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If the result of a measurement is i.e. $3.2 \pm 0.7$, what is 0.7? At which confidence level we know that the real result is inside of this interval?

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As your link indicates, it is the uncertainty in the measurement. The exact meaning of this can depend on context, but most of the time it is safe to assume that this is the standard deviation in the value that you should expect if the measurement were repeated a large number of times.

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In our undergraduate lab we teach that if it's an engineering text and the confidence level is not specified, assume 95%. Although I never check that against anything other than a decades old teaching tradition – Slaviks Mar 9 '12 at 20:57
I think it varies by field. I've heard of that 95% convention, but in experimental physics (HEP at least, since that's what I know), it's more conventional to quote the 68% confidence interval, i.e. one standard deviation. Of course the best advice is to always state the meanings of your uncertainties explicitly. – David Z Mar 9 '12 at 21:45

The two most common conventions I know are

  1. Report the standard deviation of the result: this tells you that a further measurement has 68% probability of falling inside the interval
  2. Report the standard error of the measurement (standard deviation / $\sqrt{n}$). This tell you there is a 68% chance that the actual value (the mean of the underlying population) lies within the interval

The 95% interval is often used when one is giving a "normal" range of values - for example, when reporting normal blood sugar levels etc. This is more useful, because it says "if your measurement is outside of this range, you most likely have a problem" - with the usual 5% chance of being wrong.

Without context, there is no way of knowing what the authors intended - although in the example given I would be horrified if this was the standard error (over 20% on either side of the mean)...

My advice - when in doubt, ask. And when it's your turn to report a result, try to avoid possible confusion by saying what you are doing, e.g.

3.2 ± 0.7 knorgels (sample mean ± std)

Note - always include units in your quoted result...

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