1. Correlation
S1 Maths with Liz

2. AIMS By the end of the lesson, you should be able to…
Draw scatter graphs & understand correlation
Understand causality
Calculate & use the PMCC
Understand the limitations of using the PMCC

3. Correlation Bivariate data is data that has two variables.
Scatter graphs are often used to show a relationship or correlation between two random variables. The stronger the correlation, the more closely related the two variables are likely to be.

4. Key terms

5. Product Moment Correlation Coefficient (PMCC)
The product–moment correlation coefficient gives a numerical measure of the strength of the linear association between two variables.
The PMCC is represented by the letter r and must lie between -1 and 1.
r = indicates perfect linear positive correlation;
r = indicates that there is absolutely no linear correlation between the variables.
r = indicates perfect linear negative correlation;

6. How to describe your value for r…
Some suggested descriptions of correlation are:
When r is between -0.2 and 0.2, the correlation is very weak.
When r is between 0.2 and 0.7, or between -0.2 and -0.7, the correlation is moderate.
When r is between 0.7 and 0.9, or between -0.7 and -0.9, the correlation is strong.
When r is between 0.9 or below -0.9, the correlation is very strong.

7. How to calculate the PMCC.
The product–moment correlation coefficient for n pairs of
observations is obtained using the formula:
ALL of these formulae are given on pg. 13!
where:
Usually, the second version of each formula is used because we can find these values using our calculators!

8. Example 1
A teacher measures & records the heights (in cm) and hand spans (in inches) of 10 boys in year 8.
(a) Plot the data on a scatter diagram.
(b) Calculate the product moment correlation coefficient.
(c) Describe the relationship that exists between the two variables.

9. Example 1 (a) Plot the data on a scatter diagram.
A teacher measures & records the heights (in cm) and hand spans (in inches) of 10 boys in year 8.

10. Let’s start by finding each piece separately with the use of our GDC.
Example 1
(b) Calculate the product moment correlation coefficient.
A teacher measures & records the heights (in cm) and hand spans (in inches) of 10 boys in year 8.
Open your AQA formulae booklets to pg. 13.
Let’s start by finding each piece separately with the use of our GDC.
In your GDC, press
MENU
STAT
enter data in lists 1 and 2
CALC (F2)
check settings to make sure your 2VAR Freq: 1, then EXE
2 VAR (F2)

11. Let’s start by finding each piece separately with the use of our GDC.
Example 1
(b) Calculate the product moment correlation coefficient.
A teacher measures & records the heights (in cm) and hand spans (in inches) of 10 boys in year 8.
Open your AQA formulae booklets to pg. 13.
Let’s start by finding each piece separately with the use of our GDC.

12. Let’s start by finding each piece separately with the use of our GDC.
Example 1
(b) Calculate the product moment correlation coefficient.
A teacher measures & records the heights (in cm) and hand spans (in inches) of 10 boys in year 8.
Open your AQA formulae booklets to pg. 13.
Let’s start by finding each piece separately with the use of our GDC.

13. Now, bring it all back together using the formula for r.
Example 1
(b) Calculate the product moment correlation coefficient.
A teacher measures & records the heights (in cm) and hand spans (in inches) of 10 boys in year 8.
Now, bring it all back together using the formula for r.

14. Example 1
(c) Describe the relationship that exists between the two variables.
A teacher measures & records the heights (in cm) and hand spans (in inches) of 10 boys in year 8.
There is a strong positive correlation between hand length and height, meaning that the height increases as the hand length increases.

15. Finding r directly in the GDC…
You can calculate r directly in your calculator. HOWEVER, sometimes the exam will give you summarised data where you must use the formulae to work out the answers.
In your GDC, press
MENU
STAT
enter data in lists 1 and 2
CALC (F2)
REG (F3)
X (F1)
a + bx (F2)
look for the r value

16. Plotting Scatter Graphs in your GDC…
You can plot your scatter graphs in your graphing calculator also!
In your GDC, press
MENU
STAT
enter data in lists 1 and 2
GRPH (F1)
Check the settings to make sure Graph Type: Scatter, then EXE
GPH1 (F1)

17. Limitations of the PMCC
The product–moment correlation coefficient (PMCC) measures the strength of a linear relationship.
However:
Outliers can greatly distort the PMCC;
The PMCC is not a suitable measure of correlation if the relationship is non-linear.
This graph would give a PMCC of around 0, meaning no correlation. BUT there is clearly a pattern (quadratic). It is always best to make a sketch of the data to check if there is another sort of pattern.

18. Linear Scaling – does this affect the pmcc?
Let’s go back to EXAMPLE 1.
A teacher measures & records the heights (in cm) and hand spans (in inches) of 10 boys in year 8.
(a) Calculate the product moment correlation coefficient of the new data.

19. Linear Scaling – does this affect the pmcc?
(b) How did converting the hand length from in. to cm. affect the scatter plot and product moment correlation coefficient of the new data?
The PMCC is still

20. Example 2 – Summarised data January 2013, q.4
Possible responses could include:
identify any outliers
identify any linear or non-linear relationships or patterns
use the scatter plot to estimate the value & sign of r

21. Example 2 – Summarised data January 2013, q.4

22. Example 2 – Summarised data January 2013, q.4
There is a moderate positive correlation between the two statistics exam marks.

23. Example 2 – Summarised data January 2013, q.4
ruv should be equal to rxy because linear scaling has no affect on the value of r (the PMCC).

24. Independent Study
REVISE REVISE REVISE!!!!!!! Go through all questions related to correlation in your stats 1 past paper booklet.
HEAVILY RECOMMENDED:
mymaths – scatter graphs
mymaths – product moment correlation coefficient (just do the PMCC questions. Don’t worry about filling in the tables…they are there for folks who don’t have a graphing calculator).
Stats Textbook: Pg. 149, Ex. 6A
Stats Textbook: Pg. 155, Ex. 6B
MEI WS – “AQA Stats 1 Correlation and regression, Section 1: Correlation”