TINV in Apple Numbers: Two-Tailed Student t Critical Values

=TINV(probability, degrees-freedom)

Both arguments are mandatory: no optional argument and no default is documented. The probability bound excludes exactly 0 and 1, and Apple does not specify any integer rounding/truncation rule for fractional degrees of freedom. [A1, M1]

The English guide uses commas as argument separators. Apple’s overview says a document configured with a comma decimal separator instead uses semicolons, such as =TINV(0,05;10) in an applicable locale. [A3]

Apple’s example =TINV(0.88,2) returns 0.170940864689457. The independent Student t quantile relation that reproduces this is:

P(|T₂| ≥ t) = 0.88, so t = t₂ inverse survival(0.88 / 2).

Thus, for 0 < probability < 1, TINV gives a nonnegative two-sided critical t value. It is not the inverse of the standard one-sided cumulative distribution at p=0.88. This interpretation is grounded in the Apple example and mathematical cross-check; Apple uses the briefer phrase “probability associated with the distribution.” [A1, M1]

Create a table named TCritical, with one header row. Fill A2:B5 with the numbers below. Enter =TINV(A2,B2) in C2, then copy down through C5. Use at least six decimals for the display if comparing to the table. [A1, A4]

A two-sided significance level of 5% is supplied as 0.05 (not 95 and not 0.95). For confidence level C, the corresponding two-sided tail probability is 1−C; for a 95% interval use =TINV(1-0.95, df). [M1]

Suppose your Measurements table contains five observations in B2:B6:

In a separate summary table named Results, add these formulas referencing Measurements by Numbers’ documented TableName::Range notation:

The conventional 95% t interval is mean ± t critical × (sample SD / √n); using the sample SD rather than population SD is intentional. It describes uncertainty under appropriate sampling/model assumptions, not a guarantee that 95% of the individual observations lie in that interval. The numeric results were independently validated, not run in Numbers. [M2]

Names such as Measurements and Results are examples; use your actual unique table names. Apple’s reference help confirms cross-table TableName::Cell notation and standard B2:B6 ranges. [A4]

For a two-sided decision at α=0.05 and df=10, =TINV(0.05,10) supplies 2.22813885198628. A signed observed t magnitude greater than that cutoff lies in the two tails (assuming the reference t model is appropriate). Conceptually:

=ABS(TestInputs::B2)>TINV(0.05,TestInputs::C2)

This returns the logical comparison of a statistic stored in TestInputs B2 against a t cutoff based on df in C2; it does not calculate a p-value or verify distributional assumptions. Do not swap the probability for 0.95 expecting a 95%-confidence cutoff: that would produce a very small positive t threshold. [M1, A4]

A useful distribution consistency check is:

=TDIST(TINV(0.05,10),10,2)

Mathematically the expected result is 0.05, within ordinary floating-point tolerance. Apple’s TDIST explicitly uses a two-tail mode of 2. Neither this check nor the statistical tests were executed natively in Numbers. [A2, M1]

• Apple function source: Formulas and Functions Help 15.4, checked 13 October 2026. The current English catalogue contains one canonical TINV entry. Its date of original introduction in past Numbers releases is not established. [A1, A5]
• System versions: Apple’s Numbers Mac release page describes Numbers 15.4 and macOS 26 as the minimum for that version. This is not evidence that TINV was newly added in 15.4. This guide has not tested formula execution on Mac, iPad, iPhone or web. [A6]
• Naming: TINV is an exact uppercase English identifier; do not conflate it with inverse functions whose tail probability conventions differ. [A1, M1]

• TINV — Apple syntax and worked example [A1]
• TDIST — one- versus two-tailed probability [A2]
• Functions overview — separators and required arguments [A3]
• Refer to cells in formulas — ranges and table names [A4]

Dated claim-level evidence

• [A1] Apple, “TINV,” Function Help 15.4, checked 2026-10-13: https://support.apple.com/guide/functions/tinv-ffa5a023a7/15.4/web/1.0 — exactly two required arguments; p strictly (0,1), df≥1; official example 0.88, 2.
• [A2] Apple, “TDIST,” Function Help 15.4, checked 2026-10-13: https://support.apple.com/guide/functions/tdist-ffa5b41355/15.4/web/1.0 — tail selector 1 or 2, two published right/two-tail values, link to TINV.
• [A3] Apple, “Functions overview,” checked 2026-10-13: https://support.apple.com/guide/functions/functions-overview-ffa083bef0c/15.4/web/1.0 — decimal-comma locale separators.
• [A4] Apple, “Refer to cells in formulas,” checked 2026-10-13: https://support.apple.com/en-au/guide/functions/ffa550d691/15.4/web/1.0 — ranges and table-qualified references.
• [A5] Apple, “List of functions by category,” checked 2026-10-13: https://support.apple.com/en-au/guide/functions/ffa148e9cd0/web — listed canonical English identifier.
• [A6] Apple, “What’s new in Numbers on Mac,” checked 2026-10-13: https://support.apple.com/en-sa/guide/numbers/tan700f60676/mac — current version platform minimum and general statistical changes, not a function-by-function introduction list.
• [M1] Independent SciPy: t inverse survival scipy.stats.t.isf(p/2,df) and scipy.stats.t.sf(x,df), checked 2026-10-13. No native Numbers formula runs.
• [M2] Independent Python: sample mean, statistics.stdev, and t critical × sample standard error, checked 2026-10-13. No native Numbers formula runs.

• TDIST in Apple Numbers