Inputs
Formula reference
- Mean E[X] =
k - Variance Var[X] =
2k - Std dev σ =
√(2k) - Mode max(
k − 2, 0) - Median ≈
k(1 − 2/(9k))³(Wilson–Hilferty)
Results
| Quantity | Value | Copy |
|---|---|---|
| Mean E[X] | 4.0000 | |
| Variance Var[X] | 8.0000 | |
| Standard deviation σ | 2.8284 | |
| Mode | 2.0000 | |
| Median (approx.) | 3.3697 | |
| Skewness | 1.4142 | |
| Excess kurtosis | 3.0000 |
Values are rounded to 4 decimal places. For k < 2 the mode is set to 0 by convention.
Derivation sketch
If X = \sum_{i=1}^k Z_i^2 with Z_i ∼ 𝒩(0,1) iid then E[X] = k and Var[X] = 2k by linearity of expectation and variance additivity for independent variables.
Wilson–Hilferty approximation
The cube-root transformation yields (X/k)^{1/3} ≈ 𝒩(1 − 2/(9k), 2/(9k)), giving the median approximation k(1 − 2/(9k))^3.
FAQs
The mean equals the degrees of freedom k. It follows from linearity of expectation because the distribution is a sum of k independent squared standard normal variables.
The variance is 2k because variances of independent variables add and each squared standard normal has variance 2.
Differentiating the density and setting the derivative to zero gives the maximizer at k − 2 for k ≥ 2. For smaller k the density decreases from zero so the conventional mode is 0.
The median has no simple closed form. The tool uses the Wilson–Hilferty approximation which is accurate for moderate or large k.
Yes. The chi square family generalizes to positive real degrees of freedom via the gamma distribution parameterization. Many applications require non integer k.
The core formulas are exact. Rounding is controlled by the decimal places setting. The median uses an approximation to provide a useful quick estimate.