XVIL—SIMPLER CASES OF SAMPLING FOR VARIABLES. 353 
“To convert any standard error to the probable error multiply by 
the constant 0-674489 , . . . 
16. We need hardly restate once more the warnings given in 
Chap. XIV., and repeated in § 9 above, that a standard error can 
give no evidence as to the biassed or representative character of 
a sample, nor as to the magnitude of errors of observation, but 
we may, in conclusion, again emphasise the warnings given 
in §§ 1-3, Chap. XIV,, as to the use of standard errors when 
the number of observations in the sample is small. 
In the first place, if the sample be small, we cannot in general 
assume that the distribution of errors is approximately normal : 
it would only be normal in the case of the median (for which 
» and ¢ are equal) and in the case of the mean of a normal 
distribution. Consequently, if = be small, the rule that a 
range of three times the standard error includes the majority 
of the fluctuations of simple sampling of either sign does not 
strictly apply, and the “probable error” becomes of doubtful 
significance. 
Secondly, it will be noted that the values of o and Y, in (1), of 
Jn (2), and of o in (4) and (5), ie. the values that would be 
given for these constants by an indefinitely large sample drawn 
under the same conditions, or the values that they possess in 
the original record if the sample is unbiassed, are assumed to be 
known a priori. But this is only the case in dealing with the 
problems of artificial chance: in practical cases we have to use 
the values given us by the sample itself. If this sample is based 
on a considerable number of observations, the procedure is safe 
enough, but if it be only a small sample we may possibly mis- 
estimate the standard error to a serious extent. Following the 
procedure suggested in Chap. XIV., some rough idea as to the 
possible extent. of under-estimation or over-estimation may be 
obtained, e.g. in the case of the mean, by first working out the 
standard error of o on the assumption that the values for the 
necessary moments are correct, and then replacing o in the 
expression for the standard error of the mean by o + three times 
its standard error so obtained. 
Finally, it will be remembered that unless the number of 
observations is large, we cannot interpret the standard error of 
any constant in the inverse sense, 7.e. the standard error ceases 
to measure with reasonable accuracy the standard-deviation of 
true values of the constant round the observed value (Chap. 
XIV. § 3). If the sample be large, the direct and inverse 
standard errors are approximately the same. 
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