# Chi-Square Calculator

Analyze the goodness-of-fit or test for independence using our online chi-square calculator. By inputting the observed frequencies and expected frequencies (for goodness-of-fit) or the contingency table (for independence), you can calculate the chi-square statistic and its associated p-value.

## 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

## Chi-Square Calculator example

Certainly! Let's calculate the chi-square statistic using a contingency table for a chi-square test of independence.

Suppose we want to determine whether there is a relationship between smoking habits and the occurrence of lung cancer. We have collected data from a sample of 500 individuals and created the following contingency table:

```
| Lung Cancer | No Lung Cancer | Total
```

## Smoker | 70 | 180 | 250 Non-Smoker | 45 | 205 | 250

Total | 115 | 385 | 500

Step 1: Set up the null hypothesis and the alternative hypothesis.

Null Hypothesis (H0): There is no association between smoking habits and the occurrence of lung cancer. Alternative Hypothesis (Ha): There is an association between smoking habits and the occurrence of lung cancer.

Step 2: Calculate the expected frequencies.

To calculate the expected frequencies, we assume that there is no association between smoking habits and lung cancer. We can calculate the expected frequency for each cell using the formula:

Expected Frequency = (Row Total * Column Total) / Grand Total

The expected frequency table would look like this:

```
| Lung Cancer | No Lung Cancer | Total
```

## Smoker | 57.5 | 192.5 | 250 Non-Smoker | 57.5 | 192.5 | 250

Total | 115 | 385 | 500

Step 3: Calculate the chi-square statistic.

The chi-square statistic is calculated using the formula:

Ï‡Â² = Î£ [(O - E)Â² / E]

where O is the observed frequency and E is the expected frequency.

For each cell, calculate (O - E)Â² / E, then sum all the values to get the chi-square statistic.

Let's calculate it:

(70-57.5)Â²/ 57.5 + (180-192.5)Â²/ 192.5 + (45-57.5)Â²/ 57.5 + (205-192.5)Â²/ 192.5 â‰ˆ 4.87

Step 4: Determine the degrees of freedom.

The degrees of freedom for a chi-square test of independence are calculated as:

df = (number of rows - 1) * (number of columns - 1)

In this example, we have (2-1) * (2-1) = 1 degree of freedom.

Step 5: Determine the critical value and the p-value.

Using a chi-square distribution table or statistical software, lookup the critical value for the desired significance level and degrees of freedom. Let's assume we have a significance level of 0.05 and find that the critical value is approximately 3.841.

The p-value can also be calculated using the chi-square distribution. Assuming the calculated chi-square statistic is 4.87 and 1 degree of freedom, let's assume we find the p-value to be approximately 0.027.

Step 6: Interpret the results.

If the calculated chi-square statistic is greater than the critical value, we reject the null hypothesis. In this case, since the calculated chi-square statistic (4.87) is greater than the critical value (3.841), we reject the null hypothesis. This indicates that there is a statistically significant association between smoking habits and the occurrence of lung cancer.

Please note that this is just an example of a chi-square test of independence. The specific steps and formulas may vary depending on the type of chi-square test you are performing and the software or programming environment you are using.

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