Inverse Chi-Square Calculator

What Is the Inverse Chi-Square Calculator?

The inverse chi-square function finds the χ² value corresponding to a given probability. While the Chi-Square Distribution Calculator finds P(χ² < x), this inverse asks "What χ² value has this probability below it?"

Given: Probability (cumulative), degrees of freedom Returns: The χ² value at that percentile

Common Critical Values

For right-tail probability α = 0.05 (cumulative p = 0.95):

For α = 0.01 (cumulative p = 0.99):

Applications

Hypothesis Testing

Chi-square test of independence: "At α = 0.05 with df = 4, what's the critical value?"

χ²₀.₉₅(4) = 9.49

Reject H₀ if observed χ² > 9.49

Goodness of fit test: "Testing 6 categories at α = 0.01, what's the critical χ²?"

df = 6 - 1 = 5 χ²₀.₉₉(5) = 15.09

Confidence Interval for Variance

For a normal population, the CI for σ² uses chi-square quantiles:

Lower: (n-1)s² / χ²₁₋α/₂ Upper: (n-1)s² / χ²α/₂

Example: n = 25, s² = 16, 95% CI

df = 24 χ²₀.₉₇₅(24) = 39.36 χ²₀.₀₂₅(24) = 12.40

CI: (24×16/39.36, 24×16/12.40) = (9.76, 30.97)

Control Charts for Variability

S-charts (standard deviation charts) and R-charts (range charts) use chi-square quantiles to set control limits.

Upper vs. Lower Tail

Upper percentile (right tail): χ²₀.₉₅ = value with 5% above it

Used for: Tests where large χ² indicates significance

Lower percentile (left tail): χ²₀.₀₅ = value with 5% below it

Used for: Lower bound of variance CI, tests for variance too small

Chi-Square Distribution Properties

As df increases:

• Mean = df
• Variance = 2 × df
• Distribution becomes more symmetric
• Approaches normal: χ² ≈ N(df, √(2df)) for large df

Historical Context

Chi-square critical value tables were essential before computers.

Karl Pearson published the first chi-square tables around 1900.

Ronald Fisher expanded the tables and connected them to degrees of freedom.

The incomplete gamma function underlies chi-square computation, making numerical methods essential for accurate quantile calculation.

Computational Methods

The inverse chi-square has no closed form. Methods include:

Newton-Raphson iteration: Solve CDF(x) - p = 0

Wilson-Hilferty approximation: For large df: χ² ≈ df × [1 - 2/(9df) + z×√(2/(9df))]³

This cube transformation gives a good approximation to the inverse.

Bisection with CDF: Binary search using forward CDF calculation

Two-Tailed vs. One-Tailed

One-tailed (most chi-square tests): Compare observed χ² to χ²₁₋α

Only large χ² values indicate significance.

Two-tailed (variance tests): Test whether variance is different (not just larger or smaller):

• Lower critical value: χ²α/₂
• Upper critical value: χ²₁₋α/₂

Quantile Tables

Relationship to Other Distributions

Normal: Sum of k squared standard normals is χ²(k)

Gamma: χ²(k) = Gamma(k/2, 2)

F-distribution: F = (χ₁²/df₁) / (χ₂²/df₂)

Exponential: χ²(2) = Exponential with rate 1/2

Degrees of Freedom Selection

The Degrees of Freedom Calculator helps determine df.

Related Calculations

• Chi-Square Distribution Calculator — Forward calculation
• Chi-Square Test of Independence — Uses critical χ²
• Chi-Square Goodness of Fit — Uses critical χ²
• Variance Calculator — Sample variance
• Degrees of Freedom Calculator — Calculate df

Summary

The inverse chi-square calculator converts cumulative probabilities to χ² values for any degrees of freedom. Essential for finding critical values in chi-square tests, constructing confidence intervals for variance, and setting control chart limits, the inverse function complements the forward CDF. The chi-square distribution is always right-skewed (more so for small df), so most tests use upper-tail critical values, though variance intervals require both tails.