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10.1: Facts About the Chi-Square Distribution

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    Facts About the Chi-Square Distribution

    jkesler


    [latexpage]

    The notation for the chi-square distribution is:

    $$\chi \sim \chi_{df}^2$$

    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 $\sigma = \sqrt{2(df)}$

    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 non-symmetrical 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 $\chi \sim \chi_{1000}^2$ the mean, μ = df = 1,000 and the standard deviation, $\sigma = \sqrt{2(1000)}=44.7$. Therefore, $X\sim N(1,000, 44.7)$, 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

    10.1: Facts About the Chi-Square Distribution is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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