Correlation

Biology: Cambridge International A Level, topic 18.1.4

Correlation and Causation

Correlation means that there is a relationship between two variables. Causation means that changing one causes the other to change.

Correlation does not imply causation.
If two variables are correlated that does not mean that changes in one directly cause changes in the other.

The number of ice cream sales might be positively correlated with temperature but this does not mean that ice cream sales cause higher temperatures.

Two variables are correlated if you can draw a straight line that lies close to most of the points.

The variable x is the independent or explanatory variable.
The variable y is the dependent or response variable.
The slope of the regression line is given by the constant b.
The sign and size of this number tell us how y changes for every unit increase in x on average.

The Pearson's correlation coefficient, $r$ , provides a measure of strength and direction of the correlation between x and y.
The value of $r$ is always between –1 and +1.

Values of $r$ close to –1 or to +1 indicate a stronger linear relationship between x and y.
If $r = 0$ , there is likely no linear correlation.
If $r = 1$ , there is a perfect positive correlation so all points lie on a straight line.
If $r = \text{-}1$ , there is a perfect negative correlation so all points lie on a straight line.

The formula for the correlation coefficient is:
$r = \frac{\large 1}{\large n-1}\sum\left(\frac{\large x_i-\overline{x}}{\large s_x}\right)\left(\frac{\large y_i-\overline{y}}{\large s_y}\right)$
Where $\overline{x},\overline{y}$ are the mean and $s_x,s_y$ are the standard deviations of the explanatory and response variables respectively.
It is much more common to calculate $r$ using a calculator.

Do these two variables have a correlation or a causal relationship?
Time spent practising tennis & accuracy of tennis serve.
Do these two variables have a correlation or a causal relationship?
Time spent practising tennis. Average daily temperatures.
Do these two variables have a correlation or a causal relationship?
Lemonade sold & Average daily temperatures.
Which feature of a regression line tells us how the independent variable changes for every unit increase in the dependent variable on average?
Which variable is the response variable in the graph below?