Advanced High-School Mathematics

SECTION 6.3 Parameters and Statistics 375

This shows why it’s best to take “large” samples: the “sampling

statistic”X has variance which tends to zero as the sample size

tends to infinity.

Thesample variance: this is defined by setting

s^2 x =

1

n− 1

∑n

i=1

(xi−x)^2.

Thesample standard deviationsx=

»

s^2 x.

IfX 1 , X 2 , ...,Xnrepresent independent random variables having the
same distribution, then setting

Sx^2 =

1

n− 1

∑n

i=1

(Xi−X)^2

is a random variable. Once the sample has been taken, this random
variable has taken on a value, Sx = sx and is, of course, no longer
random. The relationship betweenSxandsxis the same as the rela-
tionship betweenX(random variable before collecting the sample) and
x(the computed average of the sample).
You might wonder why we divide by n−1 rather than n, which
perhaps seems more intuitive. The reason, ultimately, is that

E(Sx^2 ) = E

Ñ

1

n− 1

∑n

i=1

(Xi−X)^2

é

=σ^2.

A sketch of a proof is given in the footnote.^21 (We remark in passing
that many authors do define the sample variance as above, except that

(^21) First of all, note that, by definition
E((Xi−μ)^2 ) =σ^2 ,
from which it follows that
E
(∑n
i=1
(Xi−μ)^2
)
=nσ^2.
Now watch this: