I want to measure average distance between fixed metal construction and water, as shown in the picture below to predict water flood. Let's call this distance water level h. If the water level starts rising, then I need to inform the local people that the flood is coming and they have to do something, etc.
By black color, I show fixed metal construction, which is not moving. The blue color is water under this metal construction. Let say water is a lake which always has some waves and never stays calm. and waves are not right Sin shape, but random.
I have an ultrasound/laser or any other measurement device which can measure the distance between the device and the water with 0.1cm error very fast (much faster than water waves changes, for example in 1ms). I do many measurements (100-200 times), and I calculate an average water level in relation to my metal construction.
For example, I got average h=123.2 cm after 100 measurements, but because water is always moving the standard deviation is high, like 20 cm.
In this example, can I tell that the water level h=123.2±0.1 cm or I can tell only h=120±20cm because the standard deviation is 20cm?
In other words, if today I get average h=123.2cm, tomorrow I will get h=130.5cm and the standard deviation is the same 20cm, then should I inform people that flood is coming or I cannot because water level difference is less than the standard deviation, that means it is below my error and I cannot really tell if water level is going up or down.
This is just an example to demonstrate the question. There is no real task like this. It can be replaced by another example (measuring cylinder diameter when it is not ideal cylinder) or something else where the error of the device is much less than the standard deviation.