The chi-squared distribution (also written χ²) is a sampling distribution derived from the normal distribution. Suppose we have a random sample of size n from a normal(μ, σ²) distribution, with sample variance S². The statistic (n − 1)S²/σ² has a chi-squared distribution with ν = n − 1 degrees of freedom.
The function Γ(s) in the pdf and cdf denotes the gamma function, while the function γ(s, x) in the cdf denotes the lower incomplete gamma function
| Parameter | Range | Description |
|---|---|---|
| ν | ν = 1, 2, ... | Degrees of freedom |
Probability Density Function
Support
Mean
Variance
| Example | ν |
|---|---|
| Z is a standard normal random variable. Let X = Z². | 1.000 |
| A dart is thrown at a dart board. The probability distribution of hits is centered on the bullseye, and has independent standard normal distributions vertically and horizontally. Let X be the square of the distance of a hit from the bullseye. | 2.000 |
| Y₁, Y₂, ..., Y₁₀ is a sample taken from a normal(0, 4) distribution, with sample variance S². Let X = 9S²/4. | 9.000 |
X ∼ Chi-Squared(ν)
E(X) = , Var(X) =
The chi-squared distribution can be used to estimate a confidence interval for the true population variance σ² of a normally distributed population based on a sample variance S². The pdf is strictly decreasing for ν ≤ 2 and unimodel for ν > 2.
The chi-squared distribution also arises naturally as the probability distribution of the sum of the squares of n independent standard normal random variables. This sum will have a chi-squared(n) distribution.
The chi-squared distribution can be used to estimate a confidence interval for the true population variance σ² of a normally distributed population based on a sample variance S².
The chi-squared distribution also arises naturally as the probability distribution of the sum of the squares of n independent standard normal random variables. This sum will have a chi-squared(n) distribution.
The illustration above shows a sample of n points (marked red) chosen independently and at random from a normal(μ, σ2) distribution. The random variable X = (n − 1)S2/σ2 has a chi-squared(n − 1) distribution, where S2 is the sample variance.
The simulation above shows a sample of n points (marked red) chosen independently and at random from a normal(μ, σ2) distribution. The light blue line shows the value of X = (n − 1)S2/σ2, where S2 is the sample variance. The random variable X has a chi-squared(n − 1) distribution. The histogram accumulates the results of each simulation.
Y =
E(Y) = , Var(Y) =