11.1 Facts about the chi-square distribution

The notation for the chi-square distribution is:

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 χ ² distribution, the population mean is μ = df and the population standard deviation is $\sigma =\sqrt{2(df)}$ .

The random variable is shown as χ ² , 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.

χ ² = ( Z ₁ ) ² + ( Z ₂ ) ² + ... + ( Z _k ) ²

1. The curve is nonsymmetrical and skewed to the right.
2. There is a different chi-square curve for each df .

   

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 ~ ${\chi }_{\mathrm{1,000}}^2$ the mean, μ = df = 1,000 and the standard deviation, σ = $\sqrt{2(\mathrm{1,000})}$ = 44.7. Therefore, X ~ N (1,000, 44.7), approximately.
5. The mean, μ , is located just to the right of the peak.

   

References

Data from Parade Magazine .

“HIV/AIDS Epidemiology Santa Clara County.”Santa Clara County Public Health Department, May 2011.

Chapter review

The chi-square distribution is a useful tool for assessment in a series of problem categories. These problem categories include primarily (i) whether a data set fits a particular distribution, (ii) whether the distributions of two populations are the same, (iii) whether two events might be independent, and (iv) whether there is a different variability than expected within a population.

An important parameter in a chi-square distribution is the degrees of freedom df in a given problem. The random variable in the chi-square distribution is the sum of squares of df standard normal variables, which must be independent. The key characteristics of the chi-square distribution also depend directly on the degrees of freedom.

The chi-square distribution curve is skewed to the right, and its shape depends on the degrees of freedom df . For df >90, the curve approximates the normal distribution. Test statistics based on the chi-square distribution are always greater than or equal to zero. Such application tests are almost always right-tailed tests.

Formula review

χ ² = ( Z ₁ ) ² + ( Z ₂ ) ² + … ( Z _df ) ² chi-square distribution random variable

μ _{χ ²} = df chi-square distribution population mean

${\sigma }_{{\chi }^2}\text{=}\sqrt{2\left(df\right)}$ Chi-Square distribution population standard deviation

Is it more likely the df is 90, 20, or two in the graph?

df = 2

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