I will rephrase your question:: Is the chi-square positively or negatively skewed? It has to be positively skewed, as the mean > median > mode, and skewness is positive. The degrees of freedom, k, must be greater than zero, therefore: skewness = square root of 8/k, is also positive. Mean of the distribution is k, median is k- 2/3 and mode is k-2 for k > 2, therefore in all cases the mean > median > mode condition holds. You may like to view the following link for additioal info: http://www.scaweb.org/assets/papers/2004_papers/SCA2004-01.pdf
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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
There are many chi-squared tests. You may mean the chi-square goodness-of-fit test or chi-square test for independence. Here is what they are used for.A test of goodness of fit establishes if an observed frequency differs from a theoretical distribution.A test of independence looks at whether paired observations on two variables, expressed in a contingency table, are independent of each.
Fisher's exact probability test, chi-square test for independence, Kolmogorov-Smirnov test, Spearman's Rank correlation and many, many more.
* Always when the assumptions for the specific test (as there are many parametric tests) are fulfilled. * When you want to say something about a statistical parameter.
Assumed that there is no relationship between variables.