2.05 Correlation and r value

The correlation coefficient is a value that describes the strength and direction of a linear relationship between two variables.

The value of the correlation coefficient varies from -1 to +1, where -1 describes perfect negative correlation and +1 describes perfect positive correlation. Any other type of correlation corresponds to a value between these two extremes, with 0 describing no correlation.

We further divide up this range of values to indicate other strengths of correlation, using descriptions of weak, moderate and strong (for both positive and negative correlations).

If the correlation coefficient takes a value between 0 and 1, then it describes a positive correlation:

• A value of r close to +1 indicates a strong positive linear correlation

• A value of r that is positive but closer to 0 indicates a weak positive linear correlation

If the correlation coefficient takes a value between -1 and 0, then it describes a negative correlation:

• A value of r close to -1 indicates a strong negative linear correlation

• A value of r that is negative but closer to 0 indicates a weak negative linear correlation

If the correlation coefficient is 0, or very close to 0, it indicates that there is no linear correlation between the variables. This may be because the variables are unrelated, or it might be that they have a non-linear relation instead.

Even when two variables have a strong relationship and r is close to 1 or -1, we cannot say that one variable causes change in the other variable. If asked "does change in the explanatory variable cause change in response variable?" we always write "No - correlation is not causation".

For example, it has been shown that there is a strong, positive, linear relationship between sunglasses sold and ice-cream cone sales. But we cannot say that sunglasses sales cause ice-cream cone sales. There is a third variable at work here; increase in temperature causes both variables to also increase. Increase in temperature is called a confounding variable.

Coincidence is also a plausible reason an association occurs. It's possible to find variables with unlikely strong correlations. For example, per capita consumption of cheese and deaths from being strangled by a bedsheet have been shown to have a strong correlation. But we cannot say that one causes the other. It is a coincidence. A website containing graphs of variables with spurious correlations can be found here.

Examples

The calculation required to determine r is very tricky to do by hand, but can be easily done using technology. To do so, we enter the raw data into two separate lists, then perform a linear regression analysis. This will calculate a number of values, though the only one we are interested in right now is the r value.

Examples

Example 3

For the graph depicted, choose the correlation coefficient that best represents it.

5

10

15

20

x

-5

5

10

15

20

y

A

1

B

0

C

-1

D

-0.64

Worked Solution

Create a strategy

Check if you could fit a straight line to the points plotted.

Apply the idea

Based on the graph, a straight line cannot be fitted on the plotted points. This means that the two variables in data set are not linearly correlated. So, we can say that the correlation coefficient is 0, and the correct answer is Option C.

Watch question walkthrough

Example 4

Given the following data:

x	1	4	7	10	13	16	19
y	4	4.25	4.55	4.4	4.45	4.75	4.2

a

Calculate the correlation coefficient and give your answer to two decimal places.

Worked Solution

Create a strategy

Use technology to find the correlation coefficient.

Apply the idea

Using the Statistics mode in your calculator, enter each x-value along with its y-value into a data table on your calculator then find the linear regression.

Look for the correlation coefficient (r):r=0.47

Watch question walkthrough

b

Choose the best description of this correlation.

A

Moderate negative

B

Strong positive

C

Weak negative

D

Moderate positive

E

Strong negative

F

Weak positive

Worked Solution

Create a strategy

Use the figure below to identify the best description of the correlation:

Apply the idea

The value of r=0.47 is positive, and between 0 and 0.5. So there is a weak positive correlation between the variables.

The correct answer is F.

Watch question walkthrough