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where, in the last expression, the constant % has been replaced by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 symmetrical
 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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