Regarding the 2nd point sorry if I was unclear, it was more a statement than a real question. The point is that sometime we tend to identify the error of the measurement with the accuracy of the telescope. Hence if I say that I have measured a flux of <i>=1+-0.009 Jy I tend to state “My telescope is accurate at the level of 9 mJy”, whereas in this case the accuracy of the telescope (that in the end is given by a single measurement, expecially if I see that the error on a single meas. tend to be pretty stable) is mixed with the accuracy of my observation strategy (so to say).

Now things are a bit clearer

Cheers

]]>I don’t understand the second question. If you only had two observations in the original sample, then the error on the mean would be greater than 0.009. You can always combine *f11* with the other samples to obtain a better estimate of the mean. And certainly, with 3 observations the uncertainty on how well the mean can be known is much larger than if you had 11 observations.

Thanks for your reply! I have an additional question and a comment/question :

- What is the usual criterion for a significant difference that you are referring to ?

- Let’s say that now I am the astronomer that, due to the lack of time, observes the source just twice. Now my *f11*=0.03~*sigmaf* and let’s assume that my SD is always 0.02. By applying the same criterion that you explained I should report a value of *fav*+-*sigmaf* = 1.04+-0.03, simply living with the fact that my single observation is as accurate as the one made before, but my final value is intrinsically less accurate as I had not the time of doing many observations…correct ?

So at the end of the run, you have measurements {*fi*~ N(1,0.03), i=1..10}, where 0.03 is due to photon noise (i.e., just the statistical error, which I would call the measurement error). The SD of 0.02 that you calculate from the 10 measurements should theoretically match the statistical error, but in practice may not due to variations in source, calibration, etc.

From this you can compute a mean *fav*=1.0 and an error on the mean, *sigmaf*=0.009, which is your estimate of the source intensity based on multiple measurements. In principle, it would be nice to fold in the non-statistical variations into the *sigmaf*, but that is a hard calculation, and requires much more modeling.

I would report the *fav+-sigmaf* because it includes the information from all the observations. Of course, the next measurement (at the same exposure time and same observing conditions) will again be drawn from N(1,0.03), say *f11* = 1.04, with a measurement error of 0.03. You can then compare this with the previous estimate of the intensity, *fav*. You can ask whether *fav-f11* differs from zero by comparing it to the propagated error, sqrt(0.009^2+0.03^2) ~ 0.031, which results in a S/N of -1.3, much less than the usual criterion for a significant difference, so no need to panic there.

Let’s say that I’m observing a source to obtain its flux and I look at it with 10 scans or shots. The user interface at my telescope already converts the counts into Kelvin once it knows the calibration noise diode value (typically assumed constant…whreas it’s not…but ok) and gives me out a Kelvin value for each scan/shot, associated with its error.

Now I apply some additional corrections (pointing, gain, etc…) and I propagate the error according to the error propagation theory/formulae, finally I apply the K/Jy conversion using my calibrators as reference and I have a single Jy value with its error, for each scan/shot.

Now…I have more or less a measurement of 1.00+-0.03 Jy for each scan/shot and I’d like to average the values in order to give out a single value for my source. The standard average and the weighted average are similar, but regarding the error some troubles arise :

- I have a standard deviation of 0.02 Jy

- I have a weighted error of approx. 0.03/sqrt(10) =0.009 Jy

What would be an “honest” output to give out ?

- 0.02 Jy would tell how much the measurements have varied during the observation but tells nothing about the instrumental accuracy.

- 0.009 Jy would tell how accurate my telescope is BUT if one repeat the measurement with just one shot and obtain 0.03 Jy the first reaction could be “Ehi, why is my result so bad with respect to what published by others ?!?”.

Any suggestions about an honest way to mix these information (SD 0.02 Jy, Single error 0.03 Jy, WE 0.009 Jy) ?

Thanks

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