Two-way Fisher F-test for analysis of variance (Anova)
  Entirely free download of software

                                The calculator gives the probability value of an F-test, given the F-value and
                                degrees of freedom of numerator (DF1) and denominator (DF2), as well as
                                the value of the F-test given the probability.

                                                        Screenprint of the two-way Fisher F-test calculator:

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F-test calculator
F-distribution calculator





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            More examples of F-test graphics
            cumulative normal distribution               normal density distribution

MATHEMATICS

A numerical solution of Fisher's F- probability distribution is obtained when either DF1 or DF2 are even.
When both are even, use the smallest. When both are uneven (odd) an approximate solution is to be found.

DF1 is even

        Pc = 1 - sn/2 [ 1 + n.t / 2 + n (n+2) t2 / 8 + n (n+2) (n+4) t3 / 48 + n (n+2 ) (n+4) (n+6) t4 / 384 . . . . . ]

where:

s = n / (m+n.z), t = 1-s, m = DF1, n = DF2, z = reference value for the variable y following the F-distribution
      (like the F-test value itself),

Pc = cumulative probability that can also be represented by the probability P(y<z) that y is less than z.

The series of denominators 2, 8, 48, 384 . . . equals the series 2, 2x4, 2x4x6, 2x4x6x8 . . .

The number of terms between the parentheses [ ] to be used is n / 2.

DF2 is even

        Pc = tn/2 [ 1 + n.s / 2 + n (n+2) s2) / 8 + n (n+2) (n+4) s3 / 48 + n (n+2) (n+4) (n+6) s4 / 384 . . . . . ]


DF1 and DF2 are uneven (odd)

The above equations for Pc are, apart of z, a function of m and n, and can be represented as Pc(m,n).

When DF1=m and DF2=n are both uneven (odd), the cumulative probability Pc(m,n) can be approximated by non linear interpolation between Pc(m,n-1) and Pc(m,n+1).
The interpolation can be done with a weight factor (w):

        Pc(m,n) = { w.Pc(m,n+1) + Pc(m,n-1) } / (1+w)

Using w=3 one finds a reasonable approximation.


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