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When I need to find an error in the value of some real physical quantity and I can have as many number of measurements as needed, then it is clear. For example, I have a metal rod 10 cm length and 1 cm diameter and I need to measure it's length. I take this tool, make 10, 20, 50 whatever measurements. Write it down, calculate average, standard deviation, etc and I will get the average length error estimation somehow in this form $10.2cm\pm0.3cm$.

But how to calculate error in case if I cannot make enough measurements? For example, I have 100 meters length and 5 cm diameter rod and I need to measure the average diameter of this rod. Because it is very long (100 m) ideally I need to measure it every millimeter many times (at least 10) and do it over whole it's length. In this case, it will be 100000*10 = 1 million measurements. If I do these 1 million measurements, then I can use the same method to calculate the error. But I cannot do 1 million measurements in real life.

What if I have only 5 measurements after every meter, which in total gives me 500 measurements. How to calculate the error and how to take into account that it is possible, that when I do the measurement after 1 meter, in the middle can be a very big deviation from average and my error will be very very high.

Rod is just an example, because it is just a theoretical question, do not spend you time to explain, that it is not possible to make this rod, invent some clever technic, etc.

Just simple case, I have:

  1. 100m rod.
  2. This tool with 0.1mm precision.
  3. Piece of paper or Excel to write it all down.
  4. 500 measurements of the diameter of this rod, 5 per each meter.

Questions: What will be the precision of this experiment? How to calculate the precision of diameter and average diameter? In other words how to get the answer in this form: $50.1mm\pm0.5mm$?

Another example can be like this. I have a piece of land 10x10 km (area $10^8 m^2$) and I need to measure "Metres above sea level". Let say I will do it with GPS sensor and let's assume GPS "Metres above sea level" is very precise. If I divide this piece for 100x100m squares and make 1 measurement in the middle of this square, then it will be 10000 measurements which is realistic. But what if in between this 100x100m will be some hill or hole? and I want to be more precise, then I need to make a measurement inside every 1x1m square and it will be $10^8$ (10 million) measurements, which is not realistic.

My question: Is there well known scientific method to estimate the error in the cases described above?

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There is no method that can help you, unless you have evidence that the the thing you are trying to measure deviates from the model you have assumed it obeys.

In the case of the rod. If you say you can measure with 0.1mm precision and you do 500 measurements along the rod and the standard deviation of the measurements is 0.1mm, then you have reasonable evidence that the rod is of uniform thickness and you can take the average of those measurements to find the mean thickness.

On the other hand you may find that the standard deviation is larger than that. In which case, either you have underestimated how precisely you can take the measurements or the rod is not of uniform thickness. In the latter case, you can try to see what better model might represent your data. Perhaps the residuals to the mean are distributed in a particular way that might lead you to a better fitting, but more complex, model?

If the residuals are essentially random and suggest that the rod varies in thickness randomly with position, then you would still take the average and use the standard deviation of your results as an estimate of by how much the rod varies in thickness.

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  • $\begingroup$ Thanks for the answer. Does it make sense to take into account the coverage? In my example, if I make a measurement of the rod every 1 meter and let's say 1 cm around measurement place I am sure, that deviation is not big (there is no big difference in diameter), that I covered 5 m (1 cm every 1 m) out of 500 m. I can say, that coverage is 1%. Is it enough evidence in this case to tell "I have reasonable evidence that the rod is of uniform thickness"? Or I can tell that my standard deviation 0.1mm is correct with probability 1%, because I cover only 1 % of the rod with my measurements? $\endgroup$
    – Zlelik
    Oct 18, 2019 at 11:49

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