T.INV in Excel: Find a Signed Left-Tail Student t Percentile

=T.INV(probability, deg_freedom)

Both arguments are required. probability is a left-tail cumulative fraction and deg_freedom is the degree count. The result is a signed t-scale quantile.

Probabilities below one half correspond to negative quantiles; those above one half correspond to positive quantiles. The median is zero. A negative output is not an error merely because a critical-value table often prints positive magnitudes.

T.INV reverses T.DIST in cumulative=TRUE mode. It does not invert density. T.INV.2T instead accepts a combined outside-tail probability and ordinarily returns a nonnegative magnitude.

Put 0.025, 0.1, 0.5, 0.9 and 0.975 in A2:A6. In B2 enter =T.INV(A2,5) and fill down. Independent targets are -2.5705818356, -1.4758840488, 0, 1.4758840488 and 2.5705818356.

The values increase down the table. At complementary probabilities p and 1−p, the quantiles are equal in magnitude with opposite signs. =B2+B6 and =B3+B5 each target zero mathematically.

Keep input probabilities and output quantiles labeled separately. A t quantile can exceed one in magnitude; that does not violate a probability bound because the output is not a probability.

In C2 enter =T.DIST(B2,5,TRUE) and fill down. Independent targets are the original A probabilities.

Retain B’s full stored results. Retyping the rounded printed values can create a small residual that is unrelated to the native inverse’s accuracy. FALSE in the forward function would calculate density and would not recover A.

The dedicated Microsoft example uses =T.INV(0.75,2), independently 0.8164965809. It uses two degrees rather than five, so it should not be compared directly with a row from the five-degree table.

A 5% right-tail threshold at five degrees is =T.INV(0.95,5), independently 2.0150483733. Its right-tail probability is 0.05.

A symmetric two-sided 5% boundary uses =T.INV(0.975,5), independently 2.5705818356, because each outside tail receives 0.025. The corresponding lower boundary is T.INV(0.025,5).

Using probability 0.05 directly would produce a negative lower-tail percentile. Tail labels should therefore travel with formulas when importing them from another workbook or a printed table.

The endpoints at the 2.5th and 97.5th percentiles enclose 0.95 of the five-degree t distribution. Verify by subtracting their cumulative probabilities.

The interval is on the standardized t scale. A confidence interval for a mean additionally requires a center, the correct standard error and assumptions that justify the t reference model. The inverse function cannot supply those missing elements.

For example, with sample size six, five degrees may arise from a single-mean calculation. That relationship is a feature of that statistical procedure, not an automatic inference made by T.INV from its arguments.

At one degree, the t distribution is Cauchy. Its interior quantile can be calculated as =TAN(PI()*(p-0.5)). At p=0.25, 0.5 and 0.75, mathematical targets are -1, 0 and 1.

Use this only as a moderate-input mathematical control for one degree. Near endpoints, tangent values become large and floating-point precision matters. It is not a substitute for a general t inverse at other degrees.

Microsoft documents #VALUE! for nonnumeric inputs, #NUM! for probability≤0 or probability>1, and #NUM! for degrees below one. Noninteger degrees are truncated.

The dedicated remarks do not exclude probability one, even though the mathematical upper-end quantile is unbounded. Actual native handling at that endpoint is not established here; all finite examples use 0<p<1. Do not silently change the documented inequality to a stricter one and label it an observed Excel result.

Use 97.5% or 0.975 rather than 97.5. Validate justified integer degrees deliberately. These checks do not certify all text coercions, native convergence paths or extreme-tail accuracy.

• Current documentation: the dedicated Microsoft reference lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016, Excel Web App.
• Introduced: Microsoft’s alphabetical catalog marks T.INV as introduced in Excel 2010. Exact first build and platform-specific introduction dates remain unverified.
• Last functioning version: unknown; a current help listing is not a last-working-build test.
• Evidence checked: October 15, 2026. Official documentation was freshly retrieved; examples were independently calculated. No native Excel, Mac or web-client execution was performed.

• Microsoft T.INV reference.
• Microsoft function catalog and introduction-marker explanation.
• Editorial testing standards.

Examples use English function names and comma separators. Expected numbers are rounded independent mathematical calculations; documented error expectations are labeled as such. They are not captured Excel output. All illustrative data and policies are fictional.