CHISQ.DIST in Google Sheets: Left Tail and Density
Use CHISQ.DIST in Google Sheets to distinguish cumulative probability from density, calculate intervals and check degrees-of-freedom truncation.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: =CHISQ.DIST(2,2,TRUE) returns a left-tail probability with expected value 0.6321205588. =CHISQ.DIST(2,2,FALSE) returns the curve’s density instead, expected 0.1839397206. The final argument changes the meaning of the result, not just its display.
Syntax and documented constraints
=CHISQ.DIST(x, degrees_freedom, cumulative)
xis the point where the distribution is evaluated. Google’s help specifies a positive numeric value.degrees_freedomcontrols the chi-square distribution. Google requires at least 1 and no more than 10^10, and documents truncation to an integer for a fractional argument.cumulativechooses the output:TRUEfor the left-tail cumulative distribution,FALSEfor the probability density.
The individual Google reference supplies all three positions and does not establish an omitted-switch default. Supply the Boolean explicitly. Its argument prose abbreviates the name once, but the title, syntax and examples identify CHISQ.DIST; use that full name.
For a continuous random variable X, cumulative mode gives P(X ≤ x). Density is a height on the distribution curve; it is not P(X = x). A single exact point has zero probability in a continuous model.
Set up a small two-degree example
Enter the following table in A1:B3. Keep D2:D4 empty for output.
| Row | A: Parameter | B: Value |
|---|---|---|
| 1 | Parameter | Value |
| 2 | Evaluation point | 2 |
| 3 | Degrees of freedom | 2 |
In D2 enter:
=CHISQ.DIST(B2,B3,TRUE)
Expected result: 0.6321205588285577, about 63.21206% of this distribution lies at or below 2.
In D3 enter:
=CHISQ.DIST(B2,B3,FALSE)
Expected result: 0.1839397205857212. Do not describe this as an 18.39% chance of obtaining a value of 2. Density becomes a probability only when integrated over an interval.
Independent calculation at exactly two degrees of freedom
The NIST chi-square distribution reference defines the density. With exactly two degrees of freedom, it simplifies to 0.5×exp(−x/2), and the cumulative probability is 1−exp(−x/2).
For the inputs above, these formulas independently check the expected answers:
=1-EXP(-B2/2)
=0.5*EXP(-B2/2)
They give 0.6321205588 and 0.1839397206. These shortened expressions are specific to two degrees of freedom. They are not replacements for the general function when B3 changes.
Advanced application: probability between two positive cutoffs
For two degrees of freedom, find the probability between 1 and 4:
=CHISQ.DIST(4,2,TRUE)-CHISQ.DIST(1,2,TRUE)
Expected result: 0.4711953764760207. The calculation subtracts the cumulative area up to 1 from the area up to 4. Because the distribution is continuous, including or excluding an exact endpoint does not change this mathematical area.
Do not subtract the two FALSE results. Differences in density heights do not measure interval probabilities.
If the question instead concerns values above a threshold, use the separately documented CHISQ.DIST.RT. It expresses the right-tail request directly; a small right tail can lose precision when obtained by subtracting a rounded cumulative probability from 1.
Build a comparison grid
Put 1, 2, 4, 6 in F2:F5. Enter these formulas in G2 and H2 and fill down:
=CHISQ.DIST(F2,2,TRUE)
=CHISQ.DIST(F2,2,FALSE)
| x | Expected cumulative probability | Expected density |
|---|---|---|
| 1 | 0.3934693403 | 0.3032653299 |
| 2 | 0.6321205588 | 0.1839397206 |
| 4 | 0.8646647168 | 0.0676676416 |
| 6 | 0.9502129316 | 0.0248935342 |
The cumulative column increases, as accumulated area must. In this particular two-degree case the density decreases. Do not generalize that decreasing density shape to every possible degree count.
Degrees of freedom, errors and troubleshooting
Changing the original B3 from 2 to 2.9 still uses 2 degrees under Google’s documented truncation rule. The expected cumulative result remains 0.6321205588. Truncation is not rounding to the nearest integer, and it is not a reason to supply a fractional degree count without understanding the statistical model.
For a failed or surprising formula:
- Check that
TRUEversusFALSEmatches the requested quantity. - Check numeric x and degree inputs against the documented domain.
- Verify whether degrees of freedom came from categories, constraints or fitted parameters; it is not automatically the sample size.
- Use the two-degree calculation as a clean diagnostic, then restore the intended model.
- Inspect local separators and the exact dotted function name if parsing fails.
The cited help does not enumerate the exact native error code for every invalid input. It specifies positive x; this guide does not promise native behavior at x = 0, negative x, or text inputs. Similarly, a valid mathematical result does not prove that a chi-square model is appropriate for the underlying study.
Availability and verification
CHISQ.DIST is documented for the current rolling Google Sheets service. Original introduction date, introduction version and last functioning numbered release are not established by the reviewed sources. Google’s right-tail legacy name CHIDIST should not be substituted for this density/left-tail function.
Evidence status: NOT_TESTED_IN_GOOGLE_SHEETS. Function-specific Google documentation and NIST’s mathematical definition were inspected on October 17, 2026. Direct retrieval was incomplete for some Google pages, so indexed first-party bodies were used and recorded. Expected values are independent calculations, not native Sheets results or screenshots.