Experimental uncertainty (error) in using linear calibration

A typical set of linear data can be described by the plot of the absorbance, A, of a solution as a function of its concentration, C, in mg dm-3.

C / mg dm-3

0 20 40 60 80 100 120

A

0 0.267 0.583 0.824 1.120 1.313 1.499

A solution of unknown concentration gives 3 replicate measures of absorbance: 0.763, 0.741 and 0.749, and we use the calibration data to calculate a best-estimate for the true concentration of this solution.
The calculation uses the Beer-Lambert law that states that, for low concentrations, the absorbance is proportional to concentration. This means that we would expect that plotting absorbance against concentration should give a straight line.

This data is analysed using:

An analysis of Residuals and Corrrelation coefficients to  - see video Beers Law v1

  • Identify curvature of the calibration data at high concentrations (x-values)
  • Select an appropriate linear calibration range 

Functions and equations to calculate  - see video Beers Law v2 

  • Best estimate of the unknown concentration using both a free-fit trendline and a trendline forced through the origin.

Functions and equations to calculate  - see video Beers Law v3 

  • Standard uncertainty in the calculated concentration, given by

Equation gives the approximate standard uncertainty in the x-value of a calibration point

or

Equation gives the approximate standard uncertainty in the x-value of a calibration point

  • 95% Confidence Interval for the true value of  the unknown concentration 

(Refer to section 13.3.3 in the text: Essential Mathematics and Statistics for Science, 2nd Ed, by G Currell and A A Dowman)