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So in class we are doing an experiment on the period of a pendulum where you measure the period of oscillation with a stopwatch.

Now the way the lab manual shows how to do the error analysis is by computing the mean and getting the standard deviation. I get this much. However, elsewhere I read that the error in measurements are the smallest unit possible on the measurement apparatus, so for example on a metre stick whose smallest unit is 1mm the error/uncertainty on any measurement would be 1mm (or 0.5mm it says elsewhere, I suppose it's the smallest unit you feel confident estimating).

So, if I were to take 10 measurements of a piece of string, each measurement would have an uncertainty of 1mm. Now, when I computed the average of these would the uncertainty be 1mm or would I have to use the formulas for the propogation of errors to compute the uncertainty of the average. Or do we disregard this 1mm and simply compute the mean and standard deviation?

Is it the case that the standard deviation would include all sources of uncertainty, i.e. the 1mm measurement uncertainty as well as others. If so, what is the point of taking notes of the measurement error in the first place when it is included in the statistical analysis?

Can anyone help me? I know I am making some fundamental misconception but I am struggling to suss it out.

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  • $\begingroup$ "Now the way the lab manual shows how to do the error analysis is by computing the mean and getting the standard deviation. I get this much. ": This is probably the most basic example of statistical estimation, but don't take it as a universal law: not for every repeated measurement you have to take the mean and not for every repeated measurement the uncertainty can be evaluated by the standard deviation. $\endgroup$ – Massimo Ortolano Sep 7 '16 at 17:57
  • $\begingroup$ "However, elsewhere I read that the error in measurements are the smallest unit possible on the measurement apparatus": This, instead, is utterly wrong: wherever you read it, forget about it and throw the source in the fireplace. $\endgroup$ – Massimo Ortolano Sep 7 '16 at 17:58
  • $\begingroup$ www2.fiu.edu/~dbrookes/ExperimentalUncertaintiesCalculus.pdf Section 1.1 explains what I mean in case I didn't explain it clearly $\endgroup$ – RonGiant Sep 7 '16 at 18:00
  • $\begingroup$ Notice that your linked document, though incorrect in many points, doesn't say what you have reported in the question. $\endgroup$ – Massimo Ortolano Sep 7 '16 at 18:03
  • $\begingroup$ "Every measuring instrument has an inherent uncertainty that is determined by the precision of the instrument. This value is taken as a half of the smallest increment of the instrument scale. " and mine: "However, elsewhere I read that the error in measurements are the smallest unit possible on the measurement apparatus, so for example on a metre stick whose smallest unit is 1mm the error/uncertainty on any measurement would be 1mm (or 0.5mm it says elsewhere, I suppose it's the smallest unit you feel confident estimating)." $\endgroup$ – RonGiant Sep 7 '16 at 18:09
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There is a difference between uncertainty and error. Uncertainty is the maximum possible or maximum likely error. Error, is the actual error in the measurement.

In your meter stick example, the uncertainty is a matter of confidence in your measurement. Some will claim you can only measure to 1mm because that is the smallest increment on the device. In actuality, normal practice is to go one level lower, that is, it is normally considered you can accurately estimate one level more than the meter stick shows, so you can measure to the nearest tenth of a millimeter. Often to acknowledge this last digit was an estimate, you would then say the uncertainty was +/- 0.3mm rather than +/- 0.1mm. With any digital device you do not have this ability to extrapolate one more level, so you must take the reading directly from the device, then know the inherent uncertainty of the device. Just because a digital stop watch for instance gives a reading to .001 seconds, the device may actually count 0.003 at a time giving an uncertainty in the reading higher than the least significant digit. Those numbers though are the uncertainty in the device, the the actual error of the readings.

To actually calculate what the error is though, you go to a statistical analysis of the data, such as a mean and standard deviation. As an example, say you had 100 people measure the same object with the same meter stick. The uncertainty of the meter stick is the same for each, say you accept the +/- 0.3mm I proposed. But they will not all get the same measurement. Some will measure wrong. Some will estimate the last digit differently, etc. Most should be within that +/- 0.3mm though. You would then get your mean, and standard deviation to get what you considered the real measurement and calculated error in that measurement. Many times, you would also refine that by throwing out any measurements that were more than 0.3mm off from the others as measuring errors, or maybe those more than 2-standard deviations, etc. as you know everyone used the same device and measured the same object so numbers outside of that range are not real.

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