Contents

Formula Review

χ² = (Z₁)² + (Z₂)² + . . . (Z_df)² chi-square distribution random variable

μ_χ² = df chi-square distribution population mean

$σ_{χ^{2}} = \sqrt{2 (d f)} σ_{χ^{2}} = \sqrt{2 (d f)}$ chi-square distribution population standard deviation

$\sum_{k} \frac{{(O - E)}^{2}}{E} \sum_{k} \frac{{(O - E)}^{2}}{E}$ goodness-of-fit test statistic where

O: observed values
E: expected values

k: number of different data cells or categories

df = k − 1 degrees of freedom

Test of Independence

The number of degrees of freedom is equal to (number of columns–1)(number of rows–1).
The test statistic is $\underset{(i \cdot j)}{Σ} \frac{{(O - E)}^{2}}{E} \underset{(i \cdot j)}{Σ} \frac{{(O - E)}^{2}}{E}$ where O = observed values, E = expected values, i = the number of rows in the table, and j = the number of columns in the table.
If the null hypothesis is true, the expected number $E = \frac{(row total)(column total)}{total surveyed} E = \frac{(row total)(column total)}{total surveyed}$ .

$\sum_{i \cdot j} \frac{{(O - E)}^{2}}{E} \sum_{i \cdot j} \frac{{(O - E)}^{2}}{E}$ Homogeneity test statistic where O = observed values
E = expected values
i = number of rows in data contingency table
j = number of columns in data contingency table

df = (i −1)(j −1) degrees of freedom

$χ^{2} = χ^{2} =$ $\frac{(n - 1) \cdot s^{2}}{σ^{2}} \frac{(n - 1) \cdot s^{2}}{σ^{2}}$ Test of a single variance statistic where
n: sample size
s: sample standard deviation
σ: population standard deviation

df = n – 1 degrees of freedom

Test of a Single Variance

Use the test to determine variation.
The degrees of freedom is the number of samples – 1.
The test statistic is $\frac{(n - 1) \cdot s^{2}}{σ^{2}} \frac{(n - 1) \cdot s^{2}}{σ^{2}}$ , where n = the total number of data, s² = sample variance, and σ² = population variance.
The test may be left-, right-, or two-tailed.