For a set of multiple measurements, with a Gaussian distribution of errors, the arithmetic mean of the (multiple measurements) data is:
We now need to address the important question of the precision of our estimate of the mean. Obviously, the more measurements we make, the more precise our estimate of the mean should become. One estimate of the error on the mean is the Standard error:

However this by itself does not take into account the number of measurements, it simply calculates the error assuming that the number of measurements represents an accurate sample of how the measured values vary about the true value. Obviously this assumption is more likely to be invalid the fewer the number of measurements. Therefore the error usually cited is the 95% Confidence Level, defined as t₉₅*s_m, where the value of t_95% depends on (n-1), which is known as the number of degrees of freedom. The value of t_95% to use (for your particular degree of freedom) is read from a t-Values Table.

It is not always possible to make multiple observations of a given observable because of, say, lack of time or insufficient quantities of reagents. In these situations, the error on a given measurement may be estimated using common sense, e.g. a distance measured on a metre ruler is arguably precise to ± 1 mm. Such subjective estimates of uncertainty must be made and recorded at the time of measurement; they depend on details of scale marking and on the eyesight and dexterity of the observer, and cannot satisfactorily be made after leaving the laboratory.