# Mean of Chi – Square Distribution

**Mean of Chi – Square
Distribution**

The chi-square distribution is a distribution of the chi-square statistic. It is a continuous distribution with k- degrees of freedom. It is applied in the description of the sum of the squares of the random variables. The mean of chi – square distribution is applied when carrying out two types of chi-square tests. The first is the chi-square goodness of fit test. This test is used to investigate whether the characteristics of the sample data re similar to those of the population used. The second is the chi-square test of independence, which is applied to test the relationship of two variables in a contingency table. Generally, it is applied to test whether there is difference in the distribution of the categorical variables. In this test, there are two types of results:

**A very small chi-square
test statistic**: this results implies that the
observed data and the expected data fits each other perfectly. In other words,
there is a relationship between the two.

**A very large chi-square
test statistic: **this implies that there is no
relationship between the observed data and the expected data.

**Chi-Square Calculation**

The formula for calculating the chi-square statistics is shown below.

The subscript “c” denotes the degree of freedom, the subscript “o” denotes the observed value, while the subscript “E” is the expected value. The summation symbol implies that every item in the data set should be calculated and then added together. Since this is a lengthy process, it is advisable to perform the analysis using the software such as SPSS, excel or others. The mean of chi-square is similar to its degrees of freedom. Generally, the distribution are positively skewed, where the degrees of skewness diminishes with the increase in the degrees of freedom. In other words, as the degrees of freedom rises, the chi-square distribution stends towards normal distribution.

**Application of Mean of
Chi-square Distribution**

The mean of chi-square is majorly used to show the relationship between categorical variables. In this case, there are two types of variables, the countable or numerical, and the categorical or non-numerical variables. The chi-squared statistic is the value, which indicates the difference existing between the observed and expected values, if the population did not have any relationship. As well, it is applied in the estimation of the confidence interval of the standard deviation of a normally distributed population, from the standard deviation of the sample. In addition, it tests the independence of the criteria applied in the classification of qualitative variables applied.