Suppose you have measurements X={X_i, i=1..N} of a quantity whose true value is, say, X₀. One can then compute the mean and standard deviation of the measurements, E(X) and σ_X. One can also infer the value of a parameter θ(X), derive the posterior probability density p(θ|X), and obtain confidence intervals on it.

So here are the different interpretations:

1. Measurement error is σ_X, or the spread in the measurements. Astronomers tend to use the term in this manner.
2. Measurement error is X₀-E(X), or the “error made when you make the measurement”, essentially what is left over beyond mere statistical variations. This is how statisticians seem to use it, essentially the bias term. To quote David van Dyk
   

   For us it is just English. If your measurement is different from the real value. So this is not the Poisson variability of the source for effects or ARF, RMF, etc. It would disappear if you had a perfect measuring device (e.g., telescope).

3. Measurement error is the width of p(θ|X), i.e., the measurement error of the first type propagated through the analysis. Astronomers use this too to refer to measurement error.

Who am I to say which is right? But be aware of who you may be speaking with and be sure to clarify what you mean when you use the term!

When C counts are observed in a region of the image that overlaps a putative source, and B counts in an adjacent, non-overlapping region that is mostly devoid of the source, the question that is asked is, what is the intensity of a source that might exist in the source region, given that there is also background. Let us say that the source has intensity s, and the background has intensity b in the first region. Further let a fraction f of the source overlap that region, and a fraction g overlap the adjacent, “background” region. Then, if the area of the background region is r times larger, we can solve for s and b and even determine the errors:

Note that the regions do not have to be circular, nor does the source have to be centered in it. As long as the PSF fractions f and g can be calculated, these formulae can be applied. In practice, f is large, typically around 0.9, and the background region is chosen as an annulus centered on the source region, with g~0.

It always comes as a shock to statisticians, but this is not ancient history. We still determine maximum likelihood estimates of source intensities by subtracting out an estimated background and propagate error by the method of moments. To be sure, astronomers are well aware that these formulae are valid only in the high counts regime ( s,C,B>>1, b>0 ) and when the source is well defined ( f~1, g~0 ), though of course it doesn’t stop them from pushing the envelope. This, in fact, is the basis of many standard X-ray source detection algorithms (e.g., celldetect).

Furthermore, it might come as a surprise to many astronomers, but this is also the rationale behind the widely-used wavelet-based source detection algorithm, wavdetect. The Mexican Hat wavelet used with it has a central positive bump, surrounded by a negative annular moat, which is a dead ringer for the source and background regions used here. The difference is that the source intensity is not deduced from the wavelet correlations and the signal-to-noise ratio ( s/sigma_s ) is not used to determine source significance, but rather extensive simulations are used to calibrate it.