Points from detectors: when to integrate, sum or average?

Question

I'm reading data from detectors like Mass Spectrometers, DAD, etc. One measurement from these detectors gives a spectrum, so there are many data points in some range e.g. from 200 to 1000 m/z or from 200 to 400 nm. Users want to get the intensity let's say at 400m/z. But detectors measure things with a finite resolution, so we should really give them 400 ± 0.2 for instance. But now what do we do with the points in this range?

I looked into other software and they seem to be doing this differently between themselves and between detectors:

1. Some would simply sum up the points in the range
2. Others would integrate the area between 399.8 and 400.2
3. And there are those that sum and divide by the number of points

What's the ideology here - when should we do what?

Also:

1. Points are not always spaced regularly
2. Another difference between sum and integration: when doing sum point 400.3 won't be included. In case of areas - the space between let's say 400.0 and 400.2 will still be integrated - so 400.3 has an effect on the result.

Answer 1

Specific to your application, the resolution bandwidth of each detector should be considered given that the intensity is applicable to a certain frequency range, such that each point is reporting the total integrated intensity over an equivalent optical bandwidth. If the adjacent points are overlapping this bandwidth, then an overestimate can occur due to the double counting if that resolution bandwidth is not properly accounted for. Ultimately, assuming the OP is looking for the spectral density at a particular wavelength, determining this would depend on the sensitivity for each point to intensity at nearby frequencies. For that, it may be sufficient to use the closest point and confirm that the resolution bandwidth is sufficient to cover the point in question, or if estimating the power spectral density over a band of frequencies where the intensity is similar, then averaging multiple points can increase the accuracy of that estimate.

Ideally you would compute and report the standard deviation for your measurement result, and state that your $\pm$ criteria is a one sigma standard deviation.

The ideology is based on the statistics of the noise process and how that translates to your estimate. Often what is done is simpler in processing and for a White Gaussian Noise process would have a direct proportional scaling. This breaks down when the process is not White Gaussian Noise.

Examples of this are doing "Absolute Average Deviation" instead of standard deviation which for white noise is related by $\sqrt{2/\pi}$. Another example is doing post detection averaging (such as on a spectrum analyzer) and then if the results are averaged in dB or magnitude quantity. This is detailed in this great app-note 1303 by the former HP now Agilent where we see the results will be in error of the actual standard deviation by -2.51 dB when averaging log-detected signals and -1.05 dB in averaging detected signals (with the point that the average of the squares is not the same as the square of the averages).

This would apply to estimating zero mean noise. For noise from a non-zero mean as would be done in a typical measurement as the OP's, I detail how to extend this to a non-zero mean at this post here (given it's similarity to a Ricean fading channel): Rayleigh fading with frequency selective fading channel