GAMMAINV in Zoho Sheet: Find Gamma Distribution Percentiles

GAMMAINV returns the quantile (percentile threshold) associated with a gamma-distribution cumulative probability. Zoho Sheet’s official GAMMAINV article documents the exact syntax GAMMAINV(p; α; β) and identifies p as the probability, α as gamma shape, and β as gamma scale. It is the inverse of a cumulative gamma calculation: rather than asking what probability lies below this threshold?, you supply a probability and ask what threshold leaves this much probability below it? Z1

For example, for gamma shape 3 and scale 2, a mathematical gamma quantile calculation predicts that =GAMMAINV(0.9;3;2) returns about 10.644640675668. Under the assumed model, 90% of outcomes would be at or below approximately 10.64 units, while 10% would exceed it. This is an independently calculated prediction, not a screenshot or executed Zoho formula.

Evidence scope: Official sources were checked on 2026-10-13; example numbers were computed independently using SciPy. No in-app tests or plan/locale/version verification occurred. Do not interpret mathematical agreement with an Excel or SciPy result as confirmation of undocumented Zoho edge behaviour.

GAMMAINV(p; α; β)

The three arguments are in a fixed order. Zoho displays semicolon delimiters in its English Help article, not commas; verify actual workbook locale rather than transplanting punctuation from other applications. There are no optional arguments in the published signature. p=0.90 is a probability; p=90 is not “90%” to this function and Zoho documents a #NUM! domain error for p outside 0–1. The article lists p=0 and p=1 within the broad stated interval but does not specify the finite/infinite boundary output; don’t assert a particular native result at exactly 0 or 1 without testing. Z1

Mathematically, for gamma shape α and scale β, GAMMAINV(p;α;β) models F⁻¹(p), where F(x)=P(X≤x) is the gamma cumulative distribution function. A correct result should be nonnegative for an interior probability, increase when p increases, and satisfy F(F⁻¹(p))≈p, allowing numerical rounding. This last property supplies a valuable independent check using Zoho’s separately documented GAMMADIST function, whose cumulative mode is TRUE.

Consider a hypothetical support-service completion-time model with gamma shape 3 and scale 2 minutes. The mathematical mean is α×β=6 minutes and variance α×β²=12 minutes². These parameters are assumptions for this tutorial, not claims estimated from historical records. You want a reasonable cutoff for the fastest 10%, 50%, 90% and 95% of modelled jobs.

Enter p, α, β in columns A–C:

Type the labels in row 1, the numbers in rows 2–5, then formulas into D2:D5. The expected results, independently calculated rather than retrieved from Zoho, are:

The 90th percentile’s 10% upper-tail complement follows from 1−0.90. It does not by itself mean 90% of a specific real service’s historical jobs met that threshold. You must validate the gamma model against observations and account for parameter-estimation uncertainty before operational decision-making.

Cross-check the result in the same worksheet

Put a cumulative check in E2 and copy through E5:

=GAMMADIST(D2;B2;C2;TRUE)

The mathematical expectation is E2 ≈ 0.10, E3 ≈ 0.50, E4 ≈ 0.90, E5 ≈ 0.95. The official GAMMADIST reference documents its fourth argument: TRUE chooses cumulative probability. FALSE would return a density and would not properly invert GAMMAINV. This composed formula has not been executed in Zoho and is proposed as a concrete native review case. Save native results with enough digits to judge tolerances, and do not demand exact equality when displayed cells are rounded.

Assume—not infer—that lead time is distributed gamma(shape 5, scale 2 days). Its mean is 10 days, but the distribution is right-skewed. For a hypothetical buffer covering 95% of individual modelled lead times under this distribution, use:

=GAMMAINV(0.95;5;2)

Independent expected result: 18.307038053275 days. If inventory planning requires whole days and the assumptions are acceptable, one might round the planning threshold up to 19 days, but the quantile formula itself returns a numeric time, not a calendar date or a guaranteed service commitment. A different business question—such as the 95% confidence interval around the mean lead time—is not answered by a distribution’s 95th percentile.

To study sensitivity, compare =GAMMAINV(0.9;5;2) ≈ 15.987179172105 with =GAMMAINV(0.95;5;2) ≈ 18.307. Choosing the higher percentile increases modeled coverage and inventory holding exposure. Actual purchase orders, seasonal trends, serial correlation and variation in supplier performance remain outside the function’s scope.

Scale transformation. Holding α and p fixed, gamma quantiles scale linearly in β. For a fixed p and shape, doubling β doubles the theoretical quantile. You can therefore compare =GAMMAINV(0.9;3;2) (approximately 10.644640675668) with =GAMMAINV(0.9;3;4) (approximately 21.289281351336), after calculating both independently. This is an algebraic model identity, not a Zoho execution record.

Exponential special case. For gamma shape α=1, the inverse cumulative formula is −β LN(1−p). For p=.9, β=4:

=GAMMAINV(0.9;1;4)

independently predicts 9.210340371976, matching the mathematical expression =-4*LN(1-0.9). Use this comparison to detect a swapped shape/scale or probability entered as 90. It checks the mathematical model, not whether an app’s boundary precision or parsing exactly mirrors SciPy.

Chi-square relationship. A chi-square model with ν degrees of freedom is mathematically a gamma model with shape ν/2 and scale 2. For example, a six-degree-of-freedom left-tail chi-square 90th percentile corresponds to the gamma inverse with p=0.9, shape 3 and scale 2. Zoho also has CHISQINV for the chi-square family. The related function must be compared in its documented tail convention; CHIINV is traditionally right-tail, so the same p might mean a very different threshold. Never mechanically paste gamma probabilities into right-tail inverse functions without reading their docs.

Spreadsheet parameter cells. If H2=3 and H3=2, and column A stores p, then =GAMMAINV(A2;$H$2;$H$3) allows a one-parameter scenario table without accidentally changing shape and scale while filling down. If your models use rate λ instead of scale β, convert consciously as β=1/λ; entering the rate in the third argument misparameterizes the distribution. Always distinguish the symbol λ for a gamma rate from λ for a Poisson mean; they are different quantities.

The official GAMMAINV help names #NUM! when p is outside [0,1] or either parameter α or β is zero/negative. It lists generic #VALUE! for invalid types, #NAME! for malformed names and #REF! for bad cell references. Zoho’s generic #N/A! paragraph references lookup failures and does not establish a GAMMAINV-specific example. Do not invent additional error names or specific output at p=1.

Unverified native cases: p=0 and p=1, tiny positive probabilities, probabilities indistinguishable from 1 due to rounding, overflows for huge parameters, non-integer but positive shapes, omitted inputs, text-numeric coercion, arrays/dynamic spills, and any specific maximum decimal precision. The official English public help gives no minimum product version, per-plan entitlement matrix, mobile/desktop parity guarantee or locale-specific separator schedule. Review these cases in the actual Zoho account rather than extrapolating from Excel.

Zoho’s canonical documented name is GAMMAINV, with semicolon-delimited p;α;β. Microsoft Excel separately documents GAMMA.INV(probability,alpha,beta) and Google Sheets lists gamma inverse names, but neither product’s formula parser nor its edge-case behavior validates Zoho’s. Some imported spreadsheets may rewrite the function name, delimiters or results; test a simple interior example and the GAMMADIST(...;TRUE) cross-check after import. Do not claim that Zoho accepts GAMMA.INV as an alias without direct Zoho documentation or an actual app test. Z1 M1 G1

• Z1 Zoho GAMMAINV, purpose, exact argument order and error domains, checked 2026-10-13: Zoho GAMMAINV.
• Z2 Zoho GAMMADIST, CDF/density switch and parameters for numerical inverse checks, checked 2026-10-13: Zoho GAMMADIST.
• Z3 Zoho CHISQINV, separately documented left-tail inverse chi-square calculation, checked 2026-10-13: Zoho CHISQINV.
• M1 Microsoft Excel GAMMA.INV, other-product signature only, checked 2026-10-13: Excel GAMMA.INV.
• G1 Google Sheets functions, other-product naming reference only, checked 2026-10-13: Google Sheets function list.

Validation record: Quantile predictions were independently checked using numerical inversion of the integer-shape gamma CDF. No Zoho native execution, screenshots or benchmark were performed. The next review stage should use both interior and boundary values, independent parameter models, locale settings and recorded account/product environment before making stronger runtime claims.