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Expected Values. Chi-square Value:. Degrees of Freedom:. As such, you expected 25 of the students would achieve a grade 5. However, in reality, 30 students achieved a score of 5. As such, the chi-square calculation is as follows:.

The grade distribution for the students you tested were as follows: 30 received a 5, 25 received a 4, 40 received a 3, and 5 received a 2. Chi-Square Calculator. What is a Chi-square Test? Let's look at an example. These modes apply to different situations covered in detail below.

The mode of operation can be selected from the radio button below the data input field in the Chi Square calculator interface. The data in all cells should be entered as counts whole numbers, integers. For example, if you have this data in Excel:. Click here to see how this example works. If the sample data is known to be independent the result can be treated as a test of homogeneity.

If the data is based on two categorical variables measured from the same population the result can be interpreted as a test of independence between the variables. The tool can be used as a goodness-of-fit calculator by entering the observed values counts in the first column and the expected frequencies for each outcome in the second column. For example, if we are testing if a dice is fair, we would have an expected frequency of 0.

An example data set might look like so: Dice roll observed vs expected frequencies Number Times turned up Expected Frequency 1 0. In its original version it was developed by Karl Pearson in as a goodness of fit test: testing whether a particular set of observed data fits a frequency distribution from the Pearson family of distributions Pearson's Chi-Squared test.

Pearson in expanded its application to a test of independence between the rows and columns of a contingency table of categorical variables [1]. It was further expanded by R. Fisher in The statistical model behind the tests requires that the variables are the result of simple random sampling and are thus independent and identically distributed ID under the null hypothesis. Consequently, the test can be used as a test for independence or a test for homogeneity identity of distributions.

In certain restricted situations it can also function as a test for the difference in variances. This, however, also means that if one wants to test non-IID data a different test should be chosen. For a simple 2 by 2 contingency table the requirement is that each cell has a value larger than 5. Our chi-square calculator will check for some of these conditions and issue warnings where appropriate.

The formula is the same regardless if you are doing a test of goodness-of-fit, test of independence or of homogeneity. Despite the formula behind all three tests being the same, however, all three have different null hypotheses and interpretations see below.

The Chi-Square formula is simply:. Here we examine the three applications of the Chi Square test: as a test of indepdence, as a test of homogeneity identical distribution and as a goodness-of-fit test. This test is also sometimes referred to as a "test of association" and it determines whether two categorical variables for a single sample are independent or associated with each other.

For example, a survey might ask respondents to state their education level, height and net wealth in order to determine whether there is some dependence on one variable on the others. The null hypothesis H 0 would thus be: the variables education, height and wealth are independent. The alternative hypothesis H 1 is then: "some of the variables education, height and net wealth are dependent on each other".

Note that in case there are more than two variables the null will be rejected even if some variables are independent of each other: one dependence in a table is enough to potentially invalidate the null. When using the calculator as a test for independence obtaining a small p-value is to be interpreted as evidence that the two or more groups are not independent. Note that if there are more than two variables you cannot say which ones are independent and which are not: it might be all of them or just some of them.

This test refers to testing if two or more variables share the same probability distribution and is also supported by this online Chi Square calculator. The test of homogeneity is used to determine whether two or more independent samples differ in their distributions on a single variable of interest: comparing two or more groups on a categorical outcome.

For example, one can compare the educational levels of groups of people from different cities in a country to determine if the proportions between the groups are essentially the same or if there is a statistically significant difference.

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