Inferential Statistics using the Coefficient of Variation ...

Academic Forum 26 2008-09
The small p-value is evidence that the variance in the lengths of the females is greater than that
of the males. However, doing a 2-sample t-test will provide evidence that female lynx spiders
are longer on average than male lynx spiders. Since the female spiders tend to be longer, it
would be more appropriate to compare the sex difference in length for the lynx spider using a
statistic that measures variation relative to length. The coefficient of variation is such a statistic
s
and is defined to be CV = 100 . For the above data,
x
CV M = 100*.663/5.92 = 11.2 and CV F = 100*1.19/8.15 = 14.6, so CV M < CV F ,
but is this difference significant? We will need to determine the distribution of the CV random
s
variable, so for simplicity the 100 in the formula will be dropped and we redefine CV = .
x
We will assume for the remainder of this paper that X
1
, X
2,...,
X
n
is a simple random sample
from N ( µ , σ ) where µ > 0.
Recall the following theorem:
1
Theorem X =
n
n
∑ X j
j=
1
2
( n −1)
S 2
~ χ ( n −1).
2
σ
1 ⎛ σ ⎞
S = ∑ X − X j
are independent, and X ~ N⎜µ,
⎟ and
n −1
j=
1
⎝ n ⎠
n
and ( ) 2
2
We will derive the distribution of the standard deviation. Recall that the probability density
function of ~ 2 2 2
u e
U χ ( n −1)
is fU
( u)
= , u > 0, n = 2,3,...
. Write
n−1
⎛ n −1⎞
2
Γ⎜
⎟2
⎝ 2 ⎠
S =
2
σ ( n −1)
S
⋅
2
( n −1)
σ
2
=
f ( s)
= f
S
σ
n −1
U
1
=
⎛ n −1⎞
Γ⎜
⎟2
⎝ 2 ⎠
n−2
s
=
⎛ n −1⎞
Γ⎜
⎟2
⎝ 2 ⎠
n−1
2
n−3
2
U
du
( u)
ds
n−3
u
−
. Apply the change-of-variable technique to obtain
⎡(
n −1)
s
⎢ 2
⎣ σ
2
⎡(
n −1)
⎤
⎢ 2 ⎥
⎣ σ ⎦
⎤
⎥
⎦
n−1
2
n−3
2
⎡ ( n −1)
s
exp⎢−
2
⎣ 2σ
⎡ ( n −1)
s
exp⎢−
2
⎣ 2σ
2
2
⎤
⎥
⎦
d
ds
⎤
⎥,
s > 0
⎦
⎡(
n −1)
s
⎢ 2
⎣ σ
2
⎤
⎥
⎦
20