# What does it mean if my result is off the known value by one Sigma?

by content404
Tags: error analysis, standard deviation
 P: 3 I'm putting together a lab report and my result is off from the known value by 1.43 σ. According to the error function tables provided by my prof, and using the error function in my error analysis textbook, that gives me a probability of ~85%. I don't understand what this means though. Will a repeat experiment have an 85% chance of being within my standard of deviation or was there only a 15% chance that my result would be off by this much? The discrepancy between the known value and my experimental value is less than my total uncertainty so I think my result is reasonable.
 Mentor P: 9,585 If you repeat the experiment with perfect uncertainty analysis over and over again and have uncorrelated deviations, 85% of those experiments should be closer to the "true" value. Note that this is an approximation for a gaussian error where the width does not change significantly if your measured value changes a bit. If that is not true, you need something like the Feldman-Cousins method to get a similar number.
 P: 3 Ok, I can see how that makes sense but I don't understand why it works like that.
Mentor
P: 9,585

## What does it mean if my result is off the known value by one Sigma?

Let's assume we know the exact value and all sources of error in the measurement (which can depend on that exact value), so we can predict how the measurements will be distributed. In addition, let's assume a gaussian distribution of those errors. If you get one measurement 1.43σ away from this exact value, you know that just 15% of all measurements will get a larger deviation (this is just a result of the gaussian distribution).

In a real experiment, you cannot use your knowledge of the exact value - you have to estimate the uncertainty based on your measurement. In many studies, this does not matter, and you can work like you had the situation described above: A gaussian distribution around the exact value, with the uncertainty of your measurement.
P: 278
 Quote by mfb In a real experiment, you cannot use your knowledge of the exact value - you have to estimate the uncertainty based on your measurement. In many studies, this does not matter, and you can work like you had the situation described above: A gaussian distribution around the exact value, with the uncertainty of your measurement.
Perhaps to clarify: in the OP situation the described "known" value is assumed to be the true value (forming the so-called null hypothesis), and this number of 85% is cooked up using that assumption, and -as mfb says- the assumption that you understand your uncertainties.
 P: 3 Ok I get it now, thank you both. If you're curious, I was conducting a repeat of the Franck Hertz experiment with 30 year old equipment. High precision, low accuracy.
 P: 233 This really means that you cannot throw away your "known" value if you are interested in using it as the "true" value, at usual 1% or 5% levels. Because, even if the "known" is exactly the "true" value, then in the long run of your experiment you will get 15% values off by at least that much.

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