T.DIST in Google Sheets: Cumulative Probability and Density

=T.DIST(x, degrees_freedom, cumulative)

All three arguments are required. The Google reference defines TRUE as cumulative probability and FALSE as density. Its opening description instead calls the function right-tailed and equates it with TDIST’s one-tail form. That wording cannot explain the page’s own results:

The separate T.DIST.RT reference prints 0.027322464868265 and 0.9068737480782105 at the same respective inputs. Those values complement the TRUE examples to 1 within printed precision. Independently integrating the t density reproduces the cumulative examples; evaluating the density reproduces the FALSE examples.

This guide therefore uses the left-tail interpretation supported by those examples. It does not treat the contradictory introduction as a valid equivalence rule. This is a bounded documentation reconciliation, not a native Google Sheets test or a guarantee about undocumented inputs.

Enter the following signed thresholds in A2:A6: -2, -0.5, 0, 0.5, 2. Put 7 in E1. In B2 and C2 enter these formulas, then fill both down:

=T.DIST(A2,$E$1,TRUE)
=T.DIST(A2,$E$1,FALSE)

The cumulative values increase as the threshold increases. The density is symmetric about zero and is largest there. Equal densities at -2 and 2 do not mean equal cumulative areas: their left-tail probabilities sum to 1.

A density is the height of a probability curve. It is not the probability of observing exactly that value; a single point has zero probability in a continuous distribution. For probability over a range, subtract cumulative areas instead.

To find the probability between -0.5 and 2, use the cumulative value at the upper boundary minus the cumulative value at the lower boundary:

=T.DIST(2,7,TRUE)-T.DIST(-0.5,7,TRUE)

Expected result: 0.6409867679. Subtracting densities at the endpoints would have no such interval interpretation. For a continuous distribution, including or excluding either exact endpoint does not change the area.

Keep full formula precision for calculations and round only the display. Subtracting nearly equal probabilities can lose numerical precision; this example is a moderate interval and does not establish the function’s accuracy for extreme thresholds.

For a signed statistic of -2 with 7 degrees of freedom:

• Left tail: =T.DIST(-2,7,TRUE), expected 0.0428096643.
• Right tail: =T.DIST.RT(-2,7), expected 0.9571903357.
• Symmetric outside tails: =T.DIST.2T(ABS(-2),7), expected 0.0856193286.

The two-sided area is twice the smaller tail, not twice whichever cumulative value happens to be returned. Taking ABS is appropriate for the symmetric two-sided calculation; it would discard the direction needed for a one-sided hypothesis. Select that direction before examining the result.

The mathematical complement =1-T.DIST(2,7,TRUE) has expected result 0.0428096643. The dedicated T.DIST.RT guide makes the requested direction explicit. For outside versus central area, see T.DIST.2T.

The NIST t-distribution reference supplies its beta-normalized density and identifies the one-degree-of-freedom distribution as Cauchy. Integrating that density from zero to x and adding 0.5 gives the expectations above. At one degree of freedom, =T.DIST(1,1,TRUE) should give 0.75, while FALSE should give approximately 0.1591549431. These provide separate checks for area and height.

The input is a t threshold, not an unstandardized measurement or a sample range. Degrees of freedom depend on how the statistic was constructed; they are not automatically sample size minus one. An area is a p-value only when the statistic, tail selection and model assumptions justify that interpretation. It is not the probability that a null hypothesis is true.

Google documents a numeric error when degrees of freedom is below 1. Its examples explicitly include negative x. The reviewed references do not settle fractional-degree handling, text and blank coercion, every error token, or the behavior of omitted arguments. Use explicit TRUE or FALSE and positive integer degrees of freedom for the examples here.

Do not mechanically replace TDIST with T.DIST: TDIST’s third argument chooses one or two tails, while T.DIST’s third argument chooses cumulative area or density. The right-tail and two-tail guides cover the documented TDIST correspondence. Similar names in Excel or other spreadsheets do not independently establish Google’s behavior.

The current rolling Google Sheets reference documents this signature. An introduction date, first supported numbered version and historical retirement date were not established.

NOT_TESTED_IN_GOOGLE_SHEETS. Documentation and mathematical reconciliation were reviewed on October 19, 2026. The worked values were independently calculated, not copied from a native session. No screenshot or live spreadsheet result is claimed.

Related published guides: ABS for threshold magnitude and ROUND for presentation rounding.