Chi-Square Distribution Mean and Variance Calculator

Calculator Inputs

This calculator focuses on the central chi-square distribution. Enter the degrees of freedom, choose graph settings, and optionally test density values at reference x positions.

Degrees of freedom

Must be greater than zero.

Decimal places

Controls numeric display precision.

Plot points

Higher values produce smoother curves.

Graph maximum x

Optional manual graph limit.

Reference x 1

Calculates PDF at this x-value.

Reference x 2

Useful for comparing two locations.

Formula Used

For a chi-square distribution with degrees of freedom k, the calculator uses these standard relationships.

Mean: μ = k

Variance: σ² = 2k

Standard deviation: σ = √(2k)

Mode: k − 2, for k ≥ 2; otherwise the boundary is x = 0

Skewness: √(8 / k)

Excess kurtosis: 12 / k

Coefficient of variation: √(2 / k)

PDF: f(x; k) = x^(k/2 − 1)e^(−x/2) / (2^(k/2)Γ(k/2)), for x > 0

The mean equals the degrees of freedom.
The variance is always double the degrees of freedom.
The shape becomes less skewed as degrees of freedom rise.
The graph uses the PDF formula to draw the distribution curve.

How to Use This Calculator

Enter the chi-square degrees of freedom.
Choose how many decimal places you want.
Adjust plot points for a smoother or lighter graph.
Leave graph maximum x blank for automatic scaling, or enter your own value.
Optionally enter one or two reference x-values to evaluate the density.
Press Calculate Now to show the result section above the form.
Use the export buttons to save the summary table as CSV or the result block as PDF.

Example Data Table

These sample rows show how the main moment measures change as degrees of freedom increase.

Degrees of freedom	Mean	Variance	Standard deviation	Mode	Skewness
1	1.00	2.00	1.4142	0.00	2.8284
2	2.00	4.00	2.0000	0.00	2.0000
5	5.00	10.00	3.1623	3.00	1.2649
10	10.00	20.00	4.4721	8.00	0.8944
20	20.00	40.00	6.3246	18.00	0.6325

Frequently Asked Questions

1. What is the mean of a chi-square distribution?

The mean equals the degrees of freedom, written as k. If k = 7, the mean is 7. This makes the center easy to interpret.

2. Why is the variance twice the degrees of freedom?

A chi-square variable is the sum of squared standard normal variables. That construction makes its variance equal to 2k, where k is the degrees of freedom.

3. Can the degrees of freedom be non-integer?

Yes. The distribution is mathematically valid for any positive real degrees of freedom, not only whole numbers. Many theoretical and modeling settings use fractional values.

4. What happens when the degrees of freedom are below 2?

The mode sits at the boundary x = 0. The density rises sharply near zero, so the curve becomes strongly right-skewed compared with larger degrees of freedom.

5. Why does the graph shift right as degrees of freedom increase?

Because the mean is exactly k, larger degrees of freedom move the distribution center rightward. The curve also spreads out and becomes less skewed.

6. Does this calculator compute probabilities or critical values?

This page focuses on mean, variance, shape measures, density values, and visualization. It does not return full tail probabilities or hypothesis-test critical values.

7. How should I choose the graph maximum x value?

Leave it blank for automatic scaling in most cases. Use a custom value when you want to inspect a narrower or wider region of the density curve.

8. Why are skewness and excess kurtosis included?

They describe distribution shape beyond mean and variance. Skewness shows asymmetry, while excess kurtosis indicates tail heaviness and peak behavior relative to a normal shape.