Repeated measurements provide information about random variability. If measurements $x_1,\ldots,x_n$ are independent observations under the same conditions, their sample mean $$\bar x=\frac1n\sum_{i=1}^n x_i$$ is an estimate of the underlying mean value.

The sample standard deviation $$s=\sqrt{\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar x)^2}$$ describes the scatter of individual observations. The uncertainty associated with estimating the mean from this scatter is smaller: $$u_{\bar x}=\frac{s}{\sqrt n}.$$ This is the standard error of the mean under the independent-sampling model.

For example, suppose five timing measurements in seconds are $$2.01,\ 2.04,\ 1.99,\ 2.02,\ 2.04.$$ Their mean is $2.020,\mathrm{s}$. If their sample standard deviation is about $0.021,\mathrm{s}$, the standard uncertainty of the mean is $$u_{\bar x}\approx\frac{0.021}{\sqrt5}\approx0.009,\mathrm{s}.$$ A compact result is therefore approximately $2.020\pm0.009,\mathrm{s}$ when the uncertainty is intended as one standard uncertainty from repeatability alone.

The $1/\sqrt n$ improvement has conditions. Correlated samples, drift, changing experimental conditions, and systematic effects can prevent repeated data from providing the expected gain. More digits or more repetitions do not repair a biased measurement model.

Repeated-measurement statistics quantify one source of uncertainty. A complete experiment may also need uncertainty contributions from calibration, resolution, environmental variables, fitted parameters, or reference values.