Standard deviation vs measurement uncertainty

[bluemystic]

Homework Statement

Suppose I measure the length of something 5 times and average the values. Each measurement has its associated uncertainty. What is the uncertainty of the average?

Relevant Equations

SD=sqrt( sum of difference^2/(N-1) )
Standard Error=SD/sqrt(N)

Using the above formulas, we can arrive at an unbiased estimate of the standard deviation of the sample, then divide by sqrt(N) to arrive at the standard deviation of the average. What I'm confused about it where the measurement uncertainty comes into the equation. Is it being ignored? Say I take only two measurements and they turn out to be equal. Then the sample standard deviation is zero. But the true uncertainty of the average can't be 0 because of measurement uncertainty, can it?

On a side note, why can't I use error propagation of measurement uncertainties to obtain the uncertainty of the average, without considering standard deviation?

 

[FactChecker]

The sample standard deviation is only an estimate. Using only two experimental samples would be a very poor estimator, so you should not draw any conclusions from that. The measurement "uncertainty" can be constant or have random variation, or a mixture of both. The sample standard deviation is only appropriate for measuring the random variation.

 

[haruspex]

As @FactChecker points out, systematic errors will not be reflected in the variation in the measurements. Putting those to one side, we have random errors and rounding errors. If the random errors are smaller than the rounding then this can also result in, effectively, systematic error. E.g. you are using a 1mm gradation, random error is only 0.1mm, and the value to be measured is 1.7mm; you will read it as 2mm every time.

But what you are asking about is purely random errors for which you have some a priori estimate.
The usual process is that you calculate the batch error (standard error of the mean) as you describe, but use the lower of that and your a priori limit. To me, that is not really satisfactory; there ought to be a general formula that smoothly covers the transition from few samples to many. I have tried to come up with one, but it might require a Bayesian approach.

 

[bluemystic]

Thanks for the help! I didn't realize one measured random error and the other measured systematic error.