If this conformity exists in a sufficient degree, then the mere statement of the probable error suffices to characterise the arrangement of all the observed values, and at the same time its quantity gives a serviceable measure for the compactness of the distribution around the central value—i.e., for its exactness and trustworthiness.
As we have spoken of the probable error of the separate observations, (P.E.o), so can we also speak of the probable error of the measures of the central tendency, or mean values, (P.E.m). This describes in similar fashion the grouping which would arise for the separate mean values if the observation of the same phenomenon were repeated very many times and each time an equally great number of observations were combined into a central value. It furnishes a brief but sufficient characterisation of the fluctuations of the mean values resulting from repeated observations, and along with it a measure of the security and the trustworthiness of the results already found.
The P.E.m is accordingly in general included in what follows. How it is found by calculation, again, cannot be explained here; suffice it that what it means be clear. It tells us, then, that, on the basis of the character of the total observations from which a mean value has just been obtained, it may be expected with a probability of 1 to 1 that the latter value departs from the presumably correct average by not more at the most than the amount of its probable error. By the presumably correct average we mean that one which would have been obtained if the observations had been indefinitely repeated. A larger deviation than this becomes improbable in the mathematical sense—i.e., there is a greater probability against it than for it. And, as a glance at the accompanying table shows us, the improbability of larger deviations increases with extreme rapidity as their size increases. The probability that the obtained average should deviate from the true one by more than 2½ times the probable error is only 92 to 908, therefore about 1/10; the probability for its exceeding four times the probable error is very slight, 7 to 993 (1 to 142).
