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Summary of Formulas

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module provides a summary on formulas used in Chi-Square Distribution as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

The Chi-Square Probability Distribution

μ=dfμ=df and σ=2dfσ=2df

Goodness-of-Fit Hypothesis Test

  • Use goodness-of-fit to test whether a data set fits a particular probability distribution.
  • The degrees of freedom are number of cells or categories - 1number of cells or categories - 1.
  • The test statistic is Σ k ( O E ) 2 E Σ k ( O E ) 2 E , where OO = observed values (data), EE = expected values (from theory), and kk = the number of different data cells or categories.
  • The test is right-tailed.

Test of Independence

  • Use the test of independence to test whether two factors are independent or not.
  • The degrees of freedom are equal to (number of columns - 1)(number of rows - 1)(number of columns - 1)(number of rows - 1).
  • The test statistic is Σ ( i j ) ( O - E ) 2 E Σ ( i j ) ( O - E ) 2 E where OO = observed values, EE = expected values, ii = the number of rows in the table, and jj = the number of columns in the table.
  • The test is right-tailed.
  • If the null hypothesis is true, the expected number E = (row total)(column total) total surveyed E= (row total)(column total) total surveyed .

Test of Homogeneity

  • Use the test for homogeneity to decide if two populations with unknown distributions have the same distribution as each other.
  • The degrees of freedom are equal to number of columns - 1number of columns - 1.
  • The test statistic is Σ ( i j ) ( O - E ) 2 E Σ ( i j ) ( O - E ) 2 E where OO = observed values, EE = expected values, ii = the number of rows in the table, and jj = the number of columns in the table.
  • The test is right-tailed.
  • If the null hypothesis is true, the expected number E = (row total)(column total) total surveyed E= (row total)(column total) total surveyed .

Note:

The expected value for each cell needs to be at least 5 in order to use the Goodness-of-Fit, Independence and Homogeneity tests.

Test of a Single Variance

  • Use the test to determine variation.
  • The degrees of freedom are the number of samples - 1.
  • The test statistic is ( n - 1 ) s 2 σ 2 ( n - 1 ) s 2 σ 2 , where nn = the total number of data, s2s2 = sample variance, and σ2σ2 = population variance.
  • The test may be left, right, or two-tailed.

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