Chi Square Distribution Calculator

Chi-Square PDF:

\[ f(x) = \frac{1}{2^{df/2} \Gamma(df/2)} x^{(df/2)-1} e^{-x/2} \]

Probability Density (PDF):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Chi-Square Distribution?

The Chi-Square distribution is a continuous probability distribution that is widely used in statistical hypothesis testing, particularly in chi-square tests for goodness of fit and independence. It is a special case of the gamma distribution.

2. How Does the Calculator Work?

The calculator uses the Chi-Square probability density function:

\[ f(x) = \frac{1}{2^{df/2} \Gamma(df/2)} x^{(df/2)-1} e^{-x/2} \]

Where:

\( df \) — Degrees of freedom (positive integer)
\( x \) — Non-negative real number
\( \Gamma \) — Gamma function

Explanation: The PDF describes the relative likelihood for a random variable to take on a given value in a chi-square distribution with specified degrees of freedom.

3. Importance of Chi-Square Distribution

Details: The chi-square distribution is fundamental in statistics for:

Chi-square tests for goodness of fit
Tests of independence in contingency tables
Estimation of confidence intervals for population variance
Analysis of variance (ANOVA)

4. Using the Calculator

Tips: Enter degrees of freedom (must be positive integer) and x value (must be non-negative). The calculator will compute the probability density at that point.

5. Frequently Asked Questions (FAQ)

Q1: What are degrees of freedom?
A: Degrees of freedom represent the number of independent quantities in the statistic. For chi-square tests, it's typically (rows-1)*(columns-1) for contingency tables.

Q2: When is the chi-square distribution used?
A: Primarily for hypothesis testing with categorical data, goodness-of-fit tests, and tests of independence.

Q3: What's the relationship to normal distribution?
A: The chi-square distribution with k degrees of freedom is the distribution of a sum of squares of k independent standard normal variables.

Q4: What are typical values?
A: The PDF value ranges from 0 to 1. The distribution is right-skewed, with skewness decreasing as degrees of freedom increase.

Q5: How accurate is this calculator?
A: It provides good approximation but for critical statistical work, specialized statistical software may be needed.