T.DIST.RT

What it does

T.DIST.RT(x, deg_freedom)

T.DIST.RT(x, deg_freedom) returns the right-tail area of Student's t-distribution with ν degrees of freedom:

T.DIST.RT(x, ν) = P(T > x) = 1 − F(x; ν)

where F is the cumulative distribution function described on T.DIST. In terms of the regularized incomplete beta function the identity is exact and, importantly, direct — the quantity is computed as itself rather than as one minus something:

T.DIST.RT(x, ν) = ½ · I_{ν/(ν+x²)}(ν/2, ½)          for x ≥ 0
T.DIST.RT(x, ν) = 1 − ½ · I_{ν/(ν+x²)}(ν/2, ½)      for x ≤ 0

Domain and range: x over all reals, ν > 0; the value decreases strictly from 1 (as x → −∞) to 0 (as x → +∞), passing through exactly ½ at x = 0. No poles, no branch cuts.

The asymptotics are the reason this surface is separate from T.DIST rather than a convenience wrapper. As x → ∞,

P(T > x) ~ ( Γ((ν+1)/2) · ν^(ν/2) ) / ( sqrt(νπ) · Γ(ν/2) · ν/2 ) · x^(−ν)

— a power law. The probability decays polynomially, so it stays representable as a double far out into the tail, and it stays informative only if it is never formed by subtracting a near-one quantity from one. A right-tail surface that internally computes 1 − T.DIST(...) throws away the entire reason it exists.

Special cases with elementary closed forms, useful as exactness probes:

Arguments

Microsoft's T.DIST.RT page documents no restriction on the sign of x, which is a real difference from the legacy TDIST spelling. TDIST documents x ≥ 0 and returns #NUM! below it. If Excel honours the modern page as written, T.DIST.RT(−1, 5) is a number greater than a half. That is a specific, cheap probe, and it appears on the list below.

The page describes deg_freedom as an integer but does not say what a non-integer value does. The T.INV and T.INV.2T pages document truncation for the same parameter; these three T.DIST* pages do not.

The numeric slots take ordinary to-number coercion; see Coercion and lifting.

Try it

Quirks

Result and edge cases

Returns Number, a probability in (0, 1).

• x = 0 gives exactly ½ for every ν, by symmetry.
• Large positive x gives a small probability that remains meaningful — the power-law tail means T.DIST.RT keeps returning distinguishable values in a range where 1 − T.DIST(x, ν, TRUE) has already collapsed to zero. This is the intended use.
• Large negative x gives a value approaching 1 from below, where the other member of the pair (T.DIST, cumulative) carries the accuracy instead. The two surfaces are accurate in opposite halves of the line; neither is a superset of the other.
• The complement identity. T.DIST.RT(x, ν) + T.DIST(x, ν, TRUE) = 1 and T.DIST.RT(−x, ν) = T.DIST(x, ν, TRUE) are exact mathematically. Whether Excel's published bits satisfy them is exactly the kind of metamorphic question a suite should settle, and no suite exists.

Note also that this surface is one of the two on this page's module carrying an open upstream defect stream, BUG-FUNC-021, whose title names a statistical numeric exactness drift. The Handbook cites the stream by name; it does not restate its contents or its figures.

Errors

As documented on Microsoft's T.DIST.RT page:

No documented #NUM! for negative x — again, unlike the legacy TDIST. Errors in either argument propagate under the universal coercion rule. The Handbook has not verified any of this against Excel.

How to implement

Numerical notes

The whole engineering content of a right-tail distribution function is: never form it by subtraction. Written as 1 − F(x; ν) in double precision, every value below about 2^-53 relative to one is gone, so the surface would saturate at zero for moderate x and stay there. Computed directly as ½ I_{ν/(ν+x²)}(ν/2, ½), it stays accurate to relative precision as far out as the power-law tail goes.

The reduction to the incomplete beta is the standard one; the continued fraction of Numerical Recipes' betacf, with the reflection I_z(a,b) = 1 − I_{1−z}(b,a) applied so that the fraction is always evaluated on its fast-converging side, is the canonical route. Cephes' stdtr and Boost.Math's students_t_distribution (built on ibeta/ibetac, where the ibetac complement entry point exists for exactly this reason) are the two implementations to read. The existence of a separate complement entry point in a serious library is the same design decision Excel made by publishing T.DIST.RT as its own function.

For integer ν, A&S 26.7.3–26.7.4 give finite closed forms in arctan and powers of sine and cosine, and there is a companion recursion for the tail area. They are exact in exact arithmetic and lose digits for large ν in floating point; a switch on integer ν is defensible but has to name its crossover.

Argument reduction has one subtlety worth stating: the beta argument z = ν/(ν + x²) needs care when x² overflows or when ν is huge. Computing it as 1/(1 + x²/ν) — or, better, keeping the complement x²/(ν + x²) and calling the complementary beta — avoids both the overflow and the loss of the small quantity.

None of this is a claim about Excel's internals. The Handbook does not know Excel's algorithm.

Rust code

41 t_dist_rt "T.DIST.RT" (2, 2) [V "x", V "deg_freedom"] |s, a, sv| {
    let [x, df] = nums::<2>(s, a)?;
    out(s, cft::t_dist_rt_kernel(x, df))
}

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

Relationships

• TDIST is the legacy spelling, and this is the one place in the t family where the Handbook holds recorded evidence about the pairing. The Handbook's alias-pairing record EV-DIST-0011 places TDIST and T.DIST.RT among five legacy/modern forward-CDF pairs found to publish the same results everywhere the recorded battery probed them. Two cautions travel with that, and the record states both: it is an Excel-versus-Excel identity, so it transfers no pass rate onto either surface, and it covers forward CDFs only — no inverse pair has ever been probed for identity. Note also that TDIST(x, ν, 1) is the corresponding call: TDIST carries a tails argument, and tails = 2 corresponds to T.DIST.2T instead.
• T.DIST is the left-tail complement and the density surface.
• T.DIST.2T is the two-tail area; for x ≥ 0, T.DIST.2T(x, ν) = 2 · T.DIST.RT(x, ν) exactly. Whether the two Excel surfaces satisfy that in published bits is unprobed and is the sharpest cheap probe on the pair.
• T.INV inverts the left tail; T.INV.2T inverts the two-tail area. There is no T.INV.RT — the right-tail inverse is spelled T.INV(1 − p, ν), which itself throws away precision for small p and is a real gap in the surface set rather than an omission this page can paper over.
• T.TEST and the regression p-values are the main consumers of this surface in practice.

Named variants

No named variants are recorded yet.