We have to measure a period of an oscillation. We are to take the time it takes for 50 oscillations multiple times.
I know that I will have a $\Delta t = 0.1 \, \mathrm s$ because of my reaction time. If I now measure, say 40, 41 and 39 seconds in three runs, I will also have standard deviation of 1.
What is the total error then? Do I add them up, like so?
$$\sqrt{1^2 + 0.1^2}$$
Or is it just the 1 and I discard the (systematic?) error of my reaction time?
I wonder if I measure a huge number of times, the standard deviation should become tiny compared to my reaction time. Is the lower bound 0 or is it my reaction time with 0.1?
Answer
I think you're exercising an incorrect picture of statistics here - mixing the inputs and outputs. You are recording the result of a measurement, and the spread of these measurement values (we'll say they're normally distributed) is theoretically a consequence of all of the variation from all different sources.
That is, every time you do it, the length of the string might be a little different, the air temperature might be a little different. Of course, all of these are fairly small and I'm just listing them for the sake of argument. The point is that the ultimate standard deviation of the measured value $\sigma$ should be the result of all individual sources (we will index by $i$), under the assumption that all sources of variation are also normally distributed.
$$\sigma^2 = \sum_i^N{\sigma_i^2}$$
When we account for individual sources of variation in an experiment, we exercise some model that formalizes our expectation about the consistency of the experiment. Your particular model is that the length of the string (for instance) changes very little trial after trial compared to the error introduced by your stopwatch timing. Unless we introduce other errors, this is claiming $N=1$, and if the standard deviation of your reaction timing contributes $0.1 s$ to the standard deviation of the measurement, then theoretically the measurement should have that standard deviation as well.
If this conflicts with the statistics of the time you actually recorded, then the possible ways to account for this include:
- Your reaction time isn't as good as you thought it was
- There are other sources of experimental error
I would favor the latter, although it could be a combination of both of them.
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