3.2 Correlation

Correlation (noun)

In Statistics, an interdependence of two or more variable quantities such that a change in the value of one is associated with a change in the value or the expectation of the others. (OED, 2025)

If the change involves an increase in both variables, the relationship is described as a positive correlation; This can certainly be said of scatter graph 2 shown again below. An increase in the maximum temperature is associated with an increase in the minimum temperature therefore potentially we could make a prediction for the maximum temperature in Stornoway, using the value of the minimum temperature.

While the maximum temperatures are increasing with increasing minimum temperatures, there isn’t a single line that goes through all the points. Statistics provides methods for determining a single line that best approximates such data, a ‘line of best fit’. This straight line would have a positive gradient (increasing in height from left to right) and should best fit the pattern of the points across the data range. At school level this line tends to be fitted ‘by eye’ but computer graphing programs fit a linear model that minimises the vertical distances of the points from the line.

In school mathematics up to GCSE the lines of best fit used to model bivariate data are usually straight, i.e. they provide a linear model, and do not necessarily go through the origin, (0, 0). However, it is worth noting that in science up to GCSE level, learners may be required to fit curved lines of best fit and that often in science, due to the nature of the variables that are plotted, it is appropriate to join the line of best fit to the origin.

One variable decreasing as the other increases gives rise to a relationship known as a negative correlation; a ‘line of best fit’ could be drawn through the points and this line would have a negative gradient (decreasing in height from left to right).

Here is a scatter graph for some data mentioned in the previous activity. The data points follow a negative correlation, demonstrated by the line of best fit. The data points, however, are quite widely scattered around the line so this could be described as a weak negative correlation.

While a ‘line of best fit’ could mathematically be determined for all four scatter graphs in the last activity (with positive gradients for scatter graphs 2 and 4, and negative gradients for scatter graphs 1 and 3), the weaker correlation in scatter graphs 3 and 4 would make it a less meaningful approximation to the datasets shown. In scatter graphs 3 and 4, the points would be more distant from – above or below – the approximating line, so not quite a line of best fit!

For the line to ‘best fit’ the points, there needs to appear to be a relationship between them. If a relationship is not evident, it is appropriate to say there is no correlation.

Here is scatter graph 1 again with the variable labels and a trend line fitted.

Compared to the scatter graph 2 representing minimum and maximum monthly temperature in Stornoway for the same year, the points in scatter graph 1 are much more scattered although generally follow a rough downward pattern. The amount of scatter around the line of best fit shown in the points tells you about the strength of the relationship between the data variables plotted. You could say scatter graph 1 shows a weak negative correlation whilst scatter graph 2 shows a strong positive correlation. The use of these additional words for strength is somewhat subjective, and you will sometimes see ‘very strong’ or ‘very weak’ used as descriptions. There is a related statistic that can be calculated to help determine the strength of any relationship identified, this is called the correlation coefficient. The correlation coefficient is beyond the scope of this course, but you will find a short explanatory video here (The Correlation Coefficient - Explained in Three Steps [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] ) if you would like to know more.

In the previous activity you may have noticed the idea of there being perfect correlation. If a straight line can be drawn that perfectly fits all the points, then this may be called a perfect correlation.