iv.] SCHEMES OF DISTRIBUTION AND OF FREQUENCY. 45 



limits, but they will never actually touch them. A chess 

 board has eight squares in a row, and eight pieces may be 

 arranged in order on any one row, each piece occupying 

 the centre of a square. Let the divisions in the row be 

 graduated, calling the boundary to the extreme left, 

 0. Then the successive divisions between the squares 

 will be 1, 2, 3, up to 7, and the boundary to the 

 extreme right will be 8. It is clear that the position of 

 the first piece lies half-way between the grades (in a 

 scale of eight grades) of and 1 ; therefore the grade 

 occupied by the first piece would be counted on that 

 scale as 0'5; also the grade of the last piece as 7 '5. 

 Or again, if we had 800 pieces, and the same number 

 of class-places, the grade of the first piece, in a scale 

 of 800 grades, would exceed the grade 0, by an amount 

 equal to the width of one half-place on that scale, 

 while the last of them would fall short of the 800th 

 grade by an equal amount. This half-place has to be 

 attended to and allowed for when schemes are con- 

 structed from comparatively few observations, and 

 always when values that are very near to either of the 

 centesimal grades or 100 are under observation ; 

 but between the centesimal grades of 5 and 95 the 

 influence of a half class-place upon the value of the 

 corresponding observation is insignificant, and may be 

 disregarded. It will not henceforth be necessary to 

 repeat the word centesimal. It will be always implied 

 when nothing is said to the contrary, and nothing 

 henceforth will be said to the contrary. The word will 

 be used for the last time in the next paragraph. 



