Chi Distribution Calculator
Use this calculator to evaluate the chi distribution for a given degrees of freedom value k. You can compute PDF/CDF at a specific x, or find the quantile x from a cumulative probability p.
Note: The chi distribution is the distribution of the square root of a chi-square random variable with the same degrees of freedom.
What Is the Chi Distribution?
The chi distribution is a continuous probability distribution for nonnegative values. If a random variable follows a chi-square distribution with k degrees of freedom, then its square root follows a chi distribution with the same k. In practical terms, this distribution often appears when dealing with vector lengths made from independent standard normal components.
In statistics, it is frequently used in geometric interpretations of uncertainty, signal processing, and quality control contexts where magnitudes are important.
Chi Distribution Formula
Probability Density Function (PDF)
For x ≥ 0 and k > 0, the chi PDF is:
f(x; k) = xk-1 e-x²/2 / [2k/2 - 1 Γ(k/2)]
where Γ(·) is the gamma function.
Cumulative Distribution Function (CDF)
The CDF gives the probability that the random variable is less than or equal to x:
F(x; k) = P(k/2, x²/2)
where P is the regularized lower incomplete gamma function.
How to Use This Calculator
- Step 1: Enter degrees of freedom k (must be greater than 0).
- Step 2: To evaluate at a point, enter x and click Calculate PDF & CDF.
- Step 3: To find a percentile/quantile, enter p and click Calculate Quantile x.
- Step 4: Read output values including PDF, CDF, right-tail probability, and summary moments.
Interpreting the Results
The PDF value is a density, not a direct probability. It helps describe how concentrated values are around a given x.
CDF
The CDF returns the probability that the random variable is at most x. For example, a CDF of 0.95 means 95% of outcomes lie below that value.
Survival Function
The right-tail probability is 1 - CDF. This is useful for threshold exceedance and risk analysis.
Where the Chi Distribution Appears
- Magnitude of Gaussian noise vectors in engineering
- Statistical process monitoring involving Euclidean norms
- Geometry of random vectors and distances
- Derivations connected to chi-square and gamma family models
Quick Practical Notes
The shape depends strongly on the degrees of freedom. Smaller k values produce heavier concentration near zero, while larger k shift the distribution rightward and make it more symmetric.
If you need variance-modeling tools, remember that chi and chi-square are related but not interchangeable: chi is about the square root of chi-square.
FAQ
Can k be non-integer?
Yes. The formulas are valid for any real k > 0, because they are defined through the gamma function.
What is the difference between chi and chi-square?
If Y ~ χ²(k), then X = √Y ~ χ(k). Chi is the square root transformation of chi-square.
Is this calculator numerically stable?
Yes. It uses logarithmic gamma evaluation and robust incomplete-gamma routines, plus bisection for quantiles.