The chi-square distribution is a probability distribution that arises in the context of statistical hypothesis testing and is used in various statistical analyses.

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It is related to the chi-square statistic, which is commonly employed in tests of independence, goodness-of-fit, and other analyses. The chi-square distribution is characterized by a positive skewness and is a family of distributions, each associated with a specific degree of freedom.

Here are key points about the chi-square distribution:

**Definition:**

- The chi-square distribution is a continuous probability distribution that takes only non-negative values. It is denoted by χ² (chi-squared). The distribution is determined by its degrees of freedom.

**Degrees of Freedom (df):**

- The degrees of freedom in the chi-square distribution depend on the context of its application. In statistical tests, the degrees of freedom are related to the number of categories or variables involved in the analysis.

**Probability Density Function (PDF):**

- The probability density function of the chi-square distribution is given by the formula:

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

where (k) is the degrees of freedom, (x) is a non-negative value, and (\Gamma) is the gamma function.

**Skewness:**

- The chi-square distribution is positively skewed. The skewness decreases as the degrees of freedom increase.

**Special Case: Chi-Square Test for Goodness of Fit:**

- The chi-square distribution is commonly used in the chi-square test for goodness of fit. This test assesses whether the observed frequency distribution of categorical data matches an expected (theoretical) distribution.

**Special Case: Chi-Square Test of Independence:**

- Another common application is the chi-square test of independence, which examines whether there is a significant association between two categorical variables.

**Special Case: Chi-Square Test for Homogeneity:**

- The chi-square test for homogeneity is used to compare the distribution of a categorical variable across different groups or populations.

**Relationship with Normal Distribution:**

- As the degrees of freedom increase, the chi-square distribution approaches a normal distribution. This property is often utilized in statistical analyses involving large sample sizes.

**Critical Values:**

- Critical values for the chi-square distribution are often tabulated for different levels of significance and degrees of freedom. These critical values help in determining whether the obtained chi-square statistic is statistically significant.

**Confidence Intervals:**- Confidence intervals for the population variance (or standard deviation) can be constructed using the chi-square distribution.

**Applications in Regression Analysis:**- In regression analysis, the chi-square distribution is used in tests related to the overall significance of a regression model or the significance of specific regression coefficients.

In summary, the chi-square distribution plays a crucial role in statistical hypothesis testing, particularly in situations involving categorical data and comparisons of observed and expected frequencies. Its properties make it a versatile tool in various statistical analyses.