# Error estimation with missed data

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?