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Re: [Phys-l] Basic statistics



I would like to affirm John D's recent post, and add a footnote.

John's post explains it well, and I want to focus on the final summary in which he says:

Therefore, the bottom line is that you have a choice:
*) If you choose to consider μ to be the object of interest, then
you should report μ' ± Δμ', which tells people how well you have
estimated μ.
*) If you choose to consider the underlying distribution to be
the object of interest, then you should report μ' ± σ'.

When I explain this to my students, and I forget to explain which we are interested in, they will ask, "Which one should we do for this lab? Are we interested in the distribution or are we interested in the mean."

Sometimes your main goal is the mean. That is probably the most common goal. Your experiment is designed to measure a particular quantity, you make multiple measurements of that quantity. The mean is your answer, and the Δμ' (as John called it, or "standard error of the mean" as many call it) is what you would want to report along with the mean.

On the other hand, suppose you are testing a new instrument or a new method of analysis. In that case you are primarily interested in things like resolution and precision. In that case your primary interest is the distribution, and you are hoping your new instrument or method results in higher precision (smaller standard deviation) than existing instruments or methods.

I would also reiterate John's statement that more measurements will reduce the standard error in the mean, but not the standard deviation of the distribution. The standard deviation of the distribution is determined by the instrument and/or method, and more measurements will not improve the instrument or method. If we want to improve that, we need a different instrument or different method. More measurements will just give us a better determination of what the actual standard deviation is for our particular instrument or method.

On the other hand, more measurements using the same instrument or method does yield a better value of the mean, assuming the instrument is calibrated properly, we are using it properly, and there aren't other systematic errors.


Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu