T.INV.2T in Google Sheets: Two-Tailed Critical Values
Find two-tailed t critical values with T.INV.2T in Google Sheets, use alpha rather than confidence as input and calculate a checked mean-interval margin.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: =T.INV.2T(0.05,9) has expected result 2.2621571628. It finds the positive threshold whose two outside tails together contain probability 0.05 for 9 degrees of freedom.
Syntax and probability meaning
=T.INV.2T(probability, degrees_freedom)
The Google T.INV.2T reference requires probability strictly between 0 and 1 and numeric degrees of freedom at least 1. Noninteger degrees of freedom are truncated to an integer. Both arguments are supplied; no defaults are documented.
The first argument is the combined two-tail probability, not the central confidence level. For a central 95% interval, enter 0.05. The remaining 95% lies between the negative and positive critical values.
Google explicitly identifies TINV as a synonym of T.INV.2T. It distinguishes T.INV, which uses the signed one-tail inverse convention. The period and suffix therefore matter.
Worked example: compare confidence levels
Enter 0.90, 0.95 and 0.99 in A2:A4 as desired central confidence levels. Put 9 in D1. In B2, enter and fill down:
=T.INV.2T(1-A2,$D$1)
| Central confidence | Combined outside probability | Expected positive critical value |
|---|---|---|
| 0.90 | 0.10 | 1.8331129327 |
| 0.95 | 0.05 | 2.2621571628 |
| 0.99 | 0.01 | 3.2498355416 |
Higher confidence pushes the boundaries farther from zero. The returned positive number supplies the upper boundary; its negative supplies the corresponding lower boundary in the standardized t distribution.
Entering 0.95 directly as the first argument would request 95% in the outside tails, which is a very different question from a central 95% interval.
Independent mathematical checks
For a symmetric t distribution, the positive critical value is the left-tail quantile at 1-probability/2. With combined tail probability 0.05, that left-tail probability is 0.975.
NIST’s t-distribution reference identifies the 1-degree case as Cauchy. At threshold 1, each outside tail has probability 0.25, so:
=T.INV.2T(0.5,1)
has expected result 1. The published Google example with probability 0.05735 and 45 degrees of freedom gives approximately 1.950592863, independently reproduced by numerical t inversion.
These checks establish the intended two-tail interpretation without claiming that this guide was executed in Sheets.
Advanced application: convert the critical value to a margin
Suppose a synthetic sample has mean 50, standard deviation 4 and count 10. A standard t-based mean interval uses n minus 1, or 9, degrees of freedom.
Calculate the 95% margin:
=T.INV.2T(0.05,9)*4/SQRT(10)
Expected result: 2.8614276239. Then calculate the endpoints:
=50-T.INV.2T(0.05,9)*4/SQRT(10)
=50+T.INV.2T(0.05,9)*4/SQRT(10)
Expected results: 47.1385723761 and 52.8614276239. The t critical value has no measurement units; multiplying it by the standard error produces a margin in the measurement’s units.
The standard mean-interval equation and repeated-sampling interpretation are described by NIST. This calculation is not a guarantee about the next individual observation and does not remedy unsuitable sampling or model assumptions.
Degrees of freedom are a separate input
For 19 degrees of freedom:
=T.INV.2T(0.05,19)
Expected result: 2.0930240544, smaller than the 9-degree critical value. Degrees of freedom must come from the chosen procedure; they are not always sample size minus one.
Google’s truncation rule means =T.INV.2T(0.05,9.8) has the same expected result as the 9-degree example. Analytical workflows should supply the intended integer rather than depend on accidental truncation. Do not use this fact to round a fractional degrees-of-freedom approximation from another method without checking that method’s requirements.
Errors and troubleshooting
Probabilities 0 and 1 are outside Google’s stated domain. Check for a confidence-versus-alpha mix-up, a percentage entered as 95 rather than 0.95, degrees of freedom below 1 or nonnumeric referenced values.
A critical value is not a p-value. Use a distribution-tail function when the test statistic is already known and the desired output is its probability. Use this inverse when the probability is known and the threshold is needed.
Exact native error tokens, blank/text coercion and extreme-tail numerical limits were not established by the reviewed reference. Keep full precision until formatting the final report.
Availability and verification
T.INV.2T and its TINV synonym are currently documented for rolling Google Sheets. Original introduction date, first release and last functioning numbered version were not established.
NOT_TESTED_IN_GOOGLE_SHEETS. Google help, its linked read-only published example and NIST references were inspected on October 17, 2026. All original quantiles and interval calculations were independently checked. No native execution or screenshot is claimed.