Formula Review

11.1 Facts About the Chi-Square Distribution

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

μχ2 = df chi-square distribution population mean

σχ2=2(df)σχ2=2(df) chi-square distribution population standard deviation

11.2 Goodness-of-Fit Test

k(OE)2Ek(OE)2E 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

11.3 Test of Independence

Test of Independence
  • The number of degrees of freedom is equal to (number of columns–1)(number of rows–1).
  • The test statistic is Σ(ij)(OE)2EΣ(ij)(OE)2E 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=(row total)(column total)total surveyedE=(row total)(column total)total surveyed.

11.4 Test for Homogeneity

ij(OE)2Eij(OE)2E 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

11.6 Test of a Single Variance

χ2=χ2=(n1)s2σ2(n1)s2σ2 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 (n1)s2σ2(n1)s2σ2, where n = the total number of data, s2 = sample variance, and σ2 = population variance.
  • The test may be left-, right-, or two-tailed.