What error to use on the mean

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In summary, the conversation discusses the issue of calculating the error on the mean for a sample with a standard deviation of zero. The person is unsure if reporting an error of zero is accurate and is considering using a Bayesian estimation or a different estimator for the standard deviation of the random variable. The advice given is to look at papers published in the journal for accepted statistical methods, and to provide a more detailed description of the problem for expert advice.
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BillKet
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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##?
 
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  • #2
There is nothing much you can do about the statistical standard deviation. The device resolution is all you have.
 
  • #3
You could try a Bayesian estimation
 
  • #4
BillKet said:
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.
 

1. What is the purpose of using an error on the mean?

The error on the mean is used to measure the uncertainty or variability in a sample mean. It helps to determine how close the sample mean is to the true population mean.

2. How is the error on the mean calculated?

The error on the mean is calculated by dividing the standard deviation of the sample by the square root of the sample size. This results in a measure of the standard error of the mean.

3. Is the error on the mean the same as the standard deviation?

No, the error on the mean is not the same as the standard deviation. The standard deviation measures the variability within a sample, while the error on the mean measures the uncertainty in the sample mean.

4. Can the error on the mean be negative?

No, the error on the mean cannot be negative. It is always a positive value as it represents the distance between the sample mean and the true population mean.

5. How is the error on the mean used in hypothesis testing?

The error on the mean is used to calculate confidence intervals and determine the significance of the difference between two means in hypothesis testing. It helps to determine if the difference between the means is due to chance or if it is a true difference.

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