Math 221 – Principles of Statistics

Testing Claims about the Goodness-of-Fit

Purpose: To test whether an unknown distribution (or population) is the same as a "known" distribution (or population).


Requirements: Let  be a random variable having the unknown distribution. Set up a two-way table. One of the ways will be the factor “distribution” and the other will be the factor “values of ”.  Enter into the table, in the row (or column) for the unknown distribution, the observed counts of the various values of . (Note: You may need to bin the values of .)  Find the total number of observations of , and call it . For the “known” distribution, enter in each cell of the row (or column) the number of observations you could expect to find IF  had the “known” distribution.


(Example: Say you have 120 observations of , and suppose you need to compute the expected count for the bin . Use the “known” distribution to find the percentage of times  should be between 4 (inclusive) and 8 (exclusive). Say it’s 20%. Multiply that percentage by 120. The result (24) is the expected count for the bin . Now do the same sort of calculation for all the cells in the row (or column) for the “known” distribution.)


(1) The values of  are from a (simple) random sample.


(2) All expected counts are higher than 1.


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





There is no difference between the two distributions. That is, the two distributions are really the same.

The two distributions are not the same.



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.

WARNING: The  test statistic is very sensitive to round-off errors made during calculation. If you’re using a hand calculator, 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 goodness-of-fit: Click here.



Examples of testing claims about several proportions: Click here.



Related topics:


Chi-squared tests master reference page


Tests of independence


Tests of homogeneity (several proportions)


Comparing unknown populations



Statistics Reference Pages page: Click here


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