Chi-Square Distribution Calculator:
Online Statistical Table

The chi-square distribution calculator makes it easy to compute cumulative probabilities, based on the chi-square statistic. For help, read the Frequently-Asked Questions or review the Sample Problems.

To learn more about the chi-square distribution, go to Stat Trek's tutorial on the chi-square distribution.

Note: For this calculator, the cumulative probability refers to P(Χ² < S), where S is the chi-square statistic.

Instructions: To find the answer to a frequently-asked question, simply click on the question. If you don't see the answer you need, read Stat Trek's tutorial on the chi-square distribution or visit the Statistics Glossary.

• What are degrees of freedom?
• What is a cumulative probability?
• What is a chi-square statistic?
• What is a probability?

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 the degrees of freedom would be 20 minus 1 or 19.

What is a cumulative probability?

A cumulative probability is a sum of probabilities. In connection with the chi-square distribution, a cumulative probability refers to the probability that a chi-square random variable will be less than or equal to a specified value.

Suppose, for example, that we sample 100 first-graders. If we ask about the probability that the average first grader weighs exactly 70 pounds, we are asking about a simple probability - not a cumulative probability.

But if we ask about the probability that average weight is less than or equal to 70 pounds, we are really asking about a sum of probabilities (i.e., the probability that the average weight is exactly 70 pounds plus the probability that it is 69 pounds plus the probability that it is 68 pounds, etc.). Thus, we are asking about a cumulative probability.

What is a chi-square statistic?

A chi-square statistic 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.

1. 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 is equal to 9.
   • The degrees of freedom is equal to 8 (because sample size minus one = 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 statistic (7.22) into the calculator, and hit the Calculate button. The calculator reports that the cumulative probability is 0.49. Therefore, there is a 49% chance that the sample standard deviation will be no more than 0.95.
   
2. Find the chi-square statistic, if the cumulative probability is 0.75 and the sample size is 25.
   
   Solution:
   
   We know the following:
   
   
   • The cumulative probability is 0.75.
   • The sample size is 25.
   • The degrees of freedom is equal to 24 (because sample size minus one = 25 - 1 = 24).
   
   Given these data, we compute the chi-square statistic, 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 statistic is 28.2.