Introduction

The Chi-squared distribution is a vital probability distribution that finds extensive use in various statistical applications. It’s a continuous distribution that arises in situations involving the sum of squared independent standard normal random variables. Understanding the properties and characteristics of the Chi-squared distribution is crucial for making informed decisions in fields such as hypothesis testing, confidence intervals, and goodness-of-fit tests.

Properties of the Chi-squared Distribution

The Chi-squared distribution is characterized by its degrees of freedom (df). The degrees of freedom dictate the shape of the distribution. As the degrees of freedom increase, the distribution becomes more symmetric and approaches a normal distribution.

Probability Density Function (PDF)

The probability density function of the Chi-squared distribution with df degrees of freedom is given by:

$$f(x)=\frac{1}{2^{d f / 2} \cdot \Gamma(d f / 2)} \cdot x^{(d f / 2)-1} \cdot e^{-x / 2}$$

Where:

• $\Gamma$ is the gamma function

Use Cases and Applications

• Hypothesis Testing: The Chi-squared distribution is used in hypothesis testing to compare observed and expected frequencies in categorical data. For example, it can be used to assess whether the observed distribution of survey responses differs significantly from the expected distribution.

• Goodness-of-Fit Tests: The Chi-squared test for goodness-of-fit assesses how well an observed frequency distribution matches an expected theoretical distribution. This is often used in quality control to check if observed data matches the expected distribution.

• Independence Testing: In contingency table analysis, the Chi-squared test is used to determine if two categorical variables are independent of each other.

Example: Hypothesis Testing

Suppose a researcher is investigating whether a six-sided die is fair. They roll the die 100 times and record the frequencies of each number. They expect each number to appear $1/ 6$ of the time.

• Null hypothesis$\left(H_0\right)$: The die is fair.
• Alternative hypothesis $\left(H_1\right)$: The die is not fair.

Using the Chi-squared test, the researcher calculates the test statistic and compares it to the critical value to determine if there’s a significant difference between the observed and expected frequencies.