T.INV in ONLYOFFICE: Signed Student t Quantiles from Left-Tail Probability

Create TModel. Put Parameter in A1 and Value in B1. Enter Threshold in A2 and 1 in B2; Degrees of freedom in A3 and 1 in B3; Tail area in A4 and 0.05 in B4. Keep these cells unchanged when adding controls.

One degree of freedom is chosen because the t distribution then has the Cauchy closed form: its CDF is 1/2+atan(t)/π and its density is 1/[π(1+t²)]. This lets the examples be checked independently without running ONLYOFFICE. It is an arithmetic teaching case, not a recommendation to base practical inference on a tiny sample.

A t distribution is symmetric around zero. Its tail heaviness depends on degrees of freedom; a t value is not a raw measurement in minutes or grams. In a statistical test, its construction and degrees of freedom must come from the appropriate sampling model. A distribution formula alone does not establish those assumptions.

The official reference requires probability strictly between 0 and 1 and integer degrees of freedom at least 1. The returned q satisfies P(T≤q)=p mathematically. It is a t-scale cutoff, not an original-unit measurement or a p-value.

At df=1, the inverse closed form is tan(π(p−1/2)). This produces −1 at p=0.25, 0 at p=0.5 and +1 at p=0.75. It also explains why quantiles diverge as probabilities approach the endpoints.

At df=1 the tails are very heavy, so these upper quantiles are much larger than standard-normal quantiles at the same probabilities. Do not transfer a normal critical value to a t procedure without justification.

A one-sided upper-tail area of 0.05 corresponds to p=0.95 in this interface. A central 95% interval leaves 0.025 in each tail, so its upper cutoff uses p=0.975 and the lower cutoff p=0.025. The latter has target −12.7062047361747 at df=1.

The mathematical symmetry is T.INV(1-p,df)=-T.INV(p,df). A round trip =T.DIST(T.INV(0.75,1),1,TRUE) targets approximately 0.75. Use an appropriate numerical tolerance rather than exact equality, since an iterative inverse and a forward probability calculation are not exact symbolic inverses.

The t critical value can be multiplied by a separately justified standard error to form a margin of error in an appropriate procedure. It does not by itself establish a confidence interval, select the right standard error or validate assumptions about independent observations and sampling.

The inspected calculation rejects p≤0 and p≥1 with #NUM!, including a dedicated p=1 check. Thus =T.INV(0,1) and =T.INV(1,1) are source-predicted #NUM!, rather than infinite endpoints. It parses df with parseInt before checking df≥1. For an ordinary positive fraction such as 1.9, the calculation follows df=1; scientific-notation extremes are not covered by a universal truncation promise.

This differs from the forward t functions, which pass their df values to the helper without the same explicit integer conversion. Keep documented integer df in reproducible examples and do not assume a fractional-df inverse round trip reproduces a forward calculation with the unmodified fraction.

Legacy TINV is not this signed inverse. It inherits T.INV.2T, which takes a total two-tail area and returns a nonnegative cutoff. For example, TINV(0.5,1) targets 1, whereas T.INV(0.5,1) targets 0. Preserve the probability interpretation when migrating names.

The inverse implementation searches through a shared numerical helper. It can expand a bracket up to 1000 times, then refine for up to 500 iterations using interpolation and bisection. A failed bracket is reported through an error flag and becomes #NUM!, but reaching the refinement iteration cap alone does not set that flag. Do not describe every finite return as a certified convergence result.

These are implementation bounds, not a measured runtime guarantee. Very small probabilities and extreme degrees of freedom need exact-build accuracy checks; the moderate closed-form examples here do not validate all tails. Array-oriented source preparation does not prove a native spill or array-entry workflow.

Checked October 18, 2026 against current official help and SDKJS tag v10.0.0.169, commit 3679e0375a758d7dc59d31337a45461be3525ed8. Introduction dates and a final supported-version endpoint are not established. Desktop Editors, Docs, mobile releases and imported files require separate compatibility checks.

Neither the ONLYOFFICE application nor its calculation engine was executed for this guide. The numerical targets are independently calculated mathematical values; code-path and error observations are manual source traces. They are not screenshots, displayed workbook results or promises of identical final digits. Use English function names and comma separators as shown, adapting separators to the editor’s locale.

Keep inputs and intermediate probabilities numeric and unrounded. Format the display rather than replacing values with rounded strings. These moderate scalar examples do not certify extreme-tail accuracy, unusual type coercion, array entry or dynamic spill. Model assumptions must be checked separately from arithmetic correctness.

• Official T.INV help, checked October 18, 2026.
• Pinned function definition.
• T-probability helper, inverse solver and test calculations.
• T.DIST.
• T.DIST.2T.
• T.DIST.RT.
• T.INV.2T.
• TDIST.
• T.TEST.