Chi-Square "Goodness of Fit" Test
  with an Option for Monte Carlo Simulation
  of the Multinomial Sampling Distribution

The logic and computational details of chi-square tests
are described in Chapter 8 of Concepts and Applications.

This page will calculate the value of chi-square for a one-dimensional "goodness of fit" test, for up to 8 mutually exclusive categories labeled A through H. To enter an observed cell frequency, click the cursor into the appropriate cell, then type in the value. Expected values can be entered as either frequencies or proportions. If you enter the expected values as proportions, the entries can take the form of either decimal fractions such as .25, or common fractions such as 1/4. Whenever possible, it is better to enter common fractions rather than rounded decimal fractions: 1/3 rather than .3333; 1/6 rather than .1667; and so forth.

When all observed and expected values have been entered, click the «Calculate» button. To perform a new analysis with a new set of data, click the «Reset» button.
Toward the bottom of the page is an option for estimating the relevant probability via Monte Carlo simulation of the multinomial sampling distribution. This might prove useful in the case of samples with expected cell frequencies smaller than 5.0.

Cate-
gory
Observed
Frequency
Expected
Frequency
Expected
Proportion
Percentage
Deviation
Standardized
Residuals
Sums:
Observed Frequencies:
  
Expected Frequencies:
  
Expected Proportions:
  
    
[Note that for df=1, the calculated value
of chi-square is corrected for continuity.] 		[For df=1, this is the uncorrected
value of chi-square.]
chi-square =

df =
[P is non-directional]

P =
Monte Carlo Simulation of the Multinomial Sampling Distribution
Total Samples
Estimated Probability
(Cumulative)
[non-directional]

Each click of this button will draw 200 random samples, each of size n=, from a multinomial distribution that includes categorical outcomes, A, B, C, etc. The mere-chance probability for each category is defined by the corresponding value that appears in the data-entry table under the heading "Expected Proportion." For each sample, a chi-square measure is taken of the aggregate degree to which the 'observed' frequencies differ from the expected frequencies, and a running talley is kept of the proportion of such measures that are equal to or greater than the observed chi-square value obtained in the main analysis. This talley is what appears, following each batch of 200 simulated samples, in the text box labeled "Estimated Probability (Cumulative)." Note that each batch of 200 samples will take a moment to complete, depending on the speed of your computer, and that processing time increases exponentially as a function of n. Optimally, your probability estimate should be based on several thousand samples.

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©Richard Lowry 1998-
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