stats

Chi-Squared Distribution

Chi-squared distribution — PDF, CDF, p-value, critical χ², tail selector; normal approximation for df > 200.

Parameters

Degrees of freedom df

χ² value

Tail

Confidence level

Right tail: goodness-of-fit, independence, variance tests (large variance)

Results

PDF f(χ²)—

CDF P(χ² ≤ x)—

p-value (tail-tailed)—

Critical χ² (right, α=.05)—

Critical χ² (left, α=.05)—

Mean = df—

Variance = 2df—

Mode = max(df−2, 0)—

Skewness = √(8/df)—

Excess kurtosis = 12/df—

CDF via 1−Q(df/2, x/2) · Q = upper incomplete gamma
df > 200: √(2χ²) − √(2df−1) ~ N(0,1) approximation