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# 11.2: Facts About the Chi-Square Distribution

The notation for the chi-square distribution is:

$\chi \sim \chi^{2}_{df}$

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 $$\chi^{2}$$ distribution, the population mean is $$\mu = df$$ and the population standard deviation is

$\sigma = \sqrt{2(df)}.$

The random variable is shown as $$\chi^{2}$$, 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.

$\chi^{2} = (Z_{1})^{2} + ... + (Z_{k})^{2}$

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

Figure $$\PageIndex{1}$$

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

Figure $$\PageIndex{2}$$

### References

1. Data from Parade Magazine.
2. “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

$\chi^{2} = (Z_{1})^{2} + (Z_{2})^{2} + ... + (Z_{df})^{2}$ chi-square distribution random variable

$$\mu_{\chi^{2}} = df$$ chi-square distribution population mean

$$\sigma_{\chi^{2}} = \sqrt{2(df)}$$ Chi-Square distribution population standard deviation

Exercise 11.2.1

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

Answer

mean $$= 25$$ and standard deviation $$= 7.0711$$

Exercise 11.2.2

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

Exercise 11.2.3

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

Answer

when the number of degrees of freedom is greater than 90

Exercise 11.2.4

Where is $$\mu$$ located on a chi-square curve?

Exercise

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

Figure 11.2.3.

Answer

$$df = 2$$