Chi-Square Test for Goodness of Fit


Instructions: This calculator conducts a Chi-Square test for goodness of fit. Please enter the observed data, the hypothesized population proportions (expected proportions) and the significance level and the results of the Chi-Square test will be presented for you below:

Observed Values
Expected Proportions
Categories (OPTIONAL)
Significance Level (\(\alpha\)) =

Chi-Square Test for Goodness of Fit

More about the Chi-Square test for goodness of fit so that you can interpret in a better way the results delivered by this calculator: A Chi-Square for goodness of fit test is a test used to assess whether the observed data can be claimed to reasonably fit the expected data. Sometimes, a Chi-Square test for goodness of fit is referred as a test for multinomial experiments, because there is a fixed number of N categories, and each of the outcomes of the experiment falls in exactly one of those categories. Then, based on sample information, the test uses a Chi-Square statistic to assess if the expected proportions for all categories reasonably fit the sample data. The main properties of a one sample Chi-Square test for goodness of fit are:

  • The distribution of the test statistic is the Chi-Square distribution, with n-1 degrees of freedom, where n is the number of categories

  • The Chi-Square distribution is one of the most important distributions in statistics, together with the normal distribution and the F-distribution

  • The Chi-Square test of goodness of fit is right-tailed

The formula for a Chi-Square statistic is

\[\chi^2 = \sum_{i=1}^n \frac{(O_i-E_i)^2 }{E_i} \]

One of the most common uses for this test is to assess whether a sample come from a population with a specific population (this is, for example, using this test we can assess if a sample comes from a normally distributed population or not).




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