The Chi-Square Distribution

# Facts About the Chi-Square Distribution

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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 \(\sigma =\sqrt{2\left(df\right)}\).

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}

- The curve is nonsymmetrical and skewed to the right.
- There is a different chi-square curve for each
*df*. - 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*~ \({\chi }_{1,000}^{2}\) the mean,*μ*=*df*= 1,000 and the standard deviation,*σ*= \(\sqrt{2\left(1,000\right)}\) = 44.7. Therefore,*X*~*N*(1,000, 44.7), approximately. - 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 _{χ2}* =

*df*chi-square distribution population mean

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

If the number of degrees of freedom for a chi-square distribution is 25, what is the population mean and standard deviation?

mean = 25 and standard deviation = 7.0711

If *df* > 90, the distribution is _____________. If *df* = 15, the distribution is ________________.

When does the chi-square curve approximate a normal distribution?

when the number of degrees of freedom is greater than 90

Where is *μ* located on a chi-square curve?

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

*df* = 2

# Homework

*Decide whether the following statements are true or false.*

As the number of degrees of freedom increases, the graph of the chi-square distribution looks more and more symmetrical.

true

The standard deviation of the chi-square distribution is twice the mean.

The mean and the median of the chi-square distribution are the same if *df* = 24.

false