# How to estimate uncertainty of measurements of equivalent widths?

I'm measuring equivalent widths of absorption lines using a spectrum of a star. I make two or three measurements of each line by making reasonable gaussian fits of the line with IRAF's splot tool. Then I calculate the mean of the measurements, which serves as my final equivalent width estimate.

What is a good way of estimating the uncertainty of this measurement?

## My current method

I'm currently using half of the range for the uncertainty. For example, if I made two measurements 10 and 16 mA (milliangstrom), then the mean is 13 mA and uncertainty is 3 mA. This gives the estimate of equivalent width to be 13±3 mA. Do you see any problems with this method of estimating uncertainty?

Yes there is a problem. You seem to be trying to derive an uncertainty in the measurement of EW by doing repeated measurements of the same data?

This can only give you the uncertainty associated with your measurement technique (i.e. where you define the limits of the line and how you set the continuum level) - the systematic error you might call it (although there can be other systematic errors inherent to EW measurements, like whether you subtracted the sky or scattered light in your spectrograph correctly for example).

What it does not do is evaluate the uncertainty in the EW caused by the quality or signal-to-noise ratio of the data itself. You might assess this using some rule-of-thumb formulae for a Gaussian line, e.g. $$\Delta EW \simeq 1.5 \frac{\sqrt{fp}}{{\rm SNR}},$$ (eqn 6 of Cayrel de Strobel 1988) where $$f$$ is the FWHM of the spectral line (in wavelength units), $$p$$ is the size of one pixel in wavelength units and SNR is the signal-to-noise ratio of the data in an average pixel. Or you could take a synthetic spectrum and add some artificial noise to it with the appropriate properties and measure the EW of several randomisations of the same spectrum, taking the standard deviation of your EW measurements to indicate the EW uncertainty for a particular level of signal-to-noise ratio.

If this statistical uncertainty is not negligible, then you would then need to add it to any systematic uncertainties associated with your analysis of the spectrum. As far as the latter is concerned then your suggested method does give some indication of what that error might be, though I suspect it will overestimate the 1-sigma uncertainty.

Estimating uncertainty... the most preferred one would be Bayesian. However, if you follow the frequentist, MCMC is the most preferred, and it would be somewhat similar to what your advisor suggested with more complicated algorithm. The simplest method would be what your advisor suggested, but you can measure for a larger sample size and use simple statistics like mean and standard error.

There are various variations along these lines that I would not make a list here.