Chi-Square distribution calculator with graph generator

Chi-square distribution formula for the probability density function (PDF) (used in our online Chi-Square Distribution calculator)

Where: k correspond to the degrees of freedom, and Γ is the Euler Gamma function.

For the study and analysis of the relationship of the categorical data, the chi-square distribution is used, which is calculated from the sum of the squares of the degrees of freedom of the independent random variables from the standard normal distribution.
Chi square distributions vary according to the degrees of freedom. You can use our Chi-Square Distribution Calculator by entering the degree of freedom corresponding to the test that you want to perform, and obtain the correct graph representation.

Chi-square distributions always start at zero on the left and continue to infinite values on the right.
Because the chi-square distribution represents the sum of random variables squared, and a number squared cannot be negative. For this reason, it always starts at zero.

The chi-square test is a non-parametric test, it is mainly performed to assess whether there is a relevant difference between the predicted results and those obtained through observation. It is carried out in order to establish if there is a significant discrepancy between the actual and forecast data.

Using the chi-square test, the null hypothesis is verified where the observed frequencies are contrasted with the expected frequencies. That is, it is a test that is mainly used to compare the observed values with the expected ones.
There are several types of chi-square tests, however, the main ones are the following: the goodness-of-fit test and the independence test.
Chi-square goodness-of-fit test: This test is used to check how well a sample of categorical data fits theoretical data.
Independence test: allows us to know if two variables are associated or not. Once the chi-square test is done, we can determine if these variables are independent with a previously chosen level of significance.

The chi square distribution always assumes positive values and its shape depends on the degrees of freedom. Since its shape depends on the degrees of freedom, there are an infinite number of curves that can characterize this distribution. These distributions are not symmetric but are skewed to the right. However, as the degrees of freedom increase, its shape begins to approach a normal (bell-shaped) distribution.
The mean of the chi square distribution is equal to the degrees of freedom: μ = df. The variance is: σ2 = 2 * df. The mode is df - 2.

To determine how well a sample of categorical data fits a theoretical distribution, the chi-square test is generally used. With the help of this test, it can be determined whether the difference between the observed and expected data is the result of pure chance or a correlation between the variables justified under analysis. The link between our two categorical variables can be better understood and interpreted by using the chi-square test.

When the standard deviation is known and the sample size is large, the normal distribution is used to determine whether or not the population means differ.
To check if the observed results are consistent with the expected result, a chi-square test is used. The normal distribution is symmetric about the mean in cases where the data is bell-shaped, the mean, median, and mode are equal.
Instead of being symmetric, the curve of the chi-square distribution is usually skewed to the right. The left tail of the chi-square distribution does not go to infinity, in contrast to the normal distribution.