Chi-Squared Distribution Probability Calculator

Understanding the Chi-Squared Distribution

The Chi-Squared distribution is a continuous probability distribution used primarily in hypothesis testing, such as the chi-squared test for independence and goodness-of-fit tests. It is right-skewed, ranges from 0 to infinity, and is defined by a single parameter: the degrees of freedom (df).

Key Components of the Chi-Squared Distribution

• Degrees of Freedom (df): The shape of the Chi-Squared distribution depends on the degrees of freedom. As the degrees of freedom increase, the distribution approaches a normal distribution.
• Probability (P): The probability that a random variable following the Chi-Squared distribution is less than or equal to a particular value (X).
• Score (X): A value within the Chi-Squared distribution that corresponds to a specific probability, used in hypothesis testing.

Chi-Squared Distribution Formula

The probability density function (pdf) of the Chi-Squared distribution with df degrees of freedom is given by:

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

where \( \Gamma \) is the gamma function, and \( x \) represents the Chi-Squared score.

Conditions for Using the Chi-Squared Distribution

• Independence: Observations must be independent of each other.
• Expected Frequency: In a chi-squared test, each category should have an expected frequency of at least 5 to ensure accurate approximation.

Calculating Chi-Squared Probabilities and Scores

To compute Chi-Squared probabilities and scores, we can use libraries in JavaScript, Python, and R. Here’s how to perform these calculations:

1. Using JavaScript

In JavaScript, you can use the jStat library for probability and inverse probability calculations.

// Calculate probability (CDF) for a Chi-Squared score
const df = 4; // Degrees of freedom
const x = 0.85; // Chi-Squared score

// CDF: P(X ≤ x)
const cumulativeProbability = jStat.chisquare.cdf(x, df);

// Inverse CDF: Get the Chi-Squared score for a given probability (e.g., 0.95)
const probability = 0.95;
const chiScore = jStat.chisquare.inv(probability, df);

console.log('Cumulative Probability:', cumulativeProbability);
console.log('Chi-Squared Score for P(X ≤ x) = 0.95:', chiScore);

Note: Ensure you have the jStat library included in your project to perform these calculations.

2. Using Python

In Python, the SciPy library provides functions for cumulative probability and inverse calculations.

from scipy.stats import chi2

# Define degrees of freedom
df = 4  # Degrees of freedom
x = 0.85  # Chi-Squared score

# CDF: P(X ≤ x)
cumulative_probability = chi2.cdf(x, df)

# Inverse CDF: Get the Chi-Squared score for a given probability (e.g., 0.95)
probability = 0.95
chi_score = chi2.ppf(probability, df)

print("Cumulative Probability:", cumulative_probability)
print("Chi-Squared Score for P(X ≤ x) = 0.95:", chi_score)

Note: Install SciPy by running pip install scipy if it’s not already installed.

3. Using R

In R, the stats package includes functions to calculate Chi-Squared probabilities and scores.

# Define degrees of freedom
df <- 4  # Degrees of freedom
x <- 0.85  # Chi-Squared score

# CDF: P(X ≤ x)
cumulative_probability <- pchisq(x, df)

# Inverse CDF: Get the Chi-Squared score for a given probability (e.g., 0.95)
probability <- 0.95
chi_score <- qchisq(probability, df)

cat("Cumulative Probability:", cumulative_probability, "\n")
cat("Chi-Squared Score for P(X ≤ x) = 0.95:", chi_score, "\n")

Note: The pchisq function calculates the cumulative probability, while qchisq gives the Chi-Squared score for a specified probability.

Further Reading

• Chi-Squared Distribution on Wikipedia
• Explore More Calculators on Research Data Pod