Chi-Square Calculator

The Chi-Square Calculator makes it easy to find the cumulative probability associated with a specified chi-square statistic. Or you can find the chi-square statistic associated with a specified cumulative probability.

If anything is unclear, read the Frequently-Asked Questions or the Sample Problems. To learn more about the chi-square, read Stat Trek's tutorial on the chi-square distribution.

Frequently-Asked Questions

Calculator | Sample Problems

Instructions: To find the answer to a frequently-asked question, simply click on the question.

What are degrees of freedom?

Degrees of freedom can be described as the number of scores that are free to vary. For example, suppose you tossed three dice. The total score adds up to 12. If you rolled a 3 on the first die and a 5 on the second, then you know that the third die must be a 4 (otherwise, the total would not add up to 12). In this example, 2 die are free to vary while the third is not. Therefore, there are 2 degrees of freedom.

In many situations, the degrees of freedom are equal to the number of observations minus one. Thus, if the sample size were 20, there would be 20 observations; the degrees of freedom would be 20 minus 1 or 19.

What is a chi-square critical value?

The chi-square critical value (x) can be any number between zero and plus infinity. The chi-square calculator computes the probability that a chi-square statistic (Χ²) falls between 0 and the critical value.

Suppose you randomly select a sample of 10 observations from a large population. In this example, the degrees of freedom (df) would be 9, since df = n - 1 = 10 - 1 = 9. Suppose you wanted to find the probability that a chi-square statistic falls between 0 and 13. In the chi-square calculator, you would enter 9 for degrees of freedom and 13 for the chi-square value. Then, after you click the Calculate button, the calculator would show the cumulative probability to be 0.84. That is, P(Χ²≤13) = 0.84.

What is a cumulative probability?

A cumulative probability is a sum of probabilities. The chi-square calculator computes two cumulative probabilities:

P(Χ² ≤ x): The probability that a chi-square statistic falls between 0 and some critical value (x).
P(Χ² ≥ x): The probability that a chi-square statistic falls between some critical value (x) and plus infinity.

What is a chi-square statistic?

A chi-square statistic (Χ²) is a statistic whose values are given by

Χ² = [ ( n - 1 ) * s² ] / σ²

where σ is the standard deviation of the population, s is the standard deviation of the sample, and n is the sample size. The distribution of the chi-square statistic has n - 1 degrees of freedom. (For more on the chi-square statistic, see the tutorial on the chi-square distribution.)

What is a probability?

A probability is a number expressing the chances that a specific event will occur. This number can take on any value from 0 to 1. A probability of 0 means that there is zero chance that the event will occur; a probability of 1 means that the event is certain to occur. Numbers between 0 and 1 quantify the uncertainty associated with the event. For example, the probability of a coin flip resulting in Heads (rather than Tails) would be 0.50. Fifty percent of the time, the coin flip would result in Heads; and fifty percent of the time, it would result in Tails.

Sample Problems

Calculator | Frequently-Asked Questions

The Acme Widget Company claims that their widgets last 5 years, with a standard deviation of 1 year. Assume that their claims are true.

If you test a random sample of 9 Acme widgets, what is the probability that the standard deviation in your sample will be less than 0.95 years?

Solution:

We know the following:
- The population standard deviation is equal to 1.
- The sample standard deviation is equal to 0.95.
- The sample size (n) is equal to 9.
- The degrees of freedom (df) is equal to 8, because df = n - 1 = 9 - 1 = 8.
Given these data, we compute the chi-square statistic:

Χ² = [ ( n - 1 ) * s² ] / σ²
Χ² = [ ( 9 - 1 ) * (0.95)² ] / (1.0)² = 7.22

where σ is the standard deviation of the population, s is the standard deviation of the sample, and n is the sample size.

Now, using the Chi-Square Distribution Calculator, we can determine the cumulative probability for the chi-square statistic. We enter the degrees of freedom (8) and the chi-square critical value (7.22) into the calculator, and hit the Calculate button.

The calculator reports that the P(Χ² ≤ 7.22) is 0.48691. Therefore, there is about a 49% chance that the sample standard deviation will be no more than 0.95.

Find the chi-square critical value, if the P(Χ² ≤ x) is 0.75 and the sample size is 25.

Solution:

We know the following:
- The P(Χ² ≤ x) is 0.75.
- The sample size (n) is 25.
- The degrees of freedom (df) is equal to 24,because df = n - 1 = 25 - 1 = 24.
Given these data, we compute the chi-square critical value, using the Chi-Square Distribution Calculator.

We enter the degrees of freedom (24) and the cumulative probability (0.75) into the calculator, and hit the Calculate button. The calculator reports that the chi-square critical value is 28.24115.

This means that if you select a random sample of 25 observations, there is a 75% chance that the chi-square statistic from that sample will be less than or equal to 28.24115.