# Correlation

When two sets of data are strongly linked together we say they have a High Correlation.

The word Correlation is made of Co- (meaning "together"), and Relation

• Correlation is Positive when the values increase together, and
• Correlation is Negative when one value decreases as the other increases

Here we look at linear correlations (correlations that follow a line).

Correlation can have a value:

• 1 is a perfect positive correlation
• 0 is no correlation (the values don't seem linked at all)
• -1 is a perfect negative correlation

The value shows how good the correlation is (not how steep the line is), and if it is positive or negative.

## Example: Ice Cream Sales

The local ice cream shop keeps track of how much ice cream they sell versus the temperature on that day, here are their figures for the last 12 days:

Temperature °C Ice Cream Sales Ice Cream Sales vs Temperature 14.2° \$215 16.4° \$325 11.9° \$185 15.2° \$332 18.5° \$406 22.1° \$522 19.4° \$412 25.1° \$614 23.4° \$544 18.1° \$421 22.6° \$445 17.2° \$408

And here is the same data as a Scatter Plot:

We can easily see that warmer weather and higher sales go together. The relationship is good but not perfect.

In fact the correlation is 0.9575 ... see at the end how I calculated it.

## Correlation Is Not Good at Curves

The correlation calculation only works well for relationships that follow a straight line.

### Our Ice Cream Example: there has been a heat wave!

It gets so hot that people aren't going near the shop, and sales start dropping.

Here is the latest graph:

The correlation value is now 0: "No Correlation" ... !

The calculated correlation value is 0 (I worked it out), which means "no correlation".

But we can see the data does have a correlation: it follows a nice curve that reaches a peak around 25° C.

But the linear correlation calculation is not "smart" enough to see this.

Moral of the story: make a Scatter Plot, and look at it!
You may see a correlation that the calculation does not.

## Correlation Is Not Causation

"Correlation Is Not Causation" ... which says that a correlation does not mean that one thing causes the other (there could be other reasons the data has a good correlation).

### Example: Sunglasses vs Ice Cream

Our Ice Cream shop finds how many sunglasses were sold by a big store for each day and compares them to their ice cream sales:

The correlation between Sunglasses and Ice Cream sales is high

Does this mean that sunglasses make people want ice cream?

### Example: A Real Case!

A few years ago a survey of employees found a strong positive correlation between "Studying an external course" and Sick Days.

Does this mean:

• Studying makes them sick?
• Sick people study a lot?
• Or did they lie about being sick to study more?

Without further research we can't be sure why.

## How To Calculate

How did I calculate the value 0.9575 at the top?

I used "Pearson's Correlation". There is software that can calculate it, such as the CORREL() function in Excel or LibreOffice Calc ...

### ... but here is how to calculate it yourself:

Let us call the two sets of data "x" and "y" (in our case Temperature is x and Ice Cream Sales is y):

• Step 1: Find the mean of x, and the mean of y
• Step 2: Subtract the mean of x from every x value (call them "a"), do the same for y (call them "b")
• Step 3: Calculate: a × b, a2 and b2 for every value
• Step 4: Sum up a × b, sum up a2 and sum up b2
• Step 5: Divide the sum of a × b by the square root of [(sum of a2) × (sum of b2)]

Here is how I calculated the first Ice Cream example (values rounded to 1 or 0 decimal places):

As a formula it is:

Where:

• Σ is Sigma, the symbol for "sum up"
• is each x-value minus the mean of x (called "a" above)
• is each y-value minus the mean of y (called "b" above)

You probably won't have to calculate it like that, but at least you know it is not "magic", but simply a routine set of calculations.

### Note for Programmers

You can calculate it in one pass through the data. Just sum up x, y, x2, y2 and xy (no need for a or b calculations above) then use the formula:

### Other Methods

There are other ways to calculate a correlation coefficient, such as "Spearman's rank correlation coefficient", but I prefer using a spreadsheet like above.