T.INV in Google Sheets: Left-Tail Student t Quantiles
Use T.INV in Google Sheets for signed t-distribution quantiles, choose the correct tail probability and build a checked two-sided mean-interval example.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: =T.INV(0.975,9) has expected result 2.2621571628. It is the t value with 97.5% of the distribution to its left for 9 degrees of freedom. The matching lower-tail quantile at 0.025 is negative.
Syntax and documented restrictions
=T.INV(probability, degrees_freedom)
The Google T.INV reference requires numeric inputs, probability strictly between 0 and 1, and degrees of freedom at least 1. Fractional degrees of freedom are truncated to an integer. Both arguments are supplied; no defaults are documented.
Google describes this as the negative inverse of one-tailed TDIST. With the symmetric t distribution, that corresponds to the signed left-tail quantile: small cumulative probabilities give negative values, 0.5 gives zero, and probabilities above 0.5 give positive values.
Google’s linked read-only published example gives approximately -1.608591458 for probability 0.05735 and 45 degrees of freedom. Independent left-tail t-quantile arithmetic reproduces that negative value.
The function returns a t value, not a probability or a sample mean. Its degrees-of-freedom input must be determined by the intended statistical procedure.
Worked example: lower and upper critical values
Enter 0.025, 0.05, 0.5, 0.95 and 0.975 in A2:A6. Put 9 in D1. In B2, enter and fill down:
=T.INV(A2,$D$1)
| Left-tail probability | Expected t quantile |
|---|---|
| 0.025 | -2.2621571628 |
| 0.05 | -1.8331129327 |
| 0.5 | 0 |
| 0.95 | 1.8331129327 |
| 0.975 | 2.2621571628 |
The probabilities 0.025 and 0.975 describe the two boundaries of a central 95% interval in the standardized t distribution. Using 0.95 for the positive boundary instead gives a one-sided 5% upper-tail threshold, not the same two-sided central interval.
Independent mathematical check
NIST’s t-distribution reference identifies the degrees-of-freedom-1 case as a Cauchy distribution. Its inverse CDF has the special form tangent of pi times p minus one half.
Consequently, these expected results provide simple checks:
=T.INV(0.25,1)
=T.INV(0.5,1)
=T.INV(0.75,1)
Expected values: -1, 0 and 1 respectively. This closed-form identity is specific to 1 degree of freedom; the other examples were independently checked with numerical t quantiles.
For any fixed valid degrees of freedom, symmetry also implies that the quantiles at p and 1-p sum to zero, within numerical precision. This is useful for checking the sign and tail choice.
Advanced application: a two-sided mean interval
Suppose a synthetic sample summary has mean 50, sample standard deviation 4 and sample count 10. The standard t mean-interval calculation uses 9 degrees of freedom.
The 95% margin is:
=T.INV(0.975,9)*4/SQRT(10)
Expected result: 2.8614276239. The endpoints are:
=50-T.INV(0.975,9)*4/SQRT(10)
=50+T.INV(0.975,9)*4/SQRT(10)
Expected results: 47.1385723761 and 52.8614276239. The conversion from a t quantile to a measurement-unit margin multiplies by the standard error. T.INV alone does not perform that scaling.
The formula follows the standard mean-interval procedure described in NIST’s confidence-limit reference. Appropriate sampling, independence and distribution assumptions remain necessary. This is not an interval intended to contain 95% of future individual observations.
Degrees of freedom and spelling matter
For 19 degrees of freedom, =T.INV(0.975,19) has expected result 2.0930240544, smaller than the 9-degree value. Do not reuse a critical value after changing the sample size without checking the appropriate degrees of freedom.
Google documents truncation, so =T.INV(0.975,9.8) uses 9 degrees of freedom and has the same expected result 2.2621571628 as the integer example. The guide uses whole-number degrees of freedom for analytical calculations rather than relying on truncation to choose them.
Google’s page distinguishes TINV, the inverse associated with two-tailed TDIST. Removing the period from T.INV is therefore not a cosmetic alias change. A two-tailed probability and a left-tail cumulative probability represent different inputs.
Errors and troubleshooting
Probabilities 0 and 1 are outside Google’s documented domain. Check for accidental percentages such as 95 instead of 0.95, degrees of freedom below 1 and text in referenced cells. If an expected positive critical value is negative, inspect whether you supplied a lower-tail probability rather than the complementary upper quantile.
Exact native error tokens and blank/text coercion were not established in this documentation-only check. Very extreme tail probabilities can be numerically sensitive; do not assume displayed rounding implies exact equality.
Availability and verification
T.INV is 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’s individual reference and NIST mathematical sources were inspected on October 17, 2026. Expected values were independently calculated using t quantiles, symmetry and the degrees-of-freedom-1 special case. No native execution or screenshot is claimed.