1. Correlation

2. Correlation This Chapter is on Correlation
We will look at patterns in data on a scatter graph
We will be looking at how to calculate the variance and co-variance of variables
We will see how to numerically measure the strength of correlation between two variables

3. Correlation Positive Negative None 6A Scatter Graphs
Scatter Graphs are a way of representing 2 sets of data. It is then possible to see whether they are related.
Positive Correlation
 As one variable increases, so does the other
Negative Correlation
 As one variable increases, the other decreases
No Correlation
 There seems to be no pattern linking the two variables
Positive
Negative
None 6A

4. Correlation
Scatter Graphs
In the study of a city, the population density, in people/hectare, and the distance from the city centre, in km, was investigated by choosing sample areas. The results are as follows:
Plot a scatter graph and describe the correlation. Interpret what the correlation means. 50 40
Area A B C D E
Distance
0.6
3.8
2.4
3.0
2.0
Pop. Density 50 22 14 20 33
Pop. Density (people/hectare) 30 20 10
Area F G H I J
Distance
1.5
1.8
3.4
4.0
0.9
Pop. Density 47 25 8 16 38 1 2 3 4
Distance from centre (km)
The correlation is negative, which means that as we get further from the city centre, the population density decreases.

5. Teachings for Exercise 6B and 6C

6. Correlation Variability of Bivariate Data We learnt in chapter 3 that:
In Correlation:
Similarly for y:
And you can also calculate the Co-variance of both variables
(Although remember that this formula changed to make it easier to use)
‘How x varies’
‘How y varies’
‘How x and y vary together’
6B/C

7. Correlation Variability of Bivariate Data
Like in chapter 3, we can use a formula which will make calculations easier
BUT:
6B/C

8. Correlation Variability of Bivariate Data
Multiply both sides by ‘n’
The easier formula for variance from chapter 3
For the second fraction, square the top and bottom separately
Multiplying both fractions by ‘n’ will cancel a ‘divide by n’ from each of them
6B/C

9. Correlation Variability of Bivariate Data
These are the formulae for Sxx, Syy and Sxy. You are given these in the formula booklet. You do not need to know how to derive them (like we just did!)
6B/C

10. Correlation Variability of Bivariate Data 6B/C
Calculate Sxx, Syy and Sxy, based on the following information.
6B/C

11. Correlation Variability of Bivariate Data 6B/C
The following table shows babies heads’ circumferences (cm) and the gestation period (weeks) for 6 new born babies. Calculate Sxx, Syy and Sxy.
We need
Baby A B C D E F
Head size (x) 31 33 30 31 35 30
Gestation period (y) 36 37 38 38 40 40 x2
961
1089
900
961
1225
900 y2
1296
1369
1444
1444
1600
1600 xy
1116
1221
1140
1178
1400
1200
6B/C

12. Correlation Variability of Bivariate Data 6B/C
The following table shows babies heads’ circumferences (cm) and the gestation period (weeks) for 6 new born babies. Calculate Sxx, Syy and Sxy.
We need
6B/C

13. Correlation Product Moment Correlation Coefficient
We can test the correlation of data by calculating the Product Moment Correlation Coefficient. This uses Sxx, Syy and Sxy.
The value of this number tells you what the correlation is and how strong it is.
The closer to 1, the stronger the positive correlation. The same applies for -1 and negative correlation. A value close to 0 implies no linear correlation.
Negative Correlation
No Linear Correlation
Positive Correlation -1 1
6B/C

15. There is positive correlation, as x increases, y does as well.
Product Moment Correlation Coefficient
Given the following data, calculate the Product Moment Correlation Coefficient.
There is positive correlation, as x increases, y does as well.
6B/C

16. Correlation Limitations of the Product Moment Correlation Coefficient
Sometimes it may indicate Correlation between unrelated variables
 Cars on a particular street have increased, as have the sales of DVDs in town
 The PMCC would indicate positive correlation where the two are most likely not linked
 The speed of computers has increased, as has life expectancy amongst people
 These are not directly linked, but are both due to scientific developments
6B/C

17. Correlation Using Coding with the PMCC
Calculating the PMCC from this table. x
102
103
102
103
104
103 y
320
335
345
355
360
380 x2
10404
10609
10404
10609
10816
10609 y2
102400
112225
119025
126025
129600
144400 xy
32640
34505
35190
36565
37440
39140 6D

18. Correlation Using Coding with the PMCC
Calculating the PMCC from this table. x
102
103
102
103
104
103 y
320
335
345
355
360
380 x2
10404
10609
10404
10609
10816
10609 y2
102400
112225
119025
126025
129600
144400 xy
32640
34505
35190
36565
37440
39140 6D

19. Correlation Using Coding with the PMCC
Calculating the PMCC from this table, using coding. x
102
103
102
103
104
103 y
320
335
345
355
360
380 p 2 3 2 3 4 3 q 4 7 9 11 12 16 p2 4 9 4 9 16 9 q2 16 49 81
121
144
256 pq 8 21 18 33 48 48 6D

20. So coding will not affect the PMCC!
Correlation
Using Coding with the PMCC
Calculating the PMCC from this table. x
102
103
102
103
104
103 y
320
335
345
355
360
380 p 2 3 2 3 4 3 q 4 7 9 11 12 16 p2 4 9 4 9 16 9 q2 16 49 81
121
144
256 pq 8 21 18 33 48 48
So coding will not affect the PMCC! 6D

21. Summary We have looked at plotting scatter graphs
We have looked at calculating measures of variance, Sxx, Syy and Sxy
We have also seen types of correlation and how to recognise them on a graph
We have calculated the Product Moment Correlation Coefficient, and interpreted it. It is a numerical measure of correlation.

22. Spearman’s Rank Correlation
CEV

23. Correlation
So far, we have considered the relationship of bivariate data which can be plotted directly, e.g. Results of a set of pupils in two tests, price vs age of cars etc. Sometimes we do not have data which suits this ideal, but we might still want to compare two sets of related data.

24. Example
Consider a manufacturer experimenting with different flavours of a drink. Two tasters put eight flavours, labelled A-H, in order of preference (starting with their favourite). The results are given in the table below:
Taster 1 D C G B A E F H
Taster 2

25. Example
Taster 1 D C G B A E F H
Taster 2
Unsurprisingly the two tasters do not agree exactly. However, there is clearly some consensus between them on the more pleasant flavours (C and D), and the least pleasant (e.g. F). It would be useful to measure how well the tasters agree. This is where we use the idea of “ranking” the flavours for each taster, according to where it appears in their list.

26. Example Taster 1 D C G B A E F H Taster 2 Flavour Rank for Taster 1, x
Rank for Taster 2, y A B C D E F G H

27. Example Taster 1 D C G B A E F H Taster 2 Flavour Rank for Taster 1, x
Rank for Taster 2, y A 5 B 4 C 2 D 1 E 6 F 7 G 3 H 8

28. Example Taster 1 D C G B A E F H Taster 2 Flavour Rank for Taster 1, x
Rank for Taster 2, y A 5 7 B 4 3 C 2 1 D E 6 F 8 G H

29. Example
If we plotted the scatter diagram for this information, we would see some positive correlation. It would seem reasonable to measure the degree of agreement by calculating the PMCC for the two sets of ranks. However, to emphasise that x and y are ranks rather than continuous variables, the coefficient is given a new symbol, rs. We can calculate rs using the same formula as for the PMCC. However, there is a simpler formula...

30. Example
Flavour
Rank for Taster 1 (x)
Rank for Taster 2 (y) di
di² A 5 7 B 4 3 C 2 1 D E 6 F 8 G H
Spearman’s Rank Correlation Coefficient is: where di = xi – yi for the ith item

31. Example
Flavour
Rank for Taster 1 (x)
Rank for Taster 2 (y) di
di² A 5 7 -2 B 4 3 1 C 2 D -1 E 6 F 8 G H
Spearman’s Rank Correlation Coefficient is: where di = xi – yi for the ith item

32. Note: Σdi = 0 always!
Example
Flavour
Rank for Taster 1 (x)
Rank for Taster 2 (y) di
di² A 5 7 -2 4 B 3 1 C 2 D -1 E 6 F 8 G H 9
Spearman’s Rank Correlation Coefficient is: where di = xi – yi for the ith item

33. Example Note: Σdi = 0 always! Flavour Rank for Taster 1 (x)
Rank for Taster 2 (y) di
di² A 5 7 -2 4 B 3 1 C 2 D -1 E 6 F 8 G H 9

34. Interpreting rs
When interpreting Spearman’s Rank Correlation Coefficient, it is important to remember that this value is the PMCC of the ranks. If the rank orders for the two tasters had been identical, then the points on the scatter diagram would fall exactly on the line y = x, and rs would equal 1. If one rank order was the exact reverse of the other, similarly, then all the points would fall on y = -x and rs = -1. If there was little agreement between the two rank orders, rs would be close to 0.

35. Comparing PMCC and Spearman’s
For some sets of data it would be reasonable to calculate both r and rs, for example the table below. r = rs = 0.745
Student A B C D E F G H I J
Pure 42 21 25 32 34 27 23 40 20 16
Stats 41 36 29 35 24 22 47 30
Student A B C D E F G H I J
Pure Rank 1 8 6 4 3 5 7 2 9 10
Stats Rank

36. Comparing PMCC and Spearman’s
These two values are not the same, because they are measuring different things. r is measuring how close the points on the scatter diagram are to a straight line, which is the strength of the linear relationship between x and y, whereas rs is measuring the tendency for y to increase as x increases, not necessarily in a linear way. NOTE: For negative correlation, rs measures the tendency for y to decrease as x increases.