1.2: Sampling Error (Repeated Measurements)

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In some experiments, the system being measured is intrinsically random. For example, radioactive decay occurs randomly, so counting the number of decay events yields a slightly different result from interval to interval, no matter how precise your apparatus is. Often, we find a mean value by taking many measurements and averaging them. The result is reported in the format \[\textrm{(estimated mean)} \,\pm\, \textrm{(standard error of the mean)}.\] The quantity after the \(\pm\) refers to sampling error, which is not the same as measurement error. Sampling error arises because you only took a finite number of samples, so your estimate of the mean is not completely certain.

In this type of scenario, measurement error is usually ignored, as the sampling error caused by the system’s randomness is larger than the measurement error. (In the opposite case, where measurement error is larger, there’d be no point doing repeated measurements. And if the measurement error is about as large as the system’s randomness, you’re probably doing social science, so good luck with that.)

Suppose you take \(N\) measurements, and the results are \(X_1, X_2, \dots, X_N\). Then \[\textrm{estimated mean} \equiv \bar{X} = \frac{X_1 + X_2 + \cdots + X_N}{N}.\] You can compute \(\bar{X}\) using the mean function in Python or Matlab. Also, \[\textrm{standard error of the mean} = \frac{\sigma}{\sqrt{N}},\] where \(\sigma\) is the standard deviation of the sample, defined as \[\sigma = \sqrt{\frac{\sum_{n=1}^N(X_n-\bar{X})^2}{N-1}}.\] You can compute \(\sigma\) using the std function in Python or Matlab.