# Chi-Square Calculator

You can use this chi-square calculator as part of a statistical analysis test to determine if there is a significant difference between observed and expected frequencies.

To use the calculator, simply input the true and expected values (on separate lines) and click on the "Calculate" button to generate the results.

## What is a Chi-square Test?

A chi-square test is a popular statistical analysis tool that is employed to identify the extent to which an observed frequency differs from the expected frequency.

Let's look at an example.

Let's say you are a college professor. The 100 students you teach complete a test that is graded on a scale ranging from 2 (lowest possible grade) through to 5 (highest possible grade). In advance of the test, you expect 25% of the students to achieve a 5, 45% to achieve a 4, 20% to achieve a 3, and 10% to get a 2.

After the test, you grade the papers. You can then use the chi-square test to determine the extent to which your predicted grades differed from the actual grades.

## How to Calculate a Chi-square

The chi-square value is determined using the formula below:

X^{2} = (observed value - expected value)^{2} / expected value

Returning to our example, before the test, you had anticipated that 25% of the students in the class would achieve a score of 5. As such, you expected 25 of the 100 students would achieve a grade 5. However, in reality, 30 students achieved a score of 5. As such, the chi-square calculation is as follows:

X^{2} = (30 - 25)^{2} / 25 = (5)^{2} / 25 = 25 / 25 = 1

## An In-depth Example of the Chi-square Calculator

Let's take a more in-depth look at the paper grading example.

The grade distribution for the 100 students you tested were as follows: 30 received a 5, 25 received a 4, 40 received a 3, and 5 received a 2.

- a.) We can now determine how many students were expected to receive each grade per the forecast distribution.
- Grade 2: 0.10 * 100 = 10
- Grade 3: 0.20 * 100 = 20
- Grade 4: 0.45 * 100 = 45
- Grade 5: 0.25 * 100 = 25
- b.) We can use this information to determine the chi-square value for each grade.
- Grade 2: X
^{2}= (5 - 10)^{2}/ 10 = 2.5 - Grade 3: X
^{2}= (40 - 20)^{2}/ 20 = 20 - Grade 4: X
^{2}= (25 - 45)^{2}/ 45 = 8.89 - Grade 5: X
^{2}= (30 - 25)^{2}/ 25 = 1 - c.) Finally, we can sum the chi-square values: X
^{2}= 2.5 + 20 + 8.89 + 1 = 32.39

You may also be interested in our P-Value Calculator or T-Value Calculator