Correlation and Regression
Correlation
A scatter diagram may be used to represent bivariate data (ie. data in which each data point is defined by two variables). The extent to which the points approximate to a straight line gives an indication of the strength of a linear relationship between the variables, known as linear correlation.
One way to arrive at a numerical measure of the correlation is to use the product-moment correlation coefficient, r.
For n pairs of (x, y) values:
Sxx = å x2 – n 2 Syy = å y2 – n 2 Sxy = å xy – n
And the product-moment correlation coefficient is given by:
r = Sxy___
√ ( Sxx Syy)
This gives values of r between -1 (representing a perfect negative correlation) and +1 (representing a perfect positive correlation.
Regression
Whereas correlation is determined by the strength of a linear relationship between the two variables, regression is about the form of the relationship given by the equation of a regression line.
The purpose in establishing the equation of a regression line is to make predictions about the values of one variable (known as the response variable) for some given values of the other variable (known as the explanatory variable).
Predictions should only be made within the range of readings of the explanatory variable. Extrapolation for values outside this range is unreliable. Another factor affecting the accuracy of any predictions is the influence of outliers on the equation of the regression line.
The regression line of y on x is given by:
y = a + bx
where b = Sxy and a = - b
Sxx
The regression line of x on y is given by:
x = c + dy
where d = Sxy and c = - d
Syy
Unless there is perfect correlation between the variables, the two regression lines will be different and you cannot rearrange one equation to obtain the other.
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