T.INV Function (LibreOffice Calc)

• Introduced: LibreOffice 4.3.
• Availability: Supported in LibreOffice 26.8.1.1, verified against its function registry. This identifies a supported release, not a final working version.
• Scope: Desktop Calc, using the English function name. Availability in other spreadsheet applications is separate.
• Checked: 2026-10-14 — function reference; versioned support. Documentation/source check; not a new native test. Existing example-specific test claims below retain their stated scope.

Historical validation scope: Examples and diagnostics refer to saved LibreOffice Calc 25.2.3.2 Linux evidence from 13 October 2026. Artifact hashes and all 95 saved ODS/result pairs agree. This review independently recalculated the 65 numerical assertions with SciPy distribution functions and arithmetic. Calc was not rerun; error and boundary behavior remains specific to that saved environment.

T.INV takes a cumulative probability and degrees of freedom and returns the corresponding left-tail critical value of Student’s t distribution. Use it when a t-based calculation needs the value t at which the area under the distribution to the left is the specified probability. For example, a 97.5% cumulative probability yields the positive critical t value used in an ordinary two-sided 95% confidence interval.

Unlike a normal-distribution inverse, the answer depends on the degrees of freedom: for smaller samples the t distribution has heavier tails. The function does not calculate a p-value from a t statistic; for that operation see T.DIST or T.DIST.RT. LibreOffice’s function documentation defines T.INV specifically as the inverse of the left-tailed t distribution [LO-TINV].

=T.INV(Number; DegreesFreedom)

There are no optional parameters or default arguments. In an English-locale Calc installation, semicolons separate arguments. Depending on your configured locale, translated function names, decimal separators, and argument separators may differ. The formula shown uses the English name and English-US decimal notation [LO-TINV].

What the returned number means

Mathematically, T.INV returns the quantile q satisfying

P(T ≤ q) = Number,   T ~ Student-t(DegreesFreedom).

The result is a t statistic, not a percentage or probability. For a symmetric t distribution, probabilities below 0.5 yield negative quantiles, 0.5 yields zero, and probabilities above 0.5 yield positive quantiles. For example, =T.INV(0.975;10) returns about 2.228138851986275; in a model with ten degrees of freedom, 97.5% of the distribution lies below that value. The returned t value is dimensionless.

Enter the following directly into an empty cell:

=T.INV(0.1;6)

Expected result: -1.4397557473 (the native test returned -1.4397557472651503). The negative result is appropriate: only 10% of a t distribution with six degrees of freedom lies to its left. This matches the example published in LibreOffice Help [LO-TINV]. The result is not “minus 143.98%”—format the cell as a number, not a percentage.

The following results were actually recalculated in LibreOffice Calc 25.2.3.2 by direct native cell-formula entry on 2026-10-13. Display rounding below is intentional; the saved test recorded full-precision native values.

Verify by reversing the calculation. The cumulative t-distribution function should return the input probability when given the quantile:

=T.DIST(T.INV(0.975;10);10;TRUE())

Expected result: 0.975. This round-trip succeeded in the native test. The third argument to T.DIST requests the cumulative distribution; the noncumulative version is a probability density and cannot serve as an inverse check [LO-TINV]. Tiny floating-point differences between a calculated probability and the starting constant are normal.

Reproducible cell-reference worksheet

Set up this small input table on a blank sheet:

Expected C2: 2.228138851986275. Changing A2 from 0.975 to 0.025 should switch the sign; changing B2 changes the t distribution’s shape. The T.INV(A2;B2) cell-reference formula itself was executed successfully in the native test.

Suppose a random sample has 16 independent observations, a sample mean of 72, and a sample standard deviation of 8. Under the usual assumptions for a one-sample t confidence interval (independent data and an appropriate approximately normal population model, especially at small n), the number of degrees of freedom is 16−1=15. A central 95% interval uses the 97.5th percentile, because each tail holds 2.5%.

The positive t critical value is =T.INV(1-0.05/2;15), equivalently =T.INV(0.975;15). The margin of error is:

=T.INV(0.975;15)*8/SQRT(16)

Native expected result: 4.262899091119551.

The two interval bounds can be calculated directly:

=72-T.INV(0.975;15)*8/SQRT(16)
=72+T.INV(0.975;15)*8/SQRT(16)

Expected interval: approximately [67.7371009089, 76.2628990911]. Both endpoints were independently expected and natively asserted in the saved test. This is an interval procedure with assumptions; it is not a guarantee that the particular fixed population mean falls inside the interval with probability 95% after the sample has been observed.

For a reusable sheet, enter mean, sample standard deviation, sample size, and confidence level in separate cells and calculate:

=T.INV(1-(1-ConfidenceLevel)/2;SampleSize-1)

Here the descriptive identifiers represent cell references or named ranges you would create, not literal built-in arguments. For instance, if D2=72, D3=8, D4=16, D5=0.95, use =D2-T.INV(1-(1-D5)/2;D4-1)*D3/SQRT(D4) for the lower bound. This parameterized example follows the same verified formula; this particular four-cell layout was not separately executed.

For a right-tailed t test with significance level α=0.05 and 19 degrees of freedom, the upper critical value is the left-tail quantile at 1−α:

=T.INV(1-0.05;19)

Native expected result: 1.729132811521369. The corresponding left-tailed cutoff is =T.INV(0.05;19), or approximately −1.729132811521369. These cutoffs mark rejection regions under the t-test’s null model; T.INV alone does not decide whether a real dataset supports the hypothesis. You still need an appropriately calculated t statistic, tail choice, and test assumptions.

For a two-tailed 5% procedure, the positive boundary is =T.INV(1-0.05/2;19). LibreOffice also provides a separate two-tail convention through T.INV.2T. Their inputs are not interchangeable: one uses a left-tail cumulative probability while the other uses a two-tail probability [LO-TINV].

The native test confirmed the equivalence for ten degrees of freedom:

=T.INV(0.975;10)-T.INV.2T(0.05;10)

Expected result: 0 (within displayed floating-point precision). These functions use different probability conventions.

Student’s t distribution is symmetric, providing a useful check:

=T.INV(0.025;10)+T.INV(0.975;10)

Native expected result: 0. This confirms that the 2.5th and 97.5th percentiles are equal in magnitude and opposite in sign. The check is useful when auditing a workbook in which the lower and upper critical values appear to have the wrong signs.

To catch malformed input before computing a quantile, with probability in A2 and degrees of freedom in B2, use:

=IF(OR(A2<=0;A2>=1;B2<1);NA();T.INV(A2;B2))

When A2=0.975 and B2=10, the native test returned the same 2.228138851986275 critical value. A separately executed invalid-input guard produced #N/A, indicating that the check can deliberately signal a bad model input. If you require integer degrees of freedom, strengthen the guard with B2<>INT(B2); that additional condition is a documented formula proposal, not independently executed in the saved test. Check numeric input types as well if a spreadsheet may contain imported text or unexpected dates.

Observed in Calc 25.2.3.2, Linux:

=T.INV(0.95;2.5)

Native result: 2.919985580353723, matching the result at 2, not 2.5, degrees of freedom. The independently computed continuous t quantile at 2.5 degrees of freedom is different. Therefore, in this tested version, fractional degrees of freedom are effectively truncated rather than evaluated as fractional distribution parameters. This can matter in an approximate Welch t procedure, which can yield fractional degrees of freedom. It is a version-specific native observation, not a statement that all LibreOffice releases behave the same way. For a method requiring fractional df, independently verify that the chosen software implements the intended distribution; silently dropping the fraction changes the result.

The table records what Calc 25.2.3.2 on Linux actually returned under direct native formula evaluation. Differences in locale, import route, or newer releases are possible.

For unexpected error messages, inspect whether cells contain percentages correctly entered as numeric values, whether a copied formula mixes comma and semicolon delimiters, whether TRUE()/FALSE() were mistakenly added as a third argument, and whether formatted cells contain text instead of numbers. In the published official syntax both arguments are mandatory [LO-TINV].

Numerical precision and tails

Quantiles become large in magnitude near cumulative probabilities zero and one. The native diagnostic =T.INV(0.999999999;10) returned approximately 20.14469777484425. The probability is close to one, but not one. When translating one-sided test thresholds to two-sided probabilities, perform the correct α/2 conversion before calling the function. Avoid equating a quantile to a probability or using a far-tail value as an effect size; they measure different quantities.

LibreOffice Help states that T.INV is available from LibreOffice 4.3 onward. Its documented compatibility extension namespace is COM.MICROSOFT.T.INV; LibreOffice Help classifies this spelling as not part of OpenDocument 1.3 OpenFormula’s core function names [LO-TINV]. This documents the name’s file-format context; it does not demonstrate that a formula was independently executed in Microsoft Excel, Google Sheets, OpenOffice or every older/newer LibreOffice platform. This guide’s native numerical test applies specifically to LibreOffice Calc 25.2.3.2 on Linux with directly entered UNO formulas, and the linked online documentation currently labels itself LibreOffice 26.8 Help.

TINV is a legacy function and must not be assumed identical to one-tail T.INV simply because the names look similar. The correct cumulative-versus-two-tail distinction is essential.

• T.DIST — cumulative probability or density given a t statistic.
• T.DIST.RT — direct upper-tail probability given a t statistic.
• T.DIST.2T — two-tailed probability convention.
• T.INV.2T — critical t value from a two-tailed probability rather than a left-tail cumulative probability.
• TINV — separate legacy inverse t name; check tail semantics.
• STDEV.S — sample standard deviation for a t confidence interval.
• COUNT — number of numeric sample observations when determining n−1 degrees of freedom.

See LibreOffice statistical function documentation for syntax, examples, version availability and compatibility details.