CHISQ.INV in Google Sheets: Left-Tail Quantiles
Find chi-square cutoffs from cumulative probabilities in Google Sheets, verify them with a round trip and construct a central probability interval.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: =CHISQ.INV(0.9,2) has expected result 4.605170186. It finds the cutoff with 90% of a two-degree chi-square distribution to its left. The input is a probability; the output is a value on the chi-square scale.
Syntax and argument limits
=CHISQ.INV(probability, degrees_freedom)
According to Google’s CHISQ.INV reference:
probabilityis the left-tail probability and must be greater than 0 and less than 1.degrees_freedommust be numeric and at least 1. Fractional values are truncated to an integer.- Both arguments are numeric and required; no optional defaults are documented.
The inverse page does not specify the same upper degree bound stated on some distribution pages. Do not assume another function’s detailed bounds apply simply because the names are related.
A left-tail probability of 0.9 means that 10% remains above the returned cutoff. It does not mean that the cutoff is 0.9 or that 90% is the observed value being tested.
Prepare the example inputs
Enter the following table in A1:B3:
| Row | A: Parameter | B: Value |
|---|---|---|
| 1 | Parameter | Value |
| 2 | Left-tail probability | 0.9 |
| 3 | Degrees of freedom | 2 |
In D2 enter:
=CHISQ.INV(B2,B3)
Expected result: 4.605170185988092, usually displayed with fewer digits. Format the output as a number. A percentage display would be misleading because the answer is a cutoff, not a probability.
Enter 90% or 0.9 for the probability. Entering 90 is outside the documented probability domain.
Round-trip and independent arithmetic checks
Use the inverse result in Google’s cumulative distribution function:
=CHISQ.DIST(D2,B3,TRUE)
Expected result: approximately 0.9. This forward calculation checks that the cutoff corresponds to the intended tail and degree count. Use TRUE; the density at the cutoff does not recover the original probability.
For exactly two degrees of freedom, the cumulative function is 1−exp(−x/2), so solving for x gives an independent formula:
=-2*LN(1-B2)
Expected result: 4.605170185988092. This shortcut is restricted to the two-degree example. Keep CHISQ.INV for a general degree input.
The NIST chi-square reference supplies the mathematical distribution definition. The logarithmic expression here is an independent algebraic specialization, not a claim that Sheets uses this implementation.
Choosing the right probability
Compare these calls with two degrees of freedom:
=CHISQ.INV(0.95,2)
=CHISQ.INV(0.05,2)
Expected answers:
- 0.95 left-tail probability: 5.9914645471.
- 0.05 left-tail probability: 0.1025865888.
For a test that rejects for large chi-square values with a 5% upper tail, the first cutoff is relevant. Entering 0.05 in a left-tail inverse selects the lower cutoff instead.
Google documents CHISQ.INV.RT for a right-tail probability. Mathematically, CHISQ.INV(1−p,df) and CHISQ.INV.RT(p,df) select the same cutoff for an interior probability. Use the spelling that expresses your intended tail clearly.
Advanced application: a central 95% reference interval
To leave 2.5% in each tail of a two-degree chi-square distribution, use:
=CHISQ.INV(0.025,2)
=CHISQ.INV(0.975,2)
Expected lower cutoff: 0.05063561597. Expected upper cutoff: 7.37775890823. The cumulative areas differ by 0.975−0.025 = 0.95, so the interval contains 95% of this mathematical distribution.
These are distribution cutoffs. They are not automatically a confidence interval for a population variance, a prediction interval for raw observations or a two-sided p-value. Those analyses require their own statistic, transformation and assumptions.
To audit an interval worksheet, keep lower and upper tail probabilities in labeled cells, compute both cutoffs, and confirm lower < upper. A reversed probability pair reverses the cutoffs even if each individual formula is valid.
Truncation, precision and troubleshooting
If B3 is changed from 2 to 2.9, Google’s truncation rule selects 2 degrees. The 0.9 cutoff therefore has the same documented mathematical expectation, 4.605170186. Do not interpret 2.9 as a smooth interpolation between two- and three-degree distributions.
For errors or surprises:
- Verify the probability is strictly inside 0–1 under Google’s documented contract.
- Confirm a positive, intended degree count and numeric input types.
- Check which tail the input probability describes.
- Keep unrounded probabilities and results for round-trip calculations.
- Verify formula separators if a pasted formula does not parse.
A probability extremely close to 1 can become exactly 1 if rounded or computed with insufficient numerical precision. For a small known upper-tail probability, the right-tail inverse avoids manually subtracting it from 1. No accuracy guarantee for extreme inputs is made here.
The Google page does not provide a full error-code or coercion table. Its worked example’s narrative cutoff differs from its displayed numeric solution; this guide uses original two-degree examples verified by direct mathematics rather than reproducing that inconsistent narrative. Native error behavior remains untested.
Availability and verification
CHISQ.INV is documented for the current rolling Google Sheets service. Its introduction date and last functioning numbered release were not established. The legacy CHIINV name describes a right-tail inverse, so replacing CHISQ.INV with CHIINV would change the probability convention.
Evidence status: NOT_TESTED_IN_GOOGLE_SHEETS. Individual Google help, supporting distribution references and indexed official bodies were inspected on October 17, 2026. Worked cutoffs were independently calculated; no Google Sheets execution or screenshot is presented.