T.INV

Microsoft publishes =T.INV(0.75,2) = 0.8164966 ("The left-tailed inverse of the Student's t-distribution with a probability of 75% and 2 degrees of freedom"). DERIVATION, AND THIS ONE IS EXACT: at two degrees of freedom the t CDF has the elementary closed form F(t) = 1/2 + t/(2*sqrt(2+t^2)), so F(t) = 3/4 solves to t/(2*sqrt(2+t^2)) = 1/4, i.e. 4t^2 = 2 + t^2 and t = sqrt(2/3) = 0.8164965809277260327324 -- an exact surd, no iteration and no library. The general root find on the cross-checked CDF used by the rest of this family lands on the same digits, and so does scipy.stats.t.ppf(0.75, 2). Microsoft's page was read live on 2026-08-31 at https://support.microsoft.com/en-us/excel/functions/t-inv-function. FETCH NOTE FOR THIS BATCH: batch F recorded every /en-us/office/<name>-function-<guid> URL returning Microsoft's 'Sorry, the page you're looking for can't be found' body; today BOTH paths serve the article again -- the GUID form redirects to /en-us/excel/functions/<slug>-function -- and every page cited in this batch came back complete (217-223 KB) in one or two attempts, with no 87 KB stub responses. Nothing here is sourced from a search snippet or a mirror.

Asserted at twelve places from the exact surd sqrt(2/3): 0.816496580928. Seven published decimals cannot see an iterative inverse that stops one iteration early.

The t distribution is symmetric about zero, so its 50th percentile is exactly 0 for every degrees of freedom. A structural assertion with no derived constant, and the sharpest single test of an inverse's centring: an engine whose root finder is biased returns a small non-zero number here.

T.DIST(T.INV(p, v), v, TRUE) = p by definition of an inverse. A structural round trip with no derived constant, run at a probability below one half so it exercises the negative-x branch of both functions. It is the strongest available check that the two are inverses OF EACH OTHER in the same engine, which no single published value can establish.

T.INV IS LEFT-TAILED AND T.INV.2T IS TWO-TAILED, and this case makes the difference impossible to pass by accident. The left-tailed 95th percentile at 10 degrees of freedom is 1.812461122811676413626, derived by root find on the cross-checked CDF; the two-tailed T.INV.2T(0.95,10) is 0.0642981 and T.INV.2T(0.05,10) is 2.2281389. Microsoft publishes no figure for T.INV(0.95,10) -- the value is ours -- but it is the same number the T.INV.2T and TINV pages MEANT to publish for TINV(2*0.05,10), which is exactly the documented relationship between the two conventions and is why this probability was chosen. Those pages print 1.812462; the correctly rounded value is 1.812461, and the error is recorded in T.INV.2T.json and TINV.json.

The Remarks publish: "If probability <= 0 or if probability > 1, T.INV returns the #NUM! error value." Note the asymmetry, which is Microsoft's: zero is excluded and ONE IS NOT, so probability = 1 is not listed as an error even though the left-tailed inverse there is unbounded.

The Remarks publish: "If deg_freedom < 1, T.INV returns the #NUM! error value."

Microsoft publishes =T.INV(0.75,2) = 0.8164966 ("The left-tailed inverse of the Student's t-distribution with a probability of 75% and 2 degrees of freedom"). DERIVATION, AND THIS ONE IS EXACT: at two degrees of freedom the t CDF has the elementary closed form F(t) = 1/2 + t/(2*sqrt(2+t^2)), so F(t) = 3/4 solves to t/(2*sqrt(2+t^2)) = 1/4, i.e. 4t^2 = 2 + t^2 and t = sqrt(2/3) = 0.8164965809277260327324 -- an exact surd, no iteration and no library. The general root find on the cross-checked CDF used by the rest of this family lands on the same digits, and so does scipy.stats.t.ppf(0.75, 2). Microsoft's page was read live on 2026-08-31 at https://support.microsoft.com/en-us/excel/functions/t-inv-function. FETCH NOTE FOR THIS BATCH: batch F recorded every /en-us/office/<name>-function-<guid> URL returning Microsoft's 'Sorry, the page you're looking for can't be found' body; today BOTH paths serve the article again -- the GUID form redirects to /en-us/excel/functions/<slug>-function -- and every page cited in this batch came back complete (217-223 KB) in one or two attempts, with no 87 KB stub responses. Nothing here is sourced from a search snippet or a mirror.

Asserted at twelve places from the exact surd sqrt(2/3): 0.816496580928. Seven published decimals cannot see an iterative inverse that stops one iteration early.

The t distribution is symmetric about zero, so its 50th percentile is exactly 0 for every degrees of freedom. A structural assertion with no derived constant, and the sharpest single test of an inverse's centring: an engine whose root finder is biased returns a small non-zero number here.

T.DIST(T.INV(p, v), v, TRUE) = p by definition of an inverse. A structural round trip with no derived constant, run at a probability below one half so it exercises the negative-x branch of both functions. It is the strongest available check that the two are inverses OF EACH OTHER in the same engine, which no single published value can establish.

T.INV IS LEFT-TAILED AND T.INV.2T IS TWO-TAILED, and this case makes the difference impossible to pass by accident. The left-tailed 95th percentile at 10 degrees of freedom is 1.812461122811676413626, derived by root find on the cross-checked CDF; the two-tailed T.INV.2T(0.95,10) is 0.0642981 and T.INV.2T(0.05,10) is 2.2281389. Microsoft publishes no figure for T.INV(0.95,10) -- the value is ours -- but it is the same number the T.INV.2T and TINV pages MEANT to publish for TINV(2*0.05,10), which is exactly the documented relationship between the two conventions and is why this probability was chosen. Those pages print 1.812462; the correctly rounded value is 1.812461, and the error is recorded in T.INV.2T.json and TINV.json.

The Remarks publish: "If probability <= 0 or if probability > 1, T.INV returns the #NUM! error value." Note the asymmetry, which is Microsoft's: zero is excluded and ONE IS NOT, so probability = 1 is not listed as an error even though the left-tailed inverse there is unbounded.

The Remarks publish: "If deg_freedom < 1, T.INV returns the #NUM! error value."

Microsoft publishes =T.INV(0.75,2) = 0.8164966 ("The left-tailed inverse of the Student's t-distribution with a probability of 75% and 2 degrees of freedom"). DERIVATION, AND THIS ONE IS EXACT: at two degrees of freedom the t CDF has the elementary closed form F(t) = 1/2 + t/(2*sqrt(2+t^2)), so F(t) = 3/4 solves to t/(2*sqrt(2+t^2)) = 1/4, i.e. 4t^2 = 2 + t^2 and t = sqrt(2/3) = 0.8164965809277260327324 -- an exact surd, no iteration and no library. The general root find on the cross-checked CDF used by the rest of this family lands on the same digits, and so does scipy.stats.t.ppf(0.75, 2). Microsoft's page was read live on 2026-08-31 at https://support.microsoft.com/en-us/excel/functions/t-inv-function. FETCH NOTE FOR THIS BATCH: batch F recorded every /en-us/office/<name>-function-<guid> URL returning Microsoft's 'Sorry, the page you're looking for can't be found' body; today BOTH paths serve the article again -- the GUID form redirects to /en-us/excel/functions/<slug>-function -- and every page cited in this batch came back complete (217-223 KB) in one or two attempts, with no 87 KB stub responses. Nothing here is sourced from a search snippet or a mirror.

Asserted at twelve places from the exact surd sqrt(2/3): 0.816496580928. Seven published decimals cannot see an iterative inverse that stops one iteration early.

The t distribution is symmetric about zero, so its 50th percentile is exactly 0 for every degrees of freedom. A structural assertion with no derived constant, and the sharpest single test of an inverse's centring: an engine whose root finder is biased returns a small non-zero number here.

T.DIST(T.INV(p, v), v, TRUE) = p by definition of an inverse. A structural round trip with no derived constant, run at a probability below one half so it exercises the negative-x branch of both functions. It is the strongest available check that the two are inverses OF EACH OTHER in the same engine, which no single published value can establish.

T.INV IS LEFT-TAILED AND T.INV.2T IS TWO-TAILED, and this case makes the difference impossible to pass by accident. The left-tailed 95th percentile at 10 degrees of freedom is 1.812461122811676413626, derived by root find on the cross-checked CDF; the two-tailed T.INV.2T(0.95,10) is 0.0642981 and T.INV.2T(0.05,10) is 2.2281389. Microsoft publishes no figure for T.INV(0.95,10) -- the value is ours -- but it is the same number the T.INV.2T and TINV pages MEANT to publish for TINV(2*0.05,10), which is exactly the documented relationship between the two conventions and is why this probability was chosen. Those pages print 1.812462; the correctly rounded value is 1.812461, and the error is recorded in T.INV.2T.json and TINV.json.

The Remarks publish: "If probability <= 0 or if probability > 1, T.INV returns the #NUM! error value." Note the asymmetry, which is Microsoft's: zero is excluded and ONE IS NOT, so probability = 1 is not listed as an error even though the left-tailed inverse there is unbounded.

The Remarks publish: "If deg_freedom < 1, T.INV returns the #NUM! error value."