Product moment correlation coefficient
Posted in Maths, Statistics 2The product moment correlation coefficient (sometimes referred to as Person's product moment correlation coefficient) is a value between -1 and +1 is a measurement of correlation. A value of +1 is perfect positive correlation, a value of -1 is perfect negative correlation and a value of 0 is no correlation.
Calculating the PMCC
The product moment correlation coefficient is denoted by the letter r
Method #1
The first method involves using the following formula:
Where:
Example
Caclulate the product moment correlation coeffecient of the following set of bivarirate data:
| 5 | 9 | 15 | 18 |
| 103 | 120 | 143 | 160 |
Start by drawing up a table of the values you need to find, together with the x and y values youhave been given
| xi | yi | ||||
|---|---|---|---|---|---|
| 5 | 103 | ||||
| 9 | 120 | ||||
| 15 | 143 | ||||
| 18 | 160 | ||||
| Sum: | 47 | 526 |
Before you can calculate the other values for the other columns you need to work out the means for the x and y values of data:
Now you can return to fill in the other columns:
| xi | yi | ||||
|---|---|---|---|---|---|
| 5 | 103 | 45.5625 | 812.25 | 192.375 | |
| 9 | 120 | 7.5625 | 132.25 | 31.625 | |
| 15 | 143 | 10.5625 | 132.25 | 37.375 | |
| 18 | 160 | 39.0625 | 812.25 | 178.125 | |
| Sum: | 47 | 526 | 102.75 | 1889 | 439.5 |
Now you have the values you an input them into the formula:
Method #2
The second method involes the following formula:
Where n is the number of pairs of data.
Example
Caclulate the product moment correlation coeffecient of the following set of bivarirate data:
| 5 | 9 | 15 | 18 |
| 103 | 120 | 143 | 160 |
As before draw up a table of the values you have and those you need:
| xi | yi | ||||
|---|---|---|---|---|---|
| 5 | 103 | 25 | 10609 | 515 | |
| 9 | 120 | 81 | 14400 | 1080 | |
| 15 | 143 | 225 | 20449 | 2145 | |
| 18 | 160 | 324 | 25600 | 2880 | |
| Sum: | 47 | 526 | 655 | 71058 | 6620 |
Next calculate the means of both the x and y sets of data:
Now just plug all the values into the formula:
Performing hypothesis testing
To perform a hypothesis test using the product moment coefficient look up the critical value in your table of values for the number of pairs of data and compare this to your value for r. If the the value for r is greater than the critical value you can reject the null hypothesis and accept the alternative hypothesis.