Page 58 - Quality control of pharmaceuticals (07-PA 704)

The product-moment correlation coefficient
A common method of estimating how well the experimental points fit a straight line
is to calculate the product-moment correlation coefficient, r. this statistic is often
referred to simply as the correlation coefficient.
The value of r is given by:

r= ∑ {( − ̅ ) ( − ̅ )}
√{∑ ( − ̅)2 ] [∑ ( − ̅ )2]}

It can be shown that r can take values in the range -1≤ r ≤ 1. As indicated in the

following figure, r- values of -1 describes perfect negative correlation, i.e. all the

experimental points line on a straight line of negative slope. Similarly, when r=+1

we have perfect positive correlation. When there is no correlation between x and y,

the value of r is close to zero.

Experience shows that even quite poor-looking calibration plots give very high r-

values. Therefore, the calibration curve must always be plotted, otherwise a straight-

line relationship might wrongly be deduced from the calculation of r.

So r- values obtained in instrumental analysis are normally very high, so a

calculated value, together with the calibration plot is often sufficient to assure that a

useful linear relationship has been obtained. In some circumstances, much lower r-

values are obtained. In these cases, it will be necessary to use a proper statistical test

to see whether the correlation coefficient is significant. The simplest method of

doing this is to calculate a t- value.

To test for a significant correlation, t = ( )√ −2
√1− 2

If the calculated value of t is greater than the tabulated value, we conclude that a

significant correlation does exist.

It can be shown that the least squares straight line is given by: