On 17 Nov 2001, Myles Gartland wrote:
> In an F distribution, the critical value for the lower tail is the
> reciprocal of the the critical value of the upper tail (with the
> degrees of freedom switched).
>
> Why? I understand how to calculate it, but do not get why the math
> works.
Essentially for the same reason that in the normal distribution the
critical value for the lower tail is the negative of the critical value
for the upper tail.
Thnke about it. For F = V1/V2, where V1 and V2 are two variance
estimates with numbers of degrees of freedom n1 and n2 respectively,
the relevant F distribution is said to have "n1 and n2 degrees of
freedom", naming the numerator first and then the denominator.
For F = V2/V1, the relevant F distribution has n2 and n1 d.f. (hence the
interchange of the numbers of degrees of freedom to which you allude).
Notice that V2/V1 is the reciprocal of V1/V2. If V1/V2 is sufficiently
larger than 1 that the hypothesis of equal variances in the populations
can be rejected, then V2/V1 must be sufficiently smaller than 1 to permit
rejection. Hence the critical value for V2/V1 must be the reciprocal of
the critical value for V1/V2, and the d.f. are interchanged simply by the
choice of which direction to divide.
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Donald F. Burrill [EMAIL PROTECTED]
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