TDIST in Apple Numbers: One- and Two-Tailed t Probabilities
Apple Numbers TDIST syntax, one- versus two-tailed Student t probabilities, worked examples, critical-value checks and troubleshooting.
Verification status: Official documentation referenced; example outcomes independently reasoned or arithmetically checked where stated. Not executed in Apple Numbers. How formula examples are checked
Use TDIST when you have a nonnegative Student’s t statistic and degrees of freedom, and you want a tail probability—typically a p-value under a t-distribution reference model. The third argument selects one right tail or both tails. This distinction matters: a two-tailed probability is twice the right-tail probability for the same nonnegative statistic in the symmetric t distribution. Apple documents TDIST in Formulas and Functions Help 15.4. [A1]
Evidence boundary: The examples identified as Apple-published come from Apple’s help page. Other numbers were independently calculated with Python/SciPy, not executed in Numbers.
TDIST syntax and argument order
=TDIST(x-value, degrees-freedom, tails)
| Required argument | Apple-documented type | Meaning and restrictions |
|---|---|---|
x-value |
Number | Nonnegative t-statistic magnitude, x ≥ 0. Negative values fall outside the documented TDIST input range. |
degrees-freedom |
Number | Degrees of freedom, at least 1. Often sample count − 1 for a one-sample t test; your statistical model determines the correct value. |
tails |
Modal/numeric selector | 1 = one right-tail probability; 2 = the two-tail probability. Other modes are undocumented. |
All three arguments are required; Apple documents no optional arguments or defaults. Preserve the exact order statistic, df, tails. Apple’s page does not explicitly describe how fractional degrees of freedom are handled, so this guide makes no truncation or rounding claim for such input. [A1]
With an English decimal point, use commas as separators. Apple’s functions overview specifies semicolons between arguments if the spreadsheet uses a comma as decimal separator. Example: =TDIST(4;2;2) where that locale requires semicolons. The spelling TDIST is the English-catalogue identifier; no additional alias or edition-specific synonym has been verified. [A3]
Make a reproducible Numbers worksheet
In Numbers, create a new table named TProbabilities, leave its first row as headers, and enter the following input values in cells A2:C5. In D2, type =TDIST(A2,B2,C2) and fill the formula down to D5. Numbers references within the same table use the ordinary A2 cell notation; another table can be addressed using TableName::B2. [A4]
| Row | A: Nonnegative t statistic | B: Degrees of freedom | C: Tails | D: Formula and expected result |
|---|---|---|---|---|
| 2 | 4 |
2 |
1 |
=TDIST(A2,B2,C2) → 0.0285954792089682 |
| 3 | 4 |
2 |
2 |
=TDIST(A3,B3,C3) → 0.0571909584179364 |
| 4 | 2 |
10 |
2 |
=TDIST(A4,B4,C4) → 0.0733880347707404 |
| 5 | 2 |
10 |
1 |
=TDIST(A5,B5,C5) → 0.0366940173853702 |
Rows 2–3 reproduce Apple’s published values, independently checked to the stated precision; rows 4–5 use the same official syntax and independent t-distribution arithmetic. Display formatting might round the on-screen decimal representation. [A1, M1]
Interpretation: what does a 0.05719 result mean?
=TDIST(4,2,2) describes a two-sided tail probability of approximately 5.72%, not the probability that a scientific hypothesis is true. Under the selected null distribution with two degrees of freedom, a t-statistic at least as far from zero as ±4 has this combined tail probability. A conventional 5% two-sided cutoff would not be met in this example because 0.05719 > 0.05; that decision depends on a test whose assumptions and pre-specified threshold are appropriate. [A1, M1]
Advanced use: from a signed t statistic to a two-sided p-value
A t statistic computed from sample data may be negative. Because Apple’s TDIST requires a nonnegative x-value, use ABS when the statistical question is two-sided:
=TDIST(ABS(B2),B3,2)
Here B2 holds the signed t statistic and B3 the degrees of freedom. If B2 is −2 and B3 is 10, the independently expected probability is 0.0733880347707404. The sign becomes irrelevant only because the question is two-sided. For a one-sided hypothesis, the correct direction (left versus right) matters; do not blindly take the absolute value and use tail 1 without interpreting that direction. [A1, M1]
For a value in a different table, if TestInputs contains B2 = −2 and C2 = 10, the following is valid Apple cross-table reference syntax:
=TDIST(ABS(TestInputs::B2),TestInputs::C2,2)
Apple’s general reference help explicitly uses the TableName::Cell pattern. Confirm any duplicate table/sheet names in your own document using Numbers’ formula editor. [A4]
Advanced use: check a critical value
A common two-sided t critical value at α = 0.05 and df = 10 is approximately 2.22813885198628. The associated probability can be independently checked with:
=TDIST(2.22813885198628,10,2)
Expected: about 0.05. The inverse TINV(0.05,10) supplies that positive cutoff; Apple separately documents TINV and lists it alongside TDIST. The combined formula =TDIST(TINV(0.05,10),10,2) is a reproducible mathematical inverse-consistency check, not evidence of Numbers-native execution. [A1, A2, M2]
Errors, limits and troubleshooting
| Symptom | Check and remedy |
|---|---|
Argument/input error for =TDIST(-2,10,2) |
TDIST requires a nonnegative statistic. For a genuine two-sided question, use ABS on the signed statistic; retain directional reasoning for one-sided questions. [A1] |
| Invalid degrees of freedom | Use a numeric value ≥1. Do not treat df = 0 as a legitimate t distribution. Apple’s page does not establish fractional-df truncation. [A1] |
| Wrong or missing tails selector | Supply an explicit 1 or 2; neither a default nor TRUE/FALSE selector is documented. [A1] |
| p-value is double/half of expectation | Revisit one-sided versus two-sided hypothesis and the third argument. At a nonnegative statistic, two-tail = twice upper one-tail. [M1] |
| Formula is not accepted after pasting | Check decimal/argument separators for the workbook’s locale; use the formula editor rather than blindly changing punctuation. [A3] |
| A cell reference changes when filled down | Confirm whether you need a relative or fixed reference; Apple’s formula reference help describes $/absolute-reference controls. [A4] |
| Published sample rounds differently | Change cell number display precision. Independent examples are numerical expectations, not screenshots or native tests. [M1] |
TDIST evaluates a distribution probability; it does not test the appropriateness of a data set, infer the correct df, or choose tails automatically. For a test calculated from two data ranges, consult Apple’s separate TTEST help rather than assuming TDIST accepts arrays as the t-statistic argument. [A1, A5]
Compatibility and version scope
- Documented version: Apple Formulas and Functions Help 15.4, checked 13 October 2026. The current English catalogue lists TDIST as a statistical function. This does not establish which earlier Numbers releases first supported TDIST. [A1, A6]
- Platforms: Apple’s Numbers Mac release page says Numbers 15.4 for Mac requires macOS 26 or later and broadly mentions additional statistics features; it does not date TDIST specifically to that release. The shared functions documentation applies in supported Numbers contexts, but this guide has not tested Mac, iPad, iPhone, or iCloud.com separately. [A7]
- Aliases and spellings:
TDISTis the exact English documented identifier. Different functions can use different tail conventions. [A1, A6]
Related Apple documentation (official destinations checked 2026-10-13)
- TINV — inverse two-tailed t distribution [A2]
- TDIST — official function syntax and examples [A1]
- Functions overview — locale argument separators [A3]
- Refer to cells in formulas — cross-table references [A4]
Claim-level evidence (checked 2026-10-13)
- [A1] Apple, “TDIST,” 15.4:
https://support.apple.com/guide/functions/tdist-ffa5b41355/15.4/web/1.0— exact three arguments, x/df domains, tail modes, two worked values, and See also links. - [A2] Apple, “TINV,” 15.4:
https://support.apple.com/guide/functions/tinv-ffa5a023a7/15.4/web/1.0— inverse syntax and probability domain. - [A3] Apple, “Functions overview,” 15.4:
https://support.apple.com/guide/functions/functions-overview-ffa083bef0c/15.4/web/1.0— commas and semicolons for decimal-comma locales. - [A4] Apple, “Refer to cells in formulas,” 15.4:
https://support.apple.com/en-au/guide/functions/ffa550d691/15.4/web/1.0— same- and cross-table reference forms and absolute references. - [A5] Apple, “TTEST,” linked from official TDIST help: available through the TDIST page’s See also navigation; test-specific behavior should be verified before adding further claims.
- [A6] Apple, “List of functions by category,” 15.4:
https://support.apple.com/en-au/guide/functions/ffa148e9cd0/web— current catalogue identity, not proof of historical availability. - [A7] Apple, “What’s new in Numbers on Mac”:
https://support.apple.com/en-sa/guide/numbers/tan700f60676/mac— Numbers 15.4 general release description and macOS minimum, not a TDIST introduction date. - [M1] Independent SciPy:
scipy.stats.t.sf(x,df)for one upper tail, twice it for two tails, checked 2026-10-13; no native Numbers execution. - [M2] Independent SciPy:
scipy.stats.t.isf(p/2,df)for the positive two-tailed critical value, checked 2026-10-13; no native Numbers execution.