# Uncertainty estimation of a quantity after its calibration

I have to calibrate power output of a diode to its input voltage. The voltage source has negligible error but the power meter has significant uncertainty (a couple %). I take a number of power readings at successive voltage inputs, plot a power vs voltage graph, and do a curve fit using least squares. I have the variances in the fit parameters. Now I put away the power meter and want to use the curve fit for calculating the power output as a function of input voltage. How do I estimate the uncertainty in the power determined this way?

• What theoretical model are you fitting your data to? Commented Feb 1, 2019 at 0:34
• power propto voltage^2
– nole
Commented Feb 1, 2019 at 2:38

I am a little unclear from your comment, but if you have not done so already, you should linearize your data. In other words, if you have a theoretical equation from the literature that says $$P = \alpha V^2 + \beta$$, where $$\alpha$$ and $$\beta$$ are constants, then you should plot $$P$$ on your vertical axis, and $$V^2$$ on your horizontal axis so that the slope of the line is $$\alpha$$, and the y-intercept is $$\beta$$. If your theoretical equation is more complicated than that, you may have to work a little harder -- regardless, the first step is to put your equation in $$y=mx +b$$ form.

Perform your weighted linear least-squares fit (so that the fit takes into account the uncertainties in your measurements), and obtain an $$R^2$$ value for the fit.

The uncertainty in the slope can than be found using $$\sigma_m = m\sqrt{\frac{(1/R^2)-1}{n-2}},$$ where $$m$$ and $$\sigma_m$$ are the slope and uncertainty in the slope respectively, and $$n$$ is the number of data points used to make the fit. Careful, the $$R^2$$ is already squared! Don't square it again!

The uncertainty in y-intercept can be found using the above result: $$\sigma_b = \sigma_m \sqrt{\frac{\sum x^2}{n}}$$

So, now you have an equation such as $$P = \alpha V^2 + \beta$$, and you have uncertainties in the constants; namely, $$\sigma_\alpha$$, and $$\sigma_\beta$$. Use standard propagation of uncertainty in quadrature to propagate uncertainty from $$\alpha V^2 + \beta$$ to $$P$$ for each value of $$V$$.

• Your last sentence is the answer I was looking for. Much thanks!
– nole
Commented Feb 1, 2019 at 3:22