F.DIST

What it does

F.DIST(x, deg_freedom1, deg_freedom2, cumulative)

F.DIST(x, deg_freedom1, deg_freedom2, cumulative) returns an F-distribution left CDF (TRUE) or density (FALSE). The first degree count belongs to the numerator and the second to the denominator.

Try it

=F.DIST(3,2,2,TRUE) is about 0.75. With both degree counts two, the mathematical left CDF is x/(1+x) and the right tail is 1/(1+x). =F.DIST(3,2,2,FALSE) is about 0.0625, the density rather than a tail.

Quirks

Degree counts are truncated. Microsoft documents #VALUE! for nonnumeric arguments and #NUM! for negative x or degree counts below one. The current shared helper also rejects a nonfinite x and a truncated degree count above 10^10; that code cap is separate from the opened page’s lower-bound wording.

The left CDF rises from zero and the right tail falls from one. The density option in F.DIST is a third quantity, rather than either of those probabilities. At x=0 the current density gives #NUM! for numerator degree one, one for degree two and zero for higher degrees. Those density cases do not change the endpoint probabilities.

The current beta transformation forms the product of x and numerator degrees directly. Extreme products or beta/gamma approximations need their own numerical error and publication checks; a finite mathematical probability alone does not ensure this candidate evaluates it successfully.

How to implement

For the current cumulative route, use ordinary binary64 stores D:

1. Validate finite nonnegative x and the truncated positive degree counts d1 and d2.
2. Calculate num=D(d1*x) and den=D(num+d2).
3. Independently store z=D(num/den) and w=D(d2/den). Return the first result of bratio(d1/2,d2/2,z,w). Both beta arguments are supplied; w is not computed by subtracting z from one.
4. Apply worksheet number publication. The upper-tail function swaps the shapes and uses the complementary argument as its leading input.

For d1=d2=2 and x=3, num is six and den eight. Both beta shapes are one, so the uniform beta CDF gives z=6/8=0.75.

For the density at positive x, with a=d1/2,b=d2/2, the mathematical expression is (d1/d2)^a*x^(a-1)/(B(a,b)*(1+d1*x/d2)^(a+b)). The current code uses runtime powers and gamma products for that normalization, plus explicit x=0 cases. At d1=d2=2 the expression reduces to 1/(1+x)^2; x=3 gives density 0.0625, distinct from the right-tail probability 0.25.

A log-density or scaled-ratio implementation can improve numerical range, but changes this candidate’s stores and leaves. The paired beta wrapper does not solve every inner approximation; the problem report retains distribution mismatches.

Rust code

30 f_dist "F.DIST" (4, 4) [V "x", V "deg_freedom1", V "deg_freedom2", V "cumulative"] |s, a, sv| {
    let [x, d1, d2] = nums::<3>(s, a)?;
    out(s, cft::f_dist_kernel(x, d1, d2, flag(s, a, 3)?))
}

pub fn f_pdf_kernel(x: f64, deg1: f64, deg2: f64) -> Result<f64, WorksheetErrorCode> {
    let x = validate_nonnegative_x(x)?;
    let d1 = truncate_positive_integer(deg1)?;
    let d2 = truncate_positive_integer(deg2)?;
    if x == 0.0 {
        return if d1 < 2.0 {
            // Density diverges to +inf; Excel returns #NUM!
            Err(WorksheetErrorCode::Num)
        } else if d1 == 2.0 {
            Ok(1.0)
        } else {
            Ok(0.0)
        };
    }
    let half_d1 = d1 / 2.0;
    let half_d2 = d2 / 2.0;
    let num = crt_pow(d1 / d2, half_d1) * crt_pow(x, half_d1 - 1.0);
    let den = gamma(half_d1) * gamma(half_d2) / gamma(half_d1 + half_d2)
        * crt_pow(1.0 + d1 * x / d2, (d1 + d2) / 2.0);
    Ok(num / den)
}

pub fn f_dist_kernel(
    x: f64,
    deg1: f64,
    deg2: f64,
    cumulative: bool,
) -> Result<f64, WorksheetErrorCode> {
    let x = validate_nonnegative_x(x)?;
    let d1 = truncate_positive_integer(deg1)?;
    let d2 = truncate_positive_integer(deg2)?;
    if cumulative {
        // W109: Excel's beta wrappers pass the ACCURATE complement pair, not
        // 1-z (identified stagings; b22 check).
        let num = d1 * x;
        let den = num + d2;
        Ok(bratio(d1 / 2.0, d2 / 2.0, num / den, d2 / den).0)
    } else {
        f_pdf_kernel(x, d1, d2)
    }
}

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

Version differences

The recorded environments below distinguish a reading from a same-input comparison. Counts describe captured inputs in each environment; overlapping captures and comparisons are not totals of unique worksheet calls.

The record contains no same-input comparison between these environments. Reading different inputs on different builds does not establish either a version change or cross-build agreement.

Recorded environments and comparison details.

Alternatives

F.INV maps a left probability to a value. F.DIST.RT computes the upper tail directly, and F.TEST starts with two samples.

Named variants

No named variants are recorded yet.