Did this video help you?
Correlation Coefficients (DP IB Maths: AI HL)
Revision Note
PMCC
What is Pearson’s product-moment correlation coefficient?
- Pearson’s product-moment correlation coefficient (PMCC) is a way of giving a numerical value to a linear relationship of bivariate data
- The PMCC of a sample is denoted by the letter
- r can take any value such that
- A positive value of r describes positive correlation
- A negative value of r describes negative correlation
- r = 0 means there is no linear correlation
- r = 1 means perfect positive linear correlation
- r = -1 means perfect negative linear correlation
- The closer to 1 or -1 the stronger the correlation
How do I calculate Pearson’s product-moment correlation coefficient (PMCC)?
- You will be expected to use the statistics mode on your GDC to calculate the PMCC
- The formula can be useful to deepen your understanding
-
-
- is linked to the covariance
- and are linked to the variances
- You do not need to learn this as using your GDC will be expected
-
When does the PMCC suggest there is a linear relationship?
- Critical values of r indicate when the PMCC would suggest there is a linear relationship
- In your exam you will be given critical values where appropriate
- Critical values will depend on the size of the sample
- If the absolute value of the PMCC is bigger than the critical value then this suggests a linear model is appropriate
Did this video help you?
Spearman’s Rank
What is Spearman’s rank correlation coefficient?
- Spearman's rank correlation coefficient is a measure of how well the relationship between two variables can be described using a monotonic function
- Monotonic means the points are either always increasing or always decreasing
- This can be used as a way to measure correlation in linear models
- Though Spearman's Rank correlation coefficient can also be used to assess a non-linear relationship
- Each data is ranked, from biggest to smallest or from smallest to biggest
- For n data values, they are ranked from 1 to n
- It doesn't matter whether variables are ranked from biggest to smallest or smallest to biggest, but they must be ranked in the same order for both variables
- Spearman’s rank of a sample is denoted by
- rs can take any value such that
- A positive value of rs describes a degree of agreement between the rankings
- A negative value of rs describes a degree of disagreement between the rankings
- rs = 0 means the data shows no monotonic behaviour
- rs = 1 means the rankings are in complete agreement: the data is strictly increasing
- An increase in one variable means an increase in the other
- rs = -1 means the rankings are in complete disagreement: the data is strictly decreasing
- An increase in one variable means a decrease in the other
- The closer to 1 or -1 the stronger the correlation of the rankings
How do I calculate Spearman’s rank correlation coefficient (PMCC)?
- Rank each set of data independently
- 1 to n for the x-values
- 1 to n for the y-values
- If some values are equal then give each the average of the ranks they would occupy
- For example: if the 3rd, 4th and 5th highest values are equal then give each the ranking of 4
- For example: if the 3rd, 4th and 5th highest values are equal then give each the ranking of 4
- Calculate the PMCC of the rankings using your GDC
- This value is Spearman's rank correlation coefficient
Did this video help you?
Appropriateness & Limitations
Which correlation coefficient should I use?
- Pearson’s PMCC tests for a linear relationship between two variables
- It will not tell you if the variables have a non-linear relationship
- Such as exponential growth
- Use this if you are interested in a linear relationship
- It will not tell you if the variables have a non-linear relationship
- Spearman’s rank tests for a monotonic relationship (always increasing or always decreasing) between two variables
- It will not tell you what function can be used to model the relationship
- Both linear relationships and exponential relationships can be monotonic
- Use this if you think there is a non-linear monotonic relationship
- It will not tell you what function can be used to model the relationship
How are Pearson’s and Spearman’s correlation coefficients connected?
- If there is linear correlation then the relationship is also monotonic
- However the converse is not true
- It is possible for Spearman’s rank to be 1 (or -1) but for the PMCC to be different
- For example: data that follows an exponential growth model
- as the points are always increasing
- as the points do not lie on a straight line
- For example: data that follows an exponential growth model
Are Pearson’s and Spearman’s correlation coefficients affected by outliers?
- Pearson’s PMCC is affected by outliers
- as it uses the numerical value of each data point
- Spearman’s rank is not usually affected by outliers
- as it only uses the ranks of each data point
Examiner Tip
- You can use your GDC to plot the scatter diagram to help you visualise the data
Worked example
The table below shows the scores of eight students for a maths test and an English test.
Maths |
7 |
18 |
37 |
52 |
61 |
68 |
75 |
82 |
English |
5 |
3 |
9 |
12 |
17 |
41 |
49 |
97 |
a)
Write down the value of Pearson’s product-moment correlation coefficient, .
b)
Find the value of Spearman’s rank correlation coefficient, .
c)
Comment on the values of the two correlation coefficients.
You've read 0 of your 10 free revision notes
Unlock more, it's free!
Did this page help you?