Table of Contents

**What is a Chi-square test**?

**A Chi-square test is performed to determine if there is a difference between the theoretical population parameter and the observed data.**

- Chi-square test is a non-parametric test where the data is not assumed to be normally distributed but is distributed in a chi-square fashion.
- It allows the researcher to test factors like a number of factors like the goodness of fit, the significance of population variance, and the homogeneity or difference in population variance.
- This test is commonly used to determine if a random sample is drawn from a population with mean µ and the variance σ
^{2}.

**Chi-square test uses**

Chi-square test is performed for various purposes, some of which are:

- This method is commonly used by researchers to determine the differences between different categorical variables in a population.
- A Chi-square test can also be used as a test for goodness of fit. It enables us to observe how well the theoretical distribution fits the observed distribution.
- It also works as a test of independence where it enables the researcher to determine if two attributes of a population are associated or not.

**Chi-square test formula**

Chi-square test is symbolically written as χ^{2} and the formula of chi-square for comparing variance is given as:

where σs^{2 }is the variance of the sample,

σp^{2 }is the variance of the sample.

Similarly, when chi-square is used as a non-parametric test for testing the goodness of fit or for testing the independence, the following formula is used:

Where *Oij* is the observed frequency of the cell in the *i*^{th} row and *j*^{th} column,* Eij *is the expected frequency of the cell in the *i*^{th} row and *j*^{th} column.

**Conditions for the chi-square test**

For the chi-square test to be performed, the following conditions are to be satisfied:

- The observations are to be recorded and collected on a random basis.
- The items in the samples should all be independent.
- The frequencies of data in a group should not be less than 10. Under such conditions, regrouping of items should be done by combining frequencies.
- The total number of individual items in the sample should also be reasonably large, about 50 or more.
- The constraints in the frequencies should be linear and not containing squares or higher powers.

**Chi-square distribution**

- Chi-square distribution in statistics is the distribution of a sum of the squares of independent normal random variables.
- This distribution is a special case of the gamma distribution and is one of the most commonly used distributions in statistics.
- This distribution is used for the chi-square test for testing the goodness of fit or testing the independence.
- Chi-square distribution is a part of the t-distribution, F-distribution used for t-tests, and ANOVA.

**Chi-square table**

The following is the chi-square distribution table:

**Chi-square test of independence**

- When the chi-square test is used as a test of independence, it allows the researcher to test whether the two attributes being tested are associated or not.
- For this test, a
**null and alternative hypothesis**is formulated where the null hypothesis is that the two attributes are not associated, and the alternative hypothesis is that the attributes are associated. - From the given data, the expected frequencies are then calculated, followed by the calculation of chi-square value.
- Based on the calculated value of chi-square, either the null or alternative hypothesis is accepted.
- Here, if the calculated value of chi-square is less than the value in the table at the given level of significance, the null hypothesis is accepted, indicating that there is no relationship between the two attributes.
- However, if the calculated value of chi-square is found to be higher than the value in the table, the alternative hypothesis is accepted, indicating that there is a relationship between the two attributes.
- The chi-square test only established the existence of a relationship but not the degree of the relationship or its form.

**Chi-square test of goodness of fit**

- Chi-square test is performed as a test of goodness of fit, which helps the researcher to compare the theoretical distribution with the observed distribution.
- When the calculated value of chi-square is found to be less than the table value at a certain level of significance, the fit between the data is considered to be good.
- A good fit indicates that the variation between the observed and expected frequencies is due to fluctuations during sampling.
- However, if the calculated value of chi-square is greater than the table value, the fit is considered not to be as good.

**Chi-square test examples**

- A chi-square test performed to determine if a new medication is effective against fever or not is an example of a chi-square test as the test of independence to determine the relationship between medicine and fever.
- Another example of the chi-square test is the testing of some genetic theory that claims that children having one parent of blood type
*A*and the other of blood type*B*will always have the blood group as one of three types,*A*,*AB*,*B,*and that the proportion of three types will on an average be as 1: 2: 1. On the basis of expected and observed outcomes, the goodness of fit of the hypothesis can be determined.

**Chi-square test applications**

- A Chi-square test is used in cryptanalysis to determine the distribution of plain text and decrypted ciphertext.
- Similarly, it is also used in bioinformatics to determine the distribution of different genes like disease genes and other important genes.
- A Chi-square test is performed by various researchers of different fields to test the minor or major hypothesis.

**References and Sources**

- R. Kothari (1990) Research Methodology. Vishwa Prakasan. India.
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