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I have a question regarding what we mean when we report the error in any measurement.

Let's say I have a measuring scale which has markings in mm. Therefore any measurement I make has a limit on the precision.

We might want to report the length of the thing we are measuring as x mm +- 0.5 mm. Here we say 0.5 mm because of the discretized nature of the scale. Any value in [x-0.5,x+0.5] is reported as x mm. So in this approach, we might say that the error is 0.5 mm.

Instead, if I assume that I have a uniform distribution in [x-0.5,x+0.5] then when I calculate the standard deviation using the usual methods of probability, I get standard deviation = 1mm/sqrt(12).

The second method is telling me that my device has a smaller error than the first.

Any ideas/suggestions how to think about this?

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I think the assumption of uniform distribution might be the problem. You can't be sure you are ever precisely at x-0.5+0.0000000000000001. You effectively have a 0% chance of getting this result. Weight measurements according to the actual associated readouts of the device.

Suppose you know your length is between $\in [x,x+1]$. If you only report measurements in terms of integers, then you'll only ever note the length as x or x+1. If you always report the length as x since its the greatest integer number of millimeters less than your length, then you'll have zero standard deviation. Your weights are 100% for x and 0% for anything greater. If your length is as likely to be measured as x as (x+1), then your weights are 50% for x, 50% for (x+1) and 0 for anything in between. The associated standard deviation is 0.5.

The error is always going to be smaller than 1mm but you don't know how much smaller with much accuracy.

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I would typically estimate the "overage" as my best tenth-of-a-millimeter guess. Also, statistics normally applies to a population size that is large enough (e.g., 30 measurements). If you take just one measurement, a standard deviation calculation probably doesn't apply.

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