Chi-Square Test
What Is the Chi-Square Test?
The chi-square test is a statistical test used to determine the probability that observed frequencies in a sample differ from the frequencies that would be expected if the variables tested are not related. This test is applied to the data in order to evaluate relationships between two categorical variables. It is used to test the hypothesis whether the observed data fit a theoretical distribution, such as the normal distribution, or an expected frequency distribution in a two-way table.
The chi-square test can also be used to test whether two distributions are identical or different. One example of this use is to test whether two independent samples, such as gender and age, come from the same population. It can be used to determine if there is an association between the variables, that is, if there is an effect of one variable on the other.
The chi-square test compares the observed frequencies to the expected frequencies in a two-way contingency table. This table is made up of the categories or classes of two variables, with the frequency data arranged in rows and columns. The total frequency for each row and column is also shown.
The Chi-Square Formula
The chi-square formula is used to test whether observed frequencies differ significantly from expected frequencies. The formula for the chi-square test is:
X^2 = Σ [(O - E)^2/E]
Where:
X^2= the chi-square statistic
Σ= the sum of
O= the observed frequency
E= the expected frequency
How to Use the Chi-Square Test
In order to use the chi-square test, the following steps need to be followed:
Step 1: State the Null and Alternative Hypotheses.
The null hypothesis (H0) states that there is no relationship between the two variables being tested. The alternative hypothesis (Ha) states that there is a relationship between the two variables being tested.
Step 2: Select a level of significance.
The level of significance is the probability of making a Type I error, which is rejecting the null hypothesis when it is true. A level of significance of 0.05 or less is commonly used in chi-square testing.
Step 3: Determine the Critical Value (X^2crit).
The critical value of X^2 is determined by referring to a chi-square probability table. This table gives the critical values for various degrees of freedom (df), which is equal to the number of categories minus one.
Step 4: Calculate the Chi-Square Statistic (X^2calc).
The chi-square formula is used to calculate the observed value of X^2 from the observed and expected frequencies.
Step 5: Compare the Calculated Value to the Critical Value.
If the calculated value is greater than or equal to the critical value, then the null hypothesis is rejected. If the calculated value is less than the critical value, then the null hypothesis is not rejected.
Applications of Chi-Square Test
The chi-square test can be used to assess the significance of differences between observed and expected frequencies in the following areas:
• Goodness-of-fit: Used to determine if an observed set of data fits a theoretical distribution.
• Contingency table: Used to determine if there is a relationship between two categorical variables.
• Homogeneity: Used to determine if two or more independent samples have the same distribution of a categorical variable.
• Independence: Used to determine if two categorical variables are independent of each other.
• Likelihood quantity: Used to compare the probabilities of two or more models.
Conclusion
The chi-square test is a statistical test used to assess the significance of differences between observed and expected frequencies. It is a powerful tool for testing the relationships between two variables, as well as for assessing goodness-of-fit, homogeneity, and independence. The chi-square test is used in many areas including medicine, genetics, and psychology.
