What is Chi-Square distribution?
The chi-square distribution is a test used to test a hypothesis and is denoted by X^{2}. In probability theory and statistics, the Chi-Square distribution is also known as the Central Chi-Square distribution.
The formula of Chi-Square distribution:
Generally, we use the following formula to calculate the Chi-Square distribution:
Chi-Square distribution = X^{2} = ∑ (O_{i} – E_{i})^{2} / E_{i}
In the above formula:
- O_{i} = Original values provided in data
- E_{i} = Expected value calculated by the following formula
- E = RT × CT / GT
Here
- RT = Row Total
- CT = Column Total
- GT = Grand Total
Steps to calculate Chi-Square Distribution manually:
These are the following steps of Chi-Square distribution:
- Make a null hypothesis and also write an alternate hypothesis
- Calculate rows, columns, and Grand total
- Calculate the “expected value” E with the help of rows, columns, and grand total
- Draw a table and put the original and expected values in separate columns
- Calculate (O – E) in the next column
- Calculate (O – E)^{2} in the next column
- Calculate (O – E)^{2} / E in the next column
- The sum of the last column is our calculated value
- Find out the degree of freedom by the (r – 1) * (c – 1) formula
- Use the degree of freedom and level of significance to find out the table value
- Compare the calculated value and table value and write a conclusion
Example section:
In this section, we solve an example with the help of basic rules of Chi-square distribution.
Example 1:
Calculate Chi-Square distribution by taking null hypothesis H_{o}: µ_{1} = µ_{2} and H_{1}: µ_{1} ≠ µ_{2 }with the level of significance 5%.
| Category 1 | Category 2 |
Group 1 | 64 | 56 |
Group 2 | 42 | 28 |
Solution:
Step 1: Extract the data.
H_{o}: µ_{1} = µ_{2}
H_{1}: µ_{1} ≠ µ_{2}
Step 2: Find RT, CT, and GT
| Category 1 | Category 2 | Total |
Group 1 | 64 | 56 | 120 |
Group 2 | 42 | 28 | 70 |
Total | 106 | 84 | 190 |
Step 3: Calculate Expected values
| Category 1 | Category 2 |
Group 1 | 66.947 | 53.053 |
Group 2 | 39.053 | 30.947 |
Step 4: Draw a Table and calculate the Chi-Square distribution value
O | E | O – E | (O – E)^{2} | (O – E)^{2} / E |
64 | 66.947 | - 2.947 | 8.6848 | 0.1297 |
56 | 53.053 | 2.947 | 8.6848 | 0.1637 |
42 | 39.053 | 2.947 | 8.6848 | 0.2223 |
28 | 30.947 | - 2.947 | 8.6848 | 0.2806 |
| | | Total | 0.7963 |
Step 5: Calculate the Degree of freedom
Df = (r – 1) * (c – 1)
Df = (2 – 1) * (2 – 1)
Df = 1 * 1
Df = 1
Step 6: Find out the table value
Table value = 0.3720
Step 7: Conclusion
Because the calculated value is greater than the table and lies in the critical region. So we reject the null hypothesis and accept the alternate hypothesis.