Math 221 – Principles of Statistics

Testing Claims about Homogeneity (Equality of Several Proportions) Reference Page

Purpose: To test whether two or more population proportions are the same. (Note: If there are only two proportions to compare, we recommend using the test of two proportions, once you’ve learned about it.)


Requirements: Set up a two-way table of observed counts. One way is for the populations; the other is for "success" / "failure". Use the table to compute the expected counts for each cell in the table, thus:. The required conditions for the test are


1) The data are from simple random samples, one from each population.


2) The samples are independent of each other.


3) All expected counts are greater than 1.


4) No more than 20% of the expected counts are less than 5.





, where  is the number of groups. This hypothesis means, “All  proportions are the same.”

The alternative hypothesis is  At least one of the proportions is different from at least one of the others. (Hopefully, this reminds you of the ANOVA hypotheses.)



Test Statistic: We recommend using software such as SPSS to calculate the  test statistic, the degrees of freedom, and the p-value. However, you can use the formula , with  degrees of freedom, and get the p-value from a  table such as Table F in Moore & McCabe’s The Practice of Business Statistics. (Note: There will either be two rows and  columns or  rows and 2 columns. Either way, there will be  degrees of freedom.)



WARNING: The  test statistic is very sensitive to round-off errors made during calculation. We recommend using the above formula instead of the one usually found in textbooks, as it suffers less from this problem. Also, we recommend entering the entire calculation in your calculator all at once, rather than computing each term separately, to help minimize round-off error.



SPSS instructions for testing claims about several proportions: Click here.



Examples of testing claims about several proportions: Click here.



Related topics:


Chi-squared tests master reference page


Tests of independence


Comparing unknown populations


Goodness-of-fit (comparing populations when one population is “known”)



Statistics Reference Pages page: Click here


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