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Introductory Statistics

11.1 Facts 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.
    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
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