F.INV.RT

What it does

F.INV.RT(probability, deg_freedom1, deg_freedom2)

F.INV.RT(probability, deg_freedom1, deg_freedom2) finds an F-distribution value from a right-tail probability. The first degree count belongs to the numerator and the second to the denominator.

Try it

=F.INV.RT(0.25,2,2) is about 3. With both degree counts two, the mathematical left CDF is x/(1+x) and the right tail is 1/(1+x).

Quirks

Degree counts are truncated. Nonnumeric arguments give #VALUE!; invalid counts or probabilities give #NUM!. The current shared inverse admits each truncated degree from one through 10^10 and requires an open probability interval 0<p<1.

The Microsoft F.INV.RT page documents a denominator degree count strictly below 10^10 and #N/A after 64 unsuccessful iterations. Keep the documentation and the current code’s boundary tests distinct: a source guard alone is not an additional Excel measurement.

A left-CDF probability and a right-tail probability select different quantiles. Subtracting an extremely small right-tail probability from one can round to one and lose the target entirely.

How to implement

The current right-tail candidate uses an accurate pair of beta arguments and solves the decreasing tail directly. Let D mean ordinary binary64 storage and I_z(a,b) the regularized lower beta ratio.

1. Truncate and validate numerator degrees d1 and denominator degrees d2; require 0<p<1.
2. At each nonnegative trial x, form num=D(d1*x) and den=D(d2+num). Compute z=D(d2/den) and w=D(num/den) separately, then the paired beta helper bratio(d2/2,d1/2,z,w).first. Do not construct w as 1-z.
3. Negate that tail to obtain an increasing function H and set target to -p. Start with lo=0,hi=1; double hi until H(hi) reaches the target, with at most 200 expansion checks.
4. Use the ordered-binary64 search: bisect the endpoint encodings until adjacent, and return the upper endpoint. The search is over the stored forward surface, not an exact real beta function.

For d1=d2=2 the mathematical tail is 1/(1+x). At p=0.25, solving 1/(1+x)=0.25 yields three.

The beta routine, its complement pair and the adjacent-value return are separate numerical stages. A beta-quantile transform x=d2*z/(d1*(1-z)) is another mathematical route with different stores. The source’s problem report retains unresolved distribution mismatches; neither route identifies every Excel approximation or certifies the full function.

Rust code

34 f_inv_rt "F.INV.RT" (3, 3) [V "probability", V "deg_freedom1", V "deg_freedom2"] |s, a, sv| {
    let [p, d1, d2] = nums::<3>(s, a)?;
    out(s, cft::f_inv_rt_kernel(p, d1, d2))
}

No implementation has been published with a Handbook test suite for this function.

Version differences

No same-input comparison between named Excel builds is recorded for this function here. The documented behaviour below is not a measurement of every Excel release.

Alternatives

F.DIST.RT supplies the forward upper tail. F.INV takes a left-CDF probability; F.TEST starts with sample data.

Named variants

No named variants are recorded yet.