12 BULLETIN" 529, U. S. DEPARTMENT OF AGRICULTURE. 
by the Office of Farm Management. The latter contained a list of 
implements, and the farmers to whom it was sent were asked to state 
in connection with each item whether he owned the implement named 
and whether he recommended its purchase by farmers. The partic- 
ular tabulation with which we are concerned here included only those 
farmers recommending the purchase, the object being to ascertain 
what proportion of them had acted on their own recommendations. 
The blanks used in tabulating the replies had spaces for entering TOO 
replies relating to a particular implement. The replies when entered 
in several cases filled two or more pages of the blank. After the 
answers had all been recorded, it was noticed that where the replies 
relating to a particular implement filled more than one page, the pro- 
portion of farmers owning the implement among those recommending 
its purchase was nearly the same on each separate page. Table IX 
has been constructed to show this interesting operation of the laws of 
chance. Take, for instance, the figures relating to the emery wheel 
(see Table IX). Of the 1,400 replies relating to it 976 were from 
farmers owning this implement. It happened that in tabulating the 
replies, exactly half of the owners were recorded on each of the two 
pages. Since this fact was not noticed until the tabulations had been 
completed, and since the replies were handled without any thought of 
the matter here under discussion, this perfectly even distribution of 
the 976 owners between the two arbitrary groups of 700 can only be 
ascribed to pure chance, or as nearly pure chance as can be imagined. 
It was an even chance whether any particular owner's reply should 
be recorded on the first page of the blank or on the second ; hence half 
fell on one page and half on the other. 
That this result is not wholly capricious but is really clue to the 
operation of a law is shown by every other case where two full pages 
of the blank were filled. There are nine such cases in the table. In 
no case where an even chance existed does the number of owners 
recorded on a page exceed 52 per cent or fall below 18 per cent of the 
total number of owners on the two pages. 
The figures relating to the set of stocks and dies are an excellent 
illustration of the importance of numbers in arriving at an average. 
The replies in this case filled slightly more than two pages. On the 
first page, containing 700 replies, 52.71 per cent were from farmers 
owning a set of stocks and dies. On the second page the percentage 
is 52.43, or practically the same. But on the third page, where there 
are only six replies, 83 J per cent are from owners. Six is too small 
a number to give a reliable average in such a case. 
In the case of most of the implements of the list there was one 
page of the tabulating blank only partially filled. In all these cases, 
excepting only the one just mentioned, the proportion of owners is 
