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154 Chapter 3 Review of Statistics

A p p e n d i x

3.3 A Proof That the Sample Variance

Is Consistent

This appendix uses the law of large numbers to prove that the sample variance sY2 is a con-
sistent estimator of the population variance sY2 , as stated in Equation (3.9), when
Y1, c, Yn are i.i.d. and E(Y4i ) 6 ∞ .

First, consider a version of the sample variance that uses n instead of n − 1 as a divisor:

1 n (Yi - Y)2 = 1 n Y2i - 2Yn1 n Yi + Y2
n n
ia= 1 ia= 1 ia= 1

= 1 n - Y 2
n
ia= 1Yi2

¡p (sY2 + mY2 ) - mY2

= sY2 , (3.29)

where the first equality uses (Yi - Y)2 = Yi2 - 2YYi + Y 2, and the second uses n1 g n 1Yi = Y.
i=

The convergence in the third line follows from (i) applying the law of large numbers to

n1 g n 1Yi2 ¡p E(Y2) (which follows because Y2i are i.i.d. and have finite variance because
i=

E(Yi4) is finite), (ii) recognizing that E(Yi2) = sY2 + m2Y (Key Concept 2.3), and (iii) noting

Y ¡p mY so that Y 2 ¡p m2Y. Finally, s2Y = 1n n 1 2 1n1 g n 1(Yi - Y)22 ¡p s2Y follows
- i=

from Equation (3.29) and 1n n 12 S 1.
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