Chi-Square Distributions: Degrees of Freedom, Shape, and Normal Approximation

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What does $X \sim \chi^2_{df}$ mean?

The random variable $X$ follows a chi-square distribution with $df$ degrees of freedom; $df$ identifies the particular chi-square curve.

How is a chi-square variable with $k$ degrees of freedom constructed?

It is the sum of the squares of $k$ independent standard normal variables: $Z_1^2 + Z_2^2 + \cdots + Z_k^2$.

What are the mean and standard deviation of a chi-square distribution?

For $df$ degrees of freedom, the population mean is $df$ and the population standard deviation is $\sqrt{2df}$.

What is the general shape of a chi-square curve?

It is nonnegative, nonsymmetrical, and skewed to the right. Its mean is just to the right of its peak.

When can a chi-square distribution be approximated by a normal distribution?

When $df > 90$. For example, with $df = 1{,}000$, its mean is $1{,}000$ and its standard deviation is about $44.7$.

Quiz questions

1. How can a chi-square random variable with $k$ degrees of freedom be constructed from standard normal variables?Answer: By summing the squares of $k$ independent standard normal variables.
2. What values can a chi-square random variable take?Answer: Values greater than or equal to zero.
3. Which description best characterizes the shape of a chi-square curve?Answer: It is nonsymmetrical and skewed to the right.
4. Where is the mean of a chi-square distribution located relative to its peak?Answer: Just to the right of the peak.
5. At what degrees of freedom does the chi-square curve approximate a normal distribution, according to the stated rule?Answer: When $df > 90$.