Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Introductory Statistics 2e

11.1 Facts About the Chi-Square Distribution

Introductory Statistics 2e11.1 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.
    Part (a) shows a chi-square curve with 2 degrees of freedom. It is nonsymmetrical and slopes downward continually. Part (b) shows a chi-square curve with 24 df. This nonsymmetrical curve does have a peak and is skewed to the right. The graphs illustrate that different degrees of freedom produce different chi-square curves.
    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.
    This is a nonsymmetrical chi-square curve which is skewed to the right. The mean, m, is labeled on the horizontal axis and is located to the right of the curve's peak.
    Figure 11.3
PreviousNext
Order a print copy
Citation/Attribution

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.

Attribution information
  • 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:
    Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
  • 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:
    Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
Citation information

© Jul 18, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.