248 REPORT— 1881. 



To work out this case, take a base line AB, divide it iuto twelve 

 equal pai'ts, and erect an ordinate (see the dark lines in tLe diagram) in 

 the middle of each of them. The first ordinate will reach to 50 inches 

 (the lower part of the ordinate is suppressed to save space) ; the next 

 three will reach to 51 inches; the nest four to 52 inches, and so on 

 according to the tabular data. Erect ordinates of suitable heights (see 

 the light lines in the diagram) at each of the graduations, and draw 

 horizontal lines through the top of each group of dark lines until it 

 meets the light lines on either side of them. A figure is thus produced 

 which consists of a series of rectangles rising in equal steps. A curved 

 line (see the dotted line) which smooths off the corners of the rectangles, 

 is the curve upon which the deciles, &c., are to be measured, and the 

 broken line formed by joining the central points of the upper boundary 

 of each rectangle may be adopted as an equivalent to the curve without 

 material error. Tl^e ordinates at these central points are those that cor- 

 respond to the successive integral heights of 50, 51, 52, &c. inches. The 

 value of their corresponding abscissas is equal to half the number of the 

 dark lines in the rectangle in question ]}lus the number of dark lines in 

 all the previous rectangles. An inspection of the figure will show this 

 more readily tiian a verbal explanatioii. 



The calculation is very easily made by appending to the tabular data 

 in A and B three other columns, C, D, and E. Column C contains in 

 each line the sum of all the heights inferior to the number of inches 

 found in A upon the same line. D contains the halves of the entries in 

 B, and E contains the sura of the entries in and D, and consequently 

 gives the abscisste corresponding to the several integral inches. 



Example : to find the lower decile in the above instance. As we know 

 the abscissa of the decile, we proceed to find from column E the two entries 

 between which it lies, and we take the corresponding ordinates from A, 

 whence we find the decile itself by simple interpolation. As there are 

 twelve observations in the example, the abscissa of the decile is 1-2, 

 which lies between the tabular entries in B of O'S and 2'5, and these are 

 the abscissiB of 50 and 51 inches respectively. Therefore the decile is 

 equal to 50 inches plus a certain fraction of an inch, x, whose value may 

 be ascertained by a simple rule of three. Thus : — 



difierence between 2-5 and 05 : 1 inch :: difference between 1-2 and 



0'5 : X inches 



a;=0-35, and the required decile=50'35 inches. 



On a first glance at the tables, a very remarkable fact is manifest. It 

 is the uniformity of range at all the ages given in it. Let us begin with 

 height, as shown in Table VIII. ; the range between the upper and lower 

 fourths is as great at ] 1, or even at 8, years of age as it is at 22 or 40 

 years, and at the intermediate ages it is much the same, viz., about 3'3 

 inches. It might have been expected that the range would vary with the 

 average height, so that the fact of boys of 11 years of age having a 

 median or average height of 53"5 inches, and an interquartile range of 

 3*2 inches, would imply that men of 22 years, having a median height of 

 Q7'Q inches, would have an interquartile range of 4'4 inches, because 

 53'6 : 3-2 ::67"6 : 4'4. The interdecile range is equally constant. It seems 

 so difficult to conceive of variation otherwise than as a fraction of the 



