For goodness of fit test using Chisquare test,
Expected frequency = Total number of observations * theoretical probability specified
or
Expected frequency = Total number of observations / Number of categories if theoretical frequencies are not given.
For contingency tables (test for independence)
Expected frequency = (Row total * Column total) / Grand total for each cell
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you do the observed-expected value and square it, then devide that by the expected you do this for each cell then you add them up also you can enter your data as a matrix on a calculator TI and go to stat, test, chi square test.
For a chi-square test there is a null hypothesis which describes some distribution for the variable that is being tested. The expected frequency for a particular cell is the number of observations that would be expected in that cell if the null hypothesis were true.
A chi-square test is often used as a "goodness-of-fit" test. You have a null hypothesis under which you expect some results. You carry out observations and get a set of results. The expected and observed results are used to calculate the chi-square statistic. This statistic is used to test how well the observations match the values expected under the null hypothesis. In other words, how good the fit between observed and expected values is.
The chi-square test is used to analyze a contingency table consisting of rows and columns to determine if the observed cell frequencies differ significantly from the expected frequencies.
The maximum likelihood estimate under the null hypothesis gives the best estimate for expected frequencies.