# Error propagation through a calibration curve

Say I have a linear fit given y = ax + b. I'm given Δa and Δb as 95% confidence intervals.

I now have several measurements y1, y2, y3, ... etc, from which of course I can gather a mean, standard deviation and confidence interval. If I want to find some x and its uncertainty from these, how should I go about this.

It's easy to get an expected value for x, from some simple algebra.

x = (y-b)/a

I use the mean of y for this calculation.

But going through the error propagation

$$\Delta x = x*\sqrt{(\frac{\sqrt{\Delta y^2 + \Delta b^2}}{y-b})^2+ (\frac{\Delta a}{a})^2}$$

I'm unsure what numbers I should be using for the Deltas in this case. Should it be the confidence intervals or standard deviations? I intuit that it should be the standard deviations but I'm unsure how to get those without any information from the original fit.

• Please see stats.stackexchange.com/a/206682/919 for an account of this problem. The formulas will reveal what additional information you need to solve this problem and how exactly to do so. The short answer is that if you must rely only on $\Delta a$ and $\Delta b,$ your confidence intervals will likely be wrong, but in which direction or by how much is impossible to say. Another related discussion appears at stats.stackexchange.com/a/31533/919 -- its reference material may be helpful.
– whuber
Feb 9 '20 at 15:54
• Part 1. Very little to add to the excellent answers by @whuber. But maybe these papers are of interest: 1.) J.O. De Beer, T.R. De Beer, L. Goeyens, “Assessment of quality performance parameters for straight line calibration curves related to the spread of the abscissa values around their mean”, Analytica Chimica Acta 584 (2007) 57–65. Eqn 9. 2.) J.-M. Mermet, “Calibration in atomic spectrometry: A tutorial review dealing with quality criteria, weighting procedures and possible curvatures”, Spectrochimica Acta Part B 65 (2010) 509–523. Eqns 23 & 24.
– Ed V
Feb 9 '20 at 16:56
• Part 2. 3.) J.P. Buonaccorsi, “Design Considerations for Calibration”, Technometrics 28 (1986) 149-155. Eqn (3.1).
– Ed V
Feb 9 '20 at 16:56