Hello all,
I simply have five quantities, each with an associated uncertainty, and wish to calculate the mean and its associated uncertainty.
The quantities are: 102.70±0.02, 102.71±0.02, 102.65±0.02, 102.68±0.02, 102.69±0.02.
Clearly the mean is 102.69 (102.686), with a standard deviation of 0.02 (0.024) and standard error of 0.01 (0.011).
When I propagate the five uncertainties in quadrature I get: uncertainty = √(0.02^2+...+0.02^2) = 0.05 (0.045), then diving by five as I do when calculating the mean gives me a final uncertainty for the mean of 0.01.
So my question is: which of these three values should I use for the uncertainty of the mean? Standard deviation, standard error or the quadrature propagated uncertainty?
Thank you in advance for your help!
Best wishes,
Robin
Dear Robin,
to propagate uncertainty correctly, you should take into account, what kind of uncertainty you deal with.
If all uncertainties of your initial data are of type B (systematical) then the better way is to calculate uncertainty of mean value for the worst possible case: the real mean value can occur between the mean of smallest possible values of initial quantities and the mean of biggest its possible values. So, the interval 102.69±0.02 is the reasonable estimate.
If all your initial data x1, x2, x3, ..., xn are the results of multiple measurements of the same quantity, then you can estimate statistical uncertainty of the mean: a ± s * t / sqrt(n), where a = (1/n) * sum(xi), i = 1, 2, ... n and s is standard deviation of set x1, x2, x3, ..., xn . Here n is measurements quantity (n = 5 for your data) and t is Student's coefficient for chosen confidence. We need to take into account the uncertainty of initial data. This uncertainty will cause errors in values of a and s and these errors will be correlated. To estimate them correctly we can propagate uncertainty of initial data through the expressions [a - s * t / sqrt(n)] and [a + s * t / sqrt(n)] and estimate its probable values. There are some sophisticated algorithms in literature for estimation the uncertainty of s but the Monte-Carlo technique is the simplest way, I think. As the interval for the mean you can take the minimum possiible value of [a - s * t / sqrt(n)] and maximum possible value of [a + s * t / sqrt(n)]. This will be confidence interval with good coverage.
It your initial data uncertainty is of type A (statistical) and it is obtained from one sensor then you can use your third variant, but you should divide the quadratic root value √(0.02^2+...+0.02^2) by √n = √5, not by n = 5.
With best regards, Konstantin.
Hi
I advice you to see these documents. You will find what you need.
Best regards
http://www.rit.edu/cos/uphysics/uncertainties/Uncertaintiespart1.html
https://www.dartmouth.edu/~chemlab/info/resources/uncertain.html
http://webpages.ursinus.edu/lriley/ref/unc/unc.html
Hi
Try to see these documents. I hope that I helped you, let us know if you have another questions or you need more details.
Best regards
https://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&cad=rja&uact=8&ved=0CDcQFjADahUKEwi07PDx4JXJAhXD2xoKHdU4Bew&url=http%3A%2F%2Fvirgo-physics.sas.upenn.edu%2Fuglabs%2Flab_manual%2FError_Analysis.pdf&usg=AFQjCNFAWEQz6X2j5VKXjUMO1zXj_KZsTw&bvm=bv.107467506,d.ZWU
https://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&cad=rja&uact=8&ved=0CD4QFjAEahUKEwi07PDx4JXJAhXD2xoKHdU4Bew&url=http%3A%2F%2Fwww.clayton.edu%2FPortals%2F498%2FContent%2FUncertainties%2520and%2520Error%2520Propagation.pdf&usg=AFQjCNGbNDyyEN7TElrS5d90ntGJ1ku-ag&bvm=bv.107467506,d.ZWU
Dear Robin,
to propagate uncertainty correctly, you should take into account, what kind of uncertainty you deal with.
If all uncertainties of your initial data are of type B (systematical) then the better way is to calculate uncertainty of mean value for the worst possible case: the real mean value can occur between the mean of smallest possible values of initial quantities and the mean of biggest its possible values. So, the interval 102.69±0.02 is the reasonable estimate.
If all your initial data x1, x2, x3, ..., xn are the results of multiple measurements of the same quantity, then you can estimate statistical uncertainty of the mean: a ± s * t / sqrt(n), where a = (1/n) * sum(xi), i = 1, 2, ... n and s is standard deviation of set x1, x2, x3, ..., xn . Here n is measurements quantity (n = 5 for your data) and t is Student's coefficient for chosen confidence. We need to take into account the uncertainty of initial data. This uncertainty will cause errors in values of a and s and these errors will be correlated. To estimate them correctly we can propagate uncertainty of initial data through the expressions [a - s * t / sqrt(n)] and [a + s * t / sqrt(n)] and estimate its probable values. There are some sophisticated algorithms in literature for estimation the uncertainty of s but the Monte-Carlo technique is the simplest way, I think. As the interval for the mean you can take the minimum possiible value of [a - s * t / sqrt(n)] and maximum possible value of [a + s * t / sqrt(n)]. This will be confidence interval with good coverage.
It your initial data uncertainty is of type A (statistical) and it is obtained from one sensor then you can use your third variant, but you should divide the quadratic root value √(0.02^2+...+0.02^2) by √n = √5, not by n = 5.
With best regards, Konstantin.
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