Increasing 'N' For Improved CRQL Recovery

To start off, there is a great book on error analysis called "An Introduction To Error Analysis" by John R. Taylor. This book has very succinct and easy to use explanations of least squares analysis, linear regression, and propagation of errors.

On the back cover are the expressions we will need to examine how the choice of 'N' in constructing a calibration curve will effect the slope and intercept of a linear regression line.

If (x₁,y₁),...,(x_N,y_N) are data points then the best fit straight line A=mc+b is given by:
m = [(sum(x²)*(sum(y)) - (sum(x)*sum(x*y))]/D
b = [(sum(x*y) - (sum(x)*sum(y))]/D
D = N*(sum(x²) - (sum(x))²)

We can see that N appears in the expression for both m and b in the denominator. Since sum(x²) >= (sum(x))² for any x we can see that increasing N will increase the value of D.

Examining the expressions for 'm' and 'b' we see that the concentration of the calibration standard added will have some effect on whether m or b are increasing or decreasing.

For now, let us focus on a calibration standard with a concentration of less than 1.0 in whatever units you are working in. Now, x² will be less than x. The value of y will depend on the choice for the value of x. For the CETAC M-7500, out of the box, the absorbance in µAbs units is approximately 10,000 times the concentration in ppb or µg/L. This means that in this specific case the values of 'y' will be very large.

The overall effect, when the additional calibration standards have a concentration of 0.5 or less, is to increase the slope as well as decrease the intercept.

Practically, this means that if your CRQL is coming in too low you may be able to make additional calibration standards at concentrations near the CRQL. Their inclusion should increase your overall recovery for low standards while not affecting the recovery very much for your high standards.

Below is an example demonstrating how adding low standards increases the slope gradually while decreasing the intercept more rapidly.

Click image for larger version. Data and graphs showing the effect of increasing N.

In this example computing percent recoveries reveals that including two standards near 0.5x the CRQL increases the percent recovery of the CRQL itself by around 2.5%.