XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 303
Y, produces a proportionate alteration in the area of the curve,
e.g. doubling 7, doubles every ordinate 7, and therefore doubles
the area: (2) any alteration in o produces a proportionate
alteration in the area, for the values of y, are the same for the
same values of z/o, and therefore doubling o doubles the distance
of every ordinate from the mean, and consequently doubles the
area. The area of the curve, or the number of observations
represented, is therefore proportional to 7,0, or we must have
N=axy,o
where a is a numerical constant. The value of a may be found
approximately by taking 7, and o both equal to unity, calculating
the values of the ordinates y, for equidistant values of z, and
taking the area, or number of observations X, as given by the
sum of the ordinates multiplied by the interval.
11. The table below gives the values of y for values of x
proceeding by fifths of a unit ; the values are, of course, the same
for positive and negative values of z. For the whole curve the
sum of the ordinates will be found to be 1253318, the interval
being 0'2 units; the area is therefore, approximately, 2:50664,
SZ
Ordinates of the Curve y=e¢ 2. (For references to more extended
lables, see list on pp. 357-8.)
Log v. = Log v.
0 100000 0 26 "03405 253209
0-2 “98020 1-99131 28 ‘01984 229757
04 92312 126526 30 ‘01111 204567
hy ‘83527 192183 32 *00598 377641
72615 1-86103 "4 *00309 348978
60653 178285 3° ‘00153 318577
; "48675 168731 ood ‘00073 4-86439
. 87531 157439 40 *00034 452564
19 "27804 144410 4-2 *00015 4°16952
193 "19790 1:20644 44 ‘00006 579603
20 13534 113141 4-6 *00003 540516
2-2 08892 2794901 48 100001 699693
2:4 ‘05614 274923 50 00000 657132
and this is the approximate value of @. The value is more than
sufficiently accurate for practical purposes, for the exact value
is ~/2r=2506627..... The proof of this value cannot be given
here, but it may be deduced from an important approximate
expression for the factorials of large numbers, due to James
a in
Zz. Y. ,
