# 11.1: Facts About the Chi-Square Distribution

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The notation for the chi-square distribution is:

$\chi \sim \chi_{d f}^{2}\nonumber$

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 $$\sigma=\sqrt{2(d f)}$$.

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

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

$\chi^{2}=\left(Z_{1}\right)^{2}+\left(Z_{2}\right)^{2}+\ldots+\left(Z_{k}\right)^{2}\nonumber$

1. The curve is non-symmetrical and skewed to the right.
2. There is a different chi-square curve for each $$df$$ ($$\PageIndex{1}$$).
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 $$\chi \sim \chi_{1,000}^{2}$$ the mean, $$\mu = df = 1,000$$ and the standard deviation, $$\sigma=\sqrt{2(1,000)}=44.7$$. Therefore, $$\chi \sim N(1,000,44.7)$$, approximately.
5. The mean, $$\mu$$, is located just to the right of the peak.

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