Zoho Sheet CHISQINV: Left-Tail Chi-Square Percentiles

This tutorial follows Zoho Sheet’s English online help reference checked on 2026-10-13. The examples use the help article’s semicolon argument separator and decimal point. In a workbook with another locale, inspect File → Spreadsheet Settings → Locale (language, decimal separator, grouping and regional formats are configurable); use the syntax accepted by your actual spreadsheet rather than guessing from a different product. Zoho locale settings.

These probability calculations assume the mathematical chi-square distribution with the specified positive integer degrees of freedom. A chi-square statistic is generally nonnegative; it is not a raw percentage or a test score. For hypothesis tests, the statistic, degrees of freedom and sampling assumptions must be justified before obtaining a tail probability or cutoff. A low p-value under a model is not the probability that a hypothesis is true, and statistical significance is not an effect size.

Zoho’s current public help article does not identify the function’s earliest-supported product release or establish exact parity across browser, mobile, desktop/offline, editions or account plans. Accordingly, no first-version claim or unsupported platform guarantee is made. No formula was executed in a signed-in Zoho Sheet instance. The values below are independently calculated using a standard chi-square model (SciPy 1.17.0), not screenshots, Zoho output or vendor certification.

=CHISQINV(p; k)

Zoho’s CHISQINV page names this the inverse of the left-tailed chi-square probability and documents the argument order (p;k). It lists no optional arguments or defaults. Mathematically, the returned cutoff q solves P(X ≤ q) = p where X ~ χ²(k).

The word inverse does not mean 1 / probability. It means recovering a statistic from its cumulative probability. For instance, p=0.95 returns a value around 11.07, not 1/0.95 = 1.0526. The statistic is dimensionless and should be compared to a statistic from the same model and degrees of freedom.

Create a blank Zoho Sheet and enter the following table beginning at A1. A and B/C below are plain values; the reference probabilities in D:F were calculated independently for checking and are not outputs of Zoho Sheet:

Format probability cells with 9–12 decimal places when inspecting a calculation. Formatting alone can make two close values look identical; rounding also affects round-trip inverse checks. Formulas belong in a new column G or farther right so you do not overwrite the comparison data. A PDF (probability density) is not the probability that a continuously distributed statistic equals a single exact number: for a continuous variable that point probability is zero.

1. Type Recovered x in G1.
2. In G2 use =CHISQINV(E2;C2), where the reference left-tail probability E2 was calculated independently from a chi-square statistic of 14 with k=5.
3. The independently expected output is about 14.000000000. Fill G2 through G5: expected 14, 4, 6, 10, allowing for rounding of the 12-decimal probabilities shown in the reference table.
4. In H2, use =CHISQDIST(G2;C2;TRUE) as the independently justified CDF round-trip. The result should approximate E2, not D2. This is a suggested Zoho validation check, not a native execution record.
5. For an explicit constant example, =CHISQINV(0.95;5) has independent expected output 11.070497694. Check it mathematically by evaluating the CDF at 11.070497694, which is approximately 0.95.

Contrast the wrong argument: =CHISQINV(0.05;5) computes a lower 5th-percentile value, independently 1.145476226. Both formulas accept plausible probabilities and produce plausible positive values, so a tail-direction mistake usually produces a wrong valid number, not an error message.

Set J1=Left-tail probability, K1=Degrees of freedom, L1=Critical value, then enter:

For a fixed k, higher left-tail probability yields a larger quantile. This is the opposite input-direction convention from CHIINV(p;k): there, a smaller right-tail p yields a larger upper cutoff.

With right-tail significance alpha = 0.05, the upper threshold uses =CHISQINV(1-0.05;5). This is mathematically the same 11.070497694 cutoff as =CHIINV(0.05;5); use the explicit 0.95 in a reviewed workbook to communicate the left-tail convention clearly.

Suppose n=16 independent observations form an approximately normal population sample, and the unbiased sample variance is s²=9. A textbook two-sided 95% chi-square interval for the population variance divides (n−1)s² = 135 by chi-square quantiles with k = n−1 = 15:

Lower variance = 135/CHISQINV(0.975;15)
Upper variance = 135/CHISQINV(0.025;15)

The independently calculated percentile cutoffs are 27.488392863443 and 6.262137795043 respectively. Thus the independently expected variance interval is approximately 4.911163802 to 21.558133088, not a confidence interval for the standard deviation. If the user needs the standard-deviation interval, take square roots of both nonnegative endpoints; the underlying assumptions remain the same.

This derivation belongs to the statistical method, not Zoho’s vendor article. An interval this wide reflects limited information from 16 observations; it does not say there’s a 95% probability that a fixed parameter is in that exact realized interval. Strong departures from population normality can invalidate the chi-square-based variance interval. Do not deploy the method on dependent or heavily skewed samples merely because the formula evaluates.

The official Zoho CHISQINV article lists #NUM when p is outside 0 to 1 or k<1. It does not document a guaranteed result at exactly p=0 or p=1; the continuous chi-square inverse approaches zero or infinity at these mathematical limits, but a finite spreadsheet implementation may have its own behavior. Do not invent an error code or an infinity display for the Zoho product. Likewise, the documentation does not establish fractional-k coercion or all implementation-specific numerical precision cases. Provide a positive integer k and an interior p for dependable worked examples, and natively test your target account for edge cases.

For p very close to 0 or 1, rounding a probability to only three decimals can materially alter a quantile. Store the underlying probability at sufficient precision; display formatting is not the same thing as value precision. Imported text such as 95 should be validated: it represents ninety-five, not 95%, unless deliberately converted.

The function-specific help page lists generic errors alongside specific #NUM conditions. A generic entry does not mean every such error necessarily occurs for every input. Neither floating-point accuracy guarantees nor boundary outcomes not explicitly documented by Zoho are asserted here. Large degrees of freedom, noninteger degrees of freedom, probability endpoints, blank inputs, logical values and high-precision tails remain native validation cases; do not infer their treatment from Excel or another calculator.

The matching upper cutoffs are not an alias guarantee: these remain separately documented names with different input probabilities. In cross-software migrations, the semantic meaning of p comes before the spelling of the function name.

Zoho’s help article supplies the behavior described in the syntax section. Modern Excel separately documents CHISQ.DIST(x,deg_freedom,cumulative) where its third argument is required, whereas the Zoho CHISQDIST help page documents an optional flag that defaults to FALSE. Microsoft’s Excel documentation is evidence for Excel only; it does not prove that Zoho has identical numeric coercions, aliases, error names or version availability. Legacy CHI... names in Excel or Google Sheets should be mapped by tail and function type rather than by visual similarity alone.

For porting a model, explicitly specify tails and cumulative modes, preserve degrees of freedom, and test representative cases in the target spreadsheet. A right-tail probability and a left-tail probability are complements for a continuous chi-square statistic, but the corresponding inverse functions receive different probability inputs for the same cutoff.

• Zoho Sheet CHIDIST tutorial
• Zoho Sheet CHIINV tutorial
• Zoho Sheet CHISQDIST tutorial
• Zoho Sheet CHISQINV tutorial

Official Zoho sources, opened and checked on 2026-10-13 (each item supports the stated syntax, direction, related function or the locale setting; none establishes first release):

• Zoho Sheet — this function reference
• CHIDIST in Zoho Sheet
• CHIINV in Zoho Sheet
• CHISQDIST in Zoho Sheet
• CHISQINV in Zoho Sheet
• CHITEST in Zoho Sheet
• Zoho Sheet — spreadsheet locale settings
• Microsoft Excel — CHISQ.DIST behavior (Excel comparison only)

Validation: Official documentation checked on 2026-10-13; worked numbers independently recalculated. Native Zoho executions: 0. Screenshots: 0. Runtime edge cases remain unverified.