Those examples are Apples & PCs; what is being averaged for the temperature “averages” are data points from multiple instruments in separate locations for a single point in time, WITHOUT A COMMON TARGET, none of which are are identical, set up the same, read the same, nor even read at the same (”correction factors” are used to “adjust” for differences reading times) time; stations added in or removed at will; stations moved spatially over time; instruments not only replaced, but replaced by different types of instrumentation; and there are other variables not controlled for. Missing data points are often “calculated” and added to the data sets.
That is very different than all instruments, all standardized and precisely arrayed, simultaneously collecting a signal from a single point.
It is also different than using the same set of equipment, though perhaps utilizing different configurations, to run an experiment several times while controlling for known variables, and accounting for known equipment limitations and error factors. Even then there are limitations to experimental results despite computational abilities.
The radio telescope example is a matter separating a signal from noise in a relatively known background; this is generating data and noise, and calling it all signal. To be analogous, each of the phased array dishes would have to be OUT of phase, and each aimed at a different object.
These people don’t even have a precise and accurate zero point; and they rather arbitrarily add in or delete equipment locations; use interpolation to “measure” huge, geographically dissimilar areas that are not instrumented, and even use that interpolated “data point” to interpolate the data point for another, contiguous, non-instrumented area; and add those into their “average”.
It’s a matter of the differences between “precise”, “accurate”, and “significant”. Garbage to 3 decimal places is still garbage, and stinks no matter how it’s sliced and diced.
And significance can not be increased by adding more insignificant “data”.
When combining measurements with different degrees of accuracy and precision, the accuracy of the final answer can be no greater than the least accurate measurement. This principle can be translated into a simple rule for addition and subtraction: When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement.
Multiplication and Division With Significant Figures
The same principle governs the use of significant figures in multiplication and division: the final result can be no more accurate than the least accurate measurement. In this case, however, we count the significant figures in each measurement, not the number of decimal places: When measurements are multiplied or divided, the answer can contain no more significant figures than the least accurate measurement.
To present a temperature average to the 100th of centigrade degree from instruments that have an accuracy of, AT MOST, of a tenth is disingenuous at best.
If you try to measure and do it really poorly, if you do it enough times, the random walk of the errors tends to average out and the average is still a decent data point.
The more errors you have will cause a wider dispersion of the data and the standard deviation will be larger.
However, as you well know, with more data points, the standard deviation narrows.
A narrowing standard deviation means higher confidence levels and greater precision of the average.