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Say I am measuring a quantity $x$ in physical system whose true value is approximately sinusoidal in time. I have an instrument to sample this quantity, for which the manufacturer gives an accuracy spec. Can I use this accuracy spec $(\Delta x?)$ to compute the uncertainty of the sample mean $\bar{x}$ of the quantity based on a finite amount of samples, $N$?

My guess is that the naive calculation would be

$$ \Delta{\bar{x}} = \frac{\Delta{x}}{\sqrt{N}}, $$

but this would assume the samples are all statistically independent. Can I use this assumption, or should $N$ be based on the number of periods sampled, or something else?

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  • $\begingroup$ How does the period of sinusoidal function compare to the time it takes to make a measurement of x? Can you fit the data to a sinusoidal function instead of just taking the mean? $\endgroup$
    – DavePhD
    Commented Mar 20, 2014 at 2:06
  • $\begingroup$ I am getting on the order of 1000 samples per period, sampling for about 10 periods. I don't have much desire to fit a sine function to the data, as I am ultimately interested in the mean value. I am just confused on how to report the error of the mean. $\endgroup$
    – petebachant
    Commented Mar 20, 2014 at 2:12
  • $\begingroup$ You haven't answer either of the important questions here. How does $\Delta x$ compare to the amplitude of the oscillation? And, how does the sampling rate compare to the frequency of oscillation? Without both those pieces of information no one can even begin to advise you, because these questions are answered by understanding the mathematics of the situation. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Mar 20, 2014 at 14:54
  • $\begingroup$ The amplitude of oscillation can range from $\approx 5 \Delta x$ to $\approx 100 \Delta x$, and as I said in the comment above, the sampling frequency is about 1000 times the oscillation frequency. $\endgroup$
    – petebachant
    Commented Mar 20, 2014 at 17:02
  • $\begingroup$ The trouble you have here is not knowing how the instrument responds when $\frac{\mathrm{d}x}{\mathrm{d}t}\Delta t > \Delta x$ ($\Delta t$ being the time it takes the instrument to fix a reading), which is possible with the parameters you state. I'm a bit beyond my comfort zone on this, but I'd guess that the way to be sure you have the right treatment with simple tools is to do an error weighted fit where the specified $\Delta x$ feeds into the error and then read the resulting parameters from the fit. You might ask on Cross Validated instead. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Mar 20, 2014 at 23:39

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I think you would underestimate the error if you used that calculation. For example, if you had a thermometer that the manufacturer said was accurate within 1 degree, and you took a reading every second for 100 years to calculate the average temperature in Chicago, the uncertainty in the average could still be about 1 degree, because the thermometer could be systematically low or high by 1 degree the whole time.

Some instruments, like pH meters, tend to drift in their calibration over time. Some instruments will vary in their inaccuracy over an amplitude range.

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  • $\begingroup$ In this case you need to distinguishing between the calibration error (roughly a constant offset) and the sampling error. Only a normal-distributed sampling error can be treated as the OP suggests. Then the systematic is added in quadrature and dominates the final result. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Mar 20, 2014 at 19:19

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