THEORY OF STATISTICS. 
Now let us approximate by assuming, as suggested in § 8, that 
k is very large, and indeed large compared with z, so that (x/k)® 
may be neglected compared with (x/k). This assumption does 
not involve any difficulty, for we need not consider values of x 
much greater than three times the standard-deviation or 3 JE2, 
and the ratio of this to % is 3/ ~ 2k, which is necessarily small if £ 
be large. On this assumption we may apply the logarithmic 
series 
S20 ST et 
log,(1+68)=20 TEE ad 
to every bracket in the fraction (3), and neglect all terms beyond 
the first. To this degree of approximation, 
z 2 re 
logle= -(1+2+3+ .. +o- 0-7 
a=)» 
5 Zk 
22 
Ro 
Therefore, finally, 
22 2 
Rr SEE Ry (4) 
Yo=Yl = =e 
where, in the last expression, the constant % has been replaced by 
the standard-deviation o, for o2="£%/2. 
The curve represented by this equation is symmetrical about 
the point = 0, which gives the greatest ordinate y=y, Mean, 
median, and mode therefore coincide, and the curve is, in fact, that 
drawn in fig. 5, p. 89, and taken as the ideal form of the symmetri- 
cal frequency-distribution in Chap. VI. The curve is generally 
known as the normal curve of errors or of frequency, or the law 
of error. 
10. A normal curve is evidently defined completely by giving 
the values of y, and o and assigning the origin of x. If we 
desire to make a normal curve fit some given distribution as near 
as may be, the last two data are given by the standard-deviation 
and the mean respectively ; the value of g, will be given by the 
fact that the areas of the two distributions, or the numbers of 
observations which these areas represent, must be the same. 
This condition does not, however, lead in any simple and 
elementary algebraic way to an expression for y, though such 
a value could be found arithmetically to any desired degree 
of approximation. For it is evident that (1) any alteration in 
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