# Calibration Curve Error Propagation

I am using ED-XRF for geochemical analysis for a geology dissertation. The ED-XRF provides a ppm reading for elements and a 2sigma error, for example 472 +- 5.11ppm Sr.

An example calibration curve made for Sr comprising 37 standards has a linear regression of y=1.26x + 32.1 (R^2 0.9991), which I can easily apply to correct my unknown analyses. I need to plot my corrected readings with their error bars, so how is the 2sigma error associated with this unknown calibrated?

Does this error value also simply get calibrated with the curve equation, or just the slope value, or is this error propagation far more complex? I have searched for a solution and keep finding an equation for uncertainty propagation with calibration curves, but this only calculates the uncertainty caused by the curve, and has no relation to the uncertainty of each analysis? So is the total error the original 2sigma error provided by the machine + calibration curve uncertainty?

Thanks

1. The theoretical best way to fit your calibration curve would be with weighted linear least-squares algorithms. In that algorithm, in addition to the {x, y} data, which in your case is $\{\ce{Sr_{known}, Sr_{meas}}\}$, you also supply weights corresponding to the uncertainty in the y values. The weight for each y data point would be $\frac{1}{\sigma^2}$, which you could calculate from the $2\sigma$ uncertainty given by your instrumentation for that data point. In practice, if you did this, I doubt that the parameters of your calibration would change very much at all from what you did previously. But formally the weighted variant of least-squares is required if the uncertainty in y depends on y. See wikipedia for more.
2. After doing the (weighted) least squares calibration regression, you can find the uncertainty in the estimation using the calibration equation. $\sigma_{est} = \sqrt{\frac{\sum{\left(y_{pred}-y_{meas} \right)^2}}{N}}$. In that equation $y_{pred}$ is the calibration-predicted Sr concentration and $y_{meas}$ is the measured Sr concentration, and for you $N$, the number of data points, is 37. This uncertainty is one part of the uncertainty you'll eventually report, and reflects the uncertainty in your calibration curve. Based on the R-squared you reported, my guess is that this uncertainty will be low.
3. After all that (!), you need to estimate the other major source of uncertainty, the uncertainty in the measurement of your unknowns ($\sigma_{meas}$). There are several OK methods for doing this, the best of which would be replicate analysis of several samples. It might be OK to use the $1\sigma$ value calculated from the $2\sigma$ uncertainty reported by your instrument too.
4. Finally, combine the uncertainties with the standard equation for combining multiple sources of uncertainty $\sigma_{total} = \sqrt{(\sigma_{meas})^2 + (\sigma_{est})^2}$.