Google Sheets CHISQ.INV Function: Syntax, Examples & Tips

What Is CHISQ.INV?

CHISQ.INV is the inverse of the left-tailed chi-square distribution. You give it a probability p and the degrees of freedom, and it returns the value x such that a chi-square variable falls at or below x with probability p. Put simply, it returns a percentile of the chi-square distribution.

The chi-square distribution describes the sum of squares of independent standard normal variables. It only takes non-negative values, is skewed to the right, and has one parameter: degrees of freedom. With few degrees of freedom the curve is bunched near zero with a long right tail; as degrees of freedom grow, it shifts right and becomes more symmetric. Its mean equals the degrees of freedom.

In practice you meet it most often in hypothesis tests: goodness-of-fit tests (does observed data match an expected pattern?), tests of independence in contingency tables (are two categories related?), and tests about a variance.

How CHISQ.INV Works

CHISQ.INV undoes CHISQ.DIST with cumulative set to TRUE. If CHISQ.DIST(x, df, TRUE) returns p, then CHISQ.INV(p, df) returns x.

Because it counts probability from the left, a probability of 0.95 returns a value near the upper end of the distribution, and 0.05 returns a value near the lower end. For right-tail critical values used in most tests, CHISQ.INV.RT is more convenient: CHISQ.INV(1 - α, df) equals CHISQ.INV.RT(α, df).

Basic Examples

95th percentile with 1 degree of freedom

=CHISQ.INV(0.95, 1)

Returns about 3.841.

5th percentile with 5 degrees of freedom

=CHISQ.INV(0.05, 5)

Returns about 1.145. Only 5% of the distribution lies below this value.

The median with 2 degrees of freedom

=CHISQ.INV(0.5, 2)

Returns about 1.386 (that is, 2 × ln 2).

Round-trip check

=CHISQ.DIST(CHISQ.INV(0.9, 7), 7, TRUE)

Returns 0.9.

Confidence Interval for a Variance

A standard use of CHISQ.INV is building a confidence interval for a population variance from a sample. With the sample variance in B1 and sample size in B2, a 95% interval is:

Lower:  =(B2-1)*B1 / CHISQ.INV(0.975, B2-1)
Upper:  =(B2-1)*B1 / CHISQ.INV(0.025, B2-1)

Take square roots of both limits for an interval on the standard deviation. This assumes the data comes from a roughly normal population.

Common Use Cases

• Variance and standard deviation intervals: as shown above
• Lower-tail tests: is a process less variable than a target?
• Critical values: when a textbook or method specifies left-tail percentiles
• Simulation: =CHISQ.INV(RAND(), df) generates chi-square random values

Common Errors

• #NUM!: probability is outside the range 0 to 1, or degrees_freedom is less than 1.
• #VALUE!: An argument is text.
• Using the wrong tail: For a 5% significance test on the right tail, use CHISQ.INV(0.95, df) or CHISQ.INV.RT(0.05, df), not CHISQ.INV(0.05, df).
• Mixing up probability and significance level: α = 0.05 corresponds to a left-tail probability of 0.95.

Tips and Best Practices

• Keep the confidence level in a cell and derive the tail probabilities from it, so the whole interval updates when you change it.
• Round critical values to three decimals to match printed tables.
• Check results against the forward function as a quick audit.

Related Functions

• CHISQ.DIST: left-tailed probability or density
• CHISQ.INV.RT: critical value from a right-tail probability
• CHIINV: legacy inverse right-tailed function
• CHISQ.DIST.RT: right-tailed probability (p-value)
• ROUND: round results for reporting