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The Chi-Square Distribution: Facts About The Chi-Square Distribution

Module by: Dr. Barbara Illowsky, Susan Dean

  1. The curve is nonsymmetrical and skewed to the right.
  2. There is a different chi-square curve for each dfdf.
    Figure 1
    Subfigure 1.1Subfigure 1.2
    Example of a nonsymmetrical chi-square curve that has a different df from the graph on the right. The curve begins at (0,∞) and slopes downwards to (∞,0).Example of a nonsymmetrical and skewed to the right, the peak is closer to the left and more values are in the tail on the right, chi-square curve which has a different df from the graph on the left.
  3. The test statistic for any test is always greater than or equal to zero.
  4. When df > 90 df>90, the chi-square curve approximates the normal. For XX ~ χ 1000 2 χ 1000 2 the mean, μ = df = 1000 μ=df=1000 and the standard deviation, σ = 2 ⋅ 1000 = 44.7 σ= 2 ⋅ 1000 =44.7. Therefore, XX ~ N ( 1000 , 44.7 ) N(1000,44.7), approximately.
  5. The mean, μ μ, is located just to the right of the peak.
    Figure 2
    Example of how the mean is located to the right of the peak with a nonsymmetrical chi-square curve skewed to the right with the mean on the x-axis.

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