251dscr_B 8/22/01 (Open this document in 'Outline' view!)
Appendix B: Explanation of Sample Formulas (Not for student consumption)
1. The Sample Variance.
 x  x   n 1  x
it must be true that E x  x    E  x  nx   n  1
The Sample Variance is defined as s 2 
value of  2
1
n  1
1
2
2
2
2
2

 nx 2 . If s 2 has an expected
2
. We can assume, without loss
 
of generality that   E x   0. Under these conditions, the Variance is defined as  2  E x   2  E x 2 .
Thus E
 x   n
2
2
. An expression like x 2 
 x has terms like n1
1
2
x x . Because of the
2 1 2
n
independence assumption on the sample, all these terms have expected values of zero except for terms with
2
2
1
1
1
 1
x  2 E
x  2 n 2   2 . Thus
two identical subscripts and E x 2  E  2
n
n
n
 n
 
 
E
 x
2
  x  nEx   n
 nx 2  E
2
2
2
 
1
 n  2  n  1 2 .
n
2
2. The Third k Statistic.
n  1n  2 
n
If the third k statistic k 3 
x  x 3


n  1n  2 
n
x 3  3x
x
2
 2nx 3

If k 3 has an expected value of  3 , it must be true that
E
 x  x    E x
3
3
 3x
x
2

 2nx 3 
n  1n  2 
3.
n
We can assume, without loss of generality that   E x   0. Under these conditions, the skewness is
 
defined as  3  E x   3  E x 3 . Thus E
like
 x   n
3
3.
An expression like x 3 
1
n
3
 x
3
has terms
1
x1 x 2 x3 . Because of the independence assumption on the sample, all these terms have expected
n3
values of zero except for terms with three identical subscripts and
3
3
1
1
1
 1
x   3 . Thus
E x 3  E 3
x  3 E
x  3 n 3  2  3 . By the same reasoning E x
n
n
n
 n
 
E  x
3
 
 
 3x  x  2nx   E  x  3E x  x  2nEx 
 n 3  3 3  2n
2
1
n2
3

3
2
n
 3   n  3   3 

2
 
3
n  1n  2  .
n 2  3n  2
3 
3
n
n
3. And now, for considerable extra credit, what can you say about the
expected value of k 4 ?