THEORY OF STATISTICS. 
the same way, which renders many statistical results based on 
samples so dubious. 
5. Thus in collecting returns as to family income and expendi- 
ure from working-class households, the families with lower 
a are almost certain to be under-represented ; they largel 
‘“ escape the sampler’s fingers” from their simple lack of abilit 
o keep the necessary accounts. It is almost impossible to say, 
however, to what extent they are under-represented, or to for 
any estimate as to the possible error when two such samples 
aken by different persons at different times, or in different places, 
are compared. Again, if estimates as to crop-production are 
formed on the basis of a limited number of voluntary returns, 
the estimates are likely to err in excess, as the persons who 
make the returns will probably include an undue proportio 
of the more intelligent farmers whose crops will tend to be 
above average. Whilst voluntary returns are in this way liable 
to lead to more or less unrepresentative samples, compulsor 
sampling does not evade the difficulty. Compulsion could not en- 
sure equally accurate and trustworthy returns from illiterate 
and well-educated workmen, from intelligent and unintelligent 
armers. The following of some definite rule in drawing the 
sample may also produce unrepresentative samples: if samples 
of fruit were taken solely from the top layers of baskets expose 
or sale, the results might be unduly favourable; if from th 
ottom layer, unduly unfavourable. 
6. In such cases we can see that any sample, taken in the 
way supposed, is likely to be definitely biassed, in the sens 
hat it will not tend to include, even in the long run, equa 
roportions of the 4’s and o’s in the original material. In othe 
ases there may be no obvious reason for presuming such bas, 
ut, on the other hand, no certainty that it does not exist. Thus 
if we noted the hair-colours of the children in, say, one 
ol in ten in a large town, the question would arise whethe 
his method would tend to give an unbiassed sample of all th 
hildren. No assured answer could be given: conjectures o 
he matter would be based in part on the way in which the 
chools were selected, e.g. the volunteering of teachers for the work 
might in itself introduce an element of bias. Again, if say 
0,000 herrings were measured as landed at various North Se 
ports, and the question were raised whether the sample was 
likely to be an unbiassed sample of North Sea herrings, no 
assured answer could be given. There may be no definite reaso 
for expecting definite bias in either case, but it may exist, an 
no mere examination of the sample itself can give any informa 
WE to whether it exists or no. 
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