What is correlation?

Correlation refers to the relationship between two variables.

The correlation can be:

1. Positive - This is when both variables increase or decrease together.
2. Negative - This is when one variable increases as the other decreases.
3. Non-existent - This is when there is no clear relationship, indicated by a correlation coefficient close to 0.

A correlation coefficient of 1 indicates a perfect, positive linear correlation, where all points lie on a straight line.

Calculating Pearson's correlation coefficient

Pearson's correlation coefficient (r) assesses the strength and direction of a linear relationship between two continuous variables that are normally distributed, x and y.

The formula is (this will be provided in an exam):

$r=\frac{\sum (x-\bar{x})(y-\bar{y})}{\sqrt{\sum {(x-\bar{x})}^2}\sqrt{\sum {(y-\bar{y})}^2}}$

Where:

• $\sum$ represents the sum of.
• $\bar{x}$ is the mean (average) of all x values.
• $\bar{y}$ is the mean (average) of all y values.

The process for calculating Pearson's correlation is:

1. Calculate the mean of each variable.
2. Subtract the mean from each observed value ($x-\bar{x}$ and $y-\bar{y}$).
3. Square the values calculated in step 2 (${(x-\bar{x})}^2$ and ${(y-\bar{y})}^2$).
4. Multiply together the values calculated in step 2 ($(x-\bar{x})(y-\bar{y})$).
5. Sum the values calculated in steps 3 and 4.

It may be helpful to carry out steps 2 to 5 in a table like shown below:

Then, substitute these values into the equation to find the Pearson’s correlation coefficient (r).

Finally, compare the calculated r value to a critical value at a 5% significance level. For this test, the $\text{degrees of freedom = }n-2$.

This determines the significance of the correlation:

• An r value near +1 indicates a strong positive correlation.
• An r value near -1 suggests a strong negative correlation.
• An r value around 0 implies there is no correlation.

Worked example - Calculating Pearson's correlation coefficient

Is there a correlation between height and weight based on the following data for 5 individuals?

Step 1: Equation

$r=\frac{\sum (x-\bar{x})(y-\bar{y})}{\sqrt{\sum {(x-\bar{x})}^2}\sqrt{\sum {(y-\bar{y})}^2}}$

Step 2: Calculate the means $\bar{x}$ and $\bar{y}$

$\bar{x}=\frac{\text{Sum of heights}}{n}=\frac{175+168+180+165+170}{5}=171.6\text{ cm}$

$\bar{y}=\frac{\text{Sum of weights}}{n}=\frac{65+60+80+55+68}{5}=65.6\text{ kg}$

Step 3: Construct a table to calculate the differences from the mean and their products

Step 4: Substitution and correct evaluation

$r=\frac{\sum (x-\bar{x})(y-\bar{y})}{\sqrt{\sum {(x-\bar{x})}^2}\sqrt{\sum {(y-\bar{y})}^2}}$

$r=\frac{205.2}{(\sqrt{141.2})(\sqrt{357.2})}$

$r=\frac{205.2}{224.58}$

$r=0.914\text{ (rounded to three decimal places)}$

Step 5: Determine significance and interpret result

for a sample size of 5, $\text{degrees of freedom = }n-2=3$, the critical value at the 5% significance level is 0.878

as the $r$ value of 0.914 exceeds this critical value, the correlation between height and weight is statistically significant, and is unlikely to be due to chance

this indicates a strong positive correlation between height and weight among the individuals in this sample, suggesting that as height increases, weight also increases