 18 16 3 19 4 20 5 6 10 7 8 9
T = 7
p0.05
One-tailed
What is the value of N?
Is it significant?
Critical value = ??
O is less than C
So yes = significant
Reject null

31. Learning Objective:
To develop an understanding of the process of statistical testing.
Success Criteria
Recall basic mathematical symbols.
Conduct two inferential statistical tests (Mann-Whitney U and Wilcoxon).
Challenge Look for critical values in a table.