How Should I Calculate Error on the Mean with Identical Measurements?

[BillKet]

Hello! I have an experiment and for some reasons I was able to do only 4 measurements and they all ended up having the same value, say for the purpose of this post $100 \pm 1$ where the error of 1 is estimated based on the measuring device resolution. The mean is obviously 100. Usually the error on the mean would be $\sigma/\sqrt{N}$, where $\sigma$ is the standard deviation of the measurements, which in this case is zero. So based on that I would have to quote an error on the mean of zero, but that seems wrong. I can't be 100% sure about my measurement. But also using 1 as the error on the mean seems too big. How should I calculate my error on the mean? Should I use $1/\sqrt{4}=1/2$?

 

[mathman]

There is nothing much you can do about the statistical standard deviation. The device resolution is all you have.

 

[Dale]

You could try a Bayesian estimation

 

[Stephen Tashi]

The mean is obviously 100. Usually the error on the mean would be $\sigma/\sqrt{N}$, where $\sigma$ is the standard deviation of the measurements, which in this case is zero. So based on that I would have to quote an error on the mean of zero, but that seems wrong.

I think you are confusing the concept of the mean and standard deviation of a sample with the concept of the mean and standard deviation of a random variable. "Standard deviation" for a sample can refer to the unbiased estimator of the population standard deviation or the biased estimator. Using the formula for either of those estimators, it is possible to get a zero result for a particular sample. Once you have chosen the estimator and have a particular sample, you don't have any choice about what value it produces. If the value is zero you shouldn't report it as something different.

The proper way to state your question is that you think the zero value of the estimator for the standard deviation of the random variable is not a correct estimate. You want create a different estimator for the standard deviation of the random variable. To create such an estimator, you are stepping outside the material found in introductory statistics textbooks.

I'll repeat my advice from other threads: Statistical methods are subjective. If your main goal is to publish a paper in a journal, look at papers published in the journal and try to find out what statistical methods were accepted for publication.

If you need advice about how to break new ground in statistics ( relative to what's found in the journal) you should describe a specific problem - including the relevant physics. It is a mistake to present only the aspects of the problem that you think are relevant to statistical issues, unless you are an expert at judging which aspects are relevant to statistics.