Statistics

# 11.1Facts About the Chi-Square Distribution

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 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 it may be any uppercase letter.

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

χ2 = (Z1)2 + (Z2)2 + ... + (Zk)2, where the following are true:

• The curve is nonsymmetrical and skewed to the right.
• There is a different chi-square curve for each df.
Figure 11.2
• The test statistic for any test is always greater than or equal to zero.
• 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.
• The mean, μ, is located just to the right of the peak.
Figure 11.3
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