Correlation Math

Correlation describes if a relationship exists between two variables, how strong this relationship is, and whether a change in one variable induces a positive or negative change in another. For example, we would expect a strong positive correlation between the amount of sunlight and the growth of flowers.

Regression

Regression is the numerical relation between two variables (we now need an independent and dependent variable). We normally look at a linear regression; if our two variables are X and Y, we seek a relationship of the form Y = mX + c. This means that this will look like a straight line on a scatter graph. We often refer to a regression line as a line of best fit.

Classification of correlation

When we describe correlation, we use two measures: strength and parity.

Strength

We can describe the strength of a correlation between two variables.

If there is a strong correlation, we have a strong relationship between the two variables (i.e. there is a strong indication that there is a dependency between variables). If we were to draw a regression line (line of best fit), all the data points would be very close to the line.

If there is a weak correlation, there is a weaker relationship between the two variables (this means there is less indication that there is a dependency between variables). If there were a regression line, the points would be more spread away from the line.

Parity

Parity is a special way of saying whether the correlation is positive or negative. If there is a positive correlation, this means that as one variable increases, the other variable increases. On a regression line, this means that the gradient is positive. If the correlation is negative, then as one variable increases, the other variable decreases. On a regression line, this means that the gradient is negative.

If the correlation is weak enough (to the extent that there is no real correlation), then we call this a zero correlation. This would mean we couldn't draw a regression line.

We often use a correlation coefficient to describe correlation, this varies from -1 to 1. The strongest positive correlation is represented by 1, the strongest negative correlation is represented by -1, and a zero correlation by zero. It is a sliding scale between these values.

Below are examples of different correlations:

Zero correlation:

Zero correlation scatter graph, Tom Maloy, StudySmarter Originals

An example of something with no correlation would be the shoe size of children in the UK and the number of sunflowers in France. These are unrelated and thus neither influence one another, and so, no correlation.

A strong positive correlation:

Strong positive correlation scatter graph, Tom Maloy, StudySmarter Originals

An example of a strong positive correlation would be hours spent in direct sunlight and the severity of sunburn. As one of these variables increases, then so does the other.

A strong negative correlation:

Strong negative correlation scatter graph, Tom Maloy, Study Smart Originals

An example of a strong negative correlation would be the speed of a train versus the time to complete a journey from one station to the next. As speed increases, time to destination decreases. As this is a physical relation, this will be strongly related.

A weak positive correlation:

Weak positive correlation scatter graph, Tom Maloy, StudySmarter Originals

An example of a weak positive correlation may be the prescription of a particular drug and increased kidney failure. The drug is unlikely to cause everyone to have kidney failure, but it will increase the likelihood for some people.

A weak negative correlation:

Weak negative correlation scatter graph, Tom Maloy, StudySmarter Originals

An example of a weak negative correlation would be lessons missed and end grades. In general, the number of lessons missed would decrease grades; however, this will differ from student to student to end up with a weaker relationship.

A real-life example of correlation and regression

There are many real-life examples of correlation and regression. In the COVID-19 pandemic, many governments will have tracked COVID deaths/hospitalisations/case numbers against various metrics, such as age, weight or location, to try and find a pattern and see where help would be needed most. There would first be a look at correlation to see if there were any apparent patterns, and then they would use regression to try and predict any missing data values.

Example question involving correlation and regression

Q: Below is a table showing the data for students in a class, where height and arm length was measured.

1. Draw a scatter graph to represent the data

2. Describe the correlation between the data sets

3. Use a regression line to predict the arm length of person who is 165cm tall.

A:

1. Let us plot this with height on the x-axis and arm length on the y-axis. This looks like

   

2. Looking at the scatter graph, as height increases, so does arm length, meaning there is a positive correlation. If we were to draw a regression line on the graph, the points would be close to this line, meaning there is a strong positive correlation between arm length and height.

3. Our first step here is to draw on our regression line. This should look something like this:

From this point, we draw a line of x = 165 and then see where this intersects with the regression line. Then the y value of this intersection point will be our estimate for arm length.

This means our estimate for arm length is 95cm.