# Chi-Square Test of Independence

The Chi-Square test of Independence is used to determine if there is a significant relationship between two nominal (categorical) variables. The frequency of one nominal variable is compared with different values of the second nominal variable. The data can be displayed in an R*C contingency table, where R is the row and C is the column. For example, a researcher wants to examine the relationship between gender (male vs. female) and empathy (high vs. low). The chi-square test of independence can be used to examine this relationship. If the null hypothesis is accepted there would be no relationship between gender and empathy. If the null hypotheses is rejected the implication would be that there is a relationship between gender and empathy (e.g. females tend to score higher on empathy and males tend to score lower on empathy).

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**How to calculate the chi-square statistic by hand. ** First we have to calculate the expected value of the two nominal variables. We can calculate the expected value of the two nominal variables by using this formula:

Where

= expected value

= Sum of the i_{th} column

= Sum of the k_{th} column

N = total number

After calculating the expected value, we will apply the following formula to calculate the value of the Chi-Square test of Independence:

= Chi-Square test of Independence

= Observed value of two nominal variables

= Expected value of two nominal variables

Degree of freedom is calculated by using the following formula:

DF = (r-1)(c-1)

Where

DF = Degree of freedom

r = number of rows

c = number of columns

**Hypothesis:**

**Null hypothesis:** Assumes that there is no association between the two variables.

**Alternative hypothesis: **Assumes that there is an association between the two variables.

**Hypothesis testing:** It is the same for the Chi-Square test of Independence as it is for other tests like ANOVA, t-test, etc. If the calculated value of the Chi-Square test is greater than the table value, we will reject the null hypothesis. If the calculated value is less, then we will accept the null hypothesis.

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