# Chi square distribution

A chi-squared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution, and the distribution of the square of the test statistic approaches a chi-squared distribution. Just as extreme values of the normal distribution have low probability and give small p-valuesextreme values of the chi-squared distribution have low probability. An additional reason that the chi-squared distribution is widely used is that it is a member of the class of likelihood ratio tests LRT. Learn How Cumulative Probability and the Chi-Square Distribution The chi-square distribution is constructed so that the total area under the curve is equal to 1. The area under the curve between 0 and a particular chi-square value is a cumulative probability associated with that chi-square value. For example, in the figure below, the shaded area represents a cumulative probability associated with a chi-square statistic equal to A; that is, it is the probability that the value of a chi-square statistic will fall between 0 and A.

Fortunately, we don't have to compute the area under the curve to find the probability. The easiest way to find the cumulative probability associated with a particular chi-square statistic is to use the Chi-Square Calculatora free tool provided by Stat Trek.

Chi-Square Calculator The Chi-Square Calculator solves common statistics problems, based on the chi-square distribution. The calculator computes cumulative probabilities, based on simple inputs. Clear instructions guide you to an accurate solution, quickly and easily. If anything is unclear, frequently-asked questions and sample problems provide straightforward explanations.

The calculator is free. It can found in the Stat Trek main menu under the Stat Tools tab. Or you can tap the button below. On average, the battery lasts 60 minutes on a single charge. The standard deviation is 4 minutes.

Suppose the manufacturing department runs a quality control test. They randomly select 7 batteries. The standard deviation of the selected batteries is 6 minutes.

## Chi-square distribution

What would be the chi-square statistic represented by this test? Solution The standard deviation of the population is 4 minutes. The standard deviation of the sample is 6 minutes.

The number of sample observations is 7. To compute the chi-square statistic, we plug these data in the chi-square equation, as shown below.

## Definition

Problem 2 Let's revisit the problem presented above. The manufacturing department ran a quality control test, using 7 randomly selected batteries. In their test, the standard deviation was 6 minutes, which equated to a chi-square statistic of Suppose they repeated the test with a new random sample of 7 batteries.The chi-square distribution (also called the chi-squared distribution) is a special case of the gamma distribution; A chi square distribution with n degrees of freedom is equal to a gamma distribution with a = n / 2 and b = (or β = 2).

We say that X follows a chi-square distribution with r degrees of freedom, denoted χ 2 (r) and read "chi-square-r." There are, of course, an infinite number of possible values for r, the degrees of freedom. The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation.

Chi-square distribution definition is - a probability density function that gives the distribution of the sum of the squares of a number of independent random variables each with a normal distribution with zero mean and unit variance, that has the property that the sum of two or more random variables with such a distribution also has one, and.

The area of a Chi Square distribution below 4 is the same as the area of a standard normal distribution below 2, since 4 is 2 2.

Consider the following problem: you sample two scores from a standard normal distribution, square each score, and sum the squares. The sum of squares of independent standard normal random variables is a Chi-square random variable.

Combining the two facts above, one trivially obtains that the sum of squares of independent standard normal random variables is a Chi-square random variable with degrees of freedom. Density plots.

Chi-Square Distributions | STAT /