Introductory Statistics

# 11.1Facts About the Chi-Square Distribution

Introductory Statistics11.1 Facts About the Chi-Square Distribution

The notation for the chi-square distribution is:

$χ∼ χ df 2 χ∼ χ df 2$

where df = degrees of freedom which depends on how chi-square is being used. (If you want to practice calculating chi-square probabilities then use df = n - 1. The degrees of freedom for the three major uses are each calculated differently.)

For the χ2 distribution, the population mean is μ = df and the population standard deviation is $σ= 2(df) σ= 2(df)$.

The random variable is shown as χ2, but may be any upper case letter.

The random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables.

χ2 = (Z1)2 + (Z2)2 + ... + (Zk)2

1. The curve is nonsymmetrical and skewed to the right.
2. There is a different chi-square curve for each df.
Figure 11.2
3. The test statistic for any test is always greater than or equal to zero.
4. When df > 90, the chi-square curve approximates the normal distribution. For X ~ $χ 1,000 2 χ 1,000 2$ the mean, μ = df = 1,000 and the standard deviation, σ = $2(1,000) 2(1,000)$ = 44.7. Therefore, X ~ N(1,000, 44.7), approximately.
5. The mean, μ, is located just to the right of the peak.
Figure 11.3

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

• If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
• If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution: