Chi-Square Distribution Calculator

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Parameters

Degrees of Freedom (df)

Calculation Type

Lower bound (x1)

Upper bound (x2)

Distribution Chart

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Chi-Square Distribution: Definition, Formula, and Applications

Chi-Square Distribution

Definition: The chi-square distribution is a probability distribution of a sum of squared standard normal random variables. It is widely used in statistical inference, particularly for tests of independence and goodness-of-fit tests.

Formula:The probability density function (PDF) is given by:

f(x; k) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2-1}e^{-x/2}, \quad x > 0

Where:

k = \text{degrees of freedom}

\Gamma(k/2) = \text{gamma function}

Where:

$k$ is the degrees of freedom (shape parameter)
$x$ is the value of the chi-square statistic

Properties

Mean: $E(X) = k$ (equals degrees of freedom)
Variance: $\text{Var}(X) = 2k$
Mode: $\max(k-2, 0)$
Support: $(0, \infty)$
Special cases:
- The sum of $k$ standard normal variables squared
- $\chi^2_1$ is the square of a standard normal
- For large $k$ , approaches normal distribution

Applications

1. Goodness of Fit Tests

Chi-square tests are used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories. This test helps determine if a sample comes from a population with a specific distribution.

2. Tests of Independence

In contingency table analysis, chi-square tests help determine whether there is a significant relationship between two categorical variables. This is crucial in fields like social sciences, market research, and medical studies.

3. Quality Control

In manufacturing and quality control, chi-square tests can be used to monitor process variability and ensure that production processes remain within acceptable limits.

4. Medical Research

The chi-square distribution is used in medical studies to analyze categorical data and assess relationships between various factors, such as treatment outcomes and patient characteristics.

R Code Example

library(tidyverse)

# Parameters
df <- 4  # degrees of freedom

# Calculate probability between two values
x1 <- 2
x2 <- 8
prob <- pchisq(x2, df = df) - pchisq(x1, df = df)
print(str_glue("P({x1} < X < {x2}) = {round(prob, 4)}"))

# Create plot
x <- seq(0, 15, length.out = 1000)
y <- dchisq(x, df = df)
data <- tibble(x = x, y = y)

ggplot(data, aes(x = x, y = y)) +
  geom_line(color = "blue") +
  geom_area(data = subset(data, x >= x1 & x <= x2), 
            aes(x = x, y = y), 
            fill = "blue", 
            alpha = 0.2) +
  labs(title = str_glue("Chi-Square Distribution (df = {df})"),
       x = "x",
       y = "Probability Density",
       caption = str_glue("P({x1} < X < {x2}) = {round(prob, 4)}")) +
  theme_minimal()

Python Code Example

import numpy as np
from scipy import stats
import matplotlib.pyplot as plt
import seaborn as sns

# Set parameters
df = 4  # degrees of freedom

# Calculate probability between two values
x1, x2 = 2, 8
prob = stats.chi2.cdf(x2, df) - stats.chi2.cdf(x1, df)
print(f"P({x1} < X < {x2}) = {prob:.4f}")

# Create plot
x = np.linspace(0, 15, 1000)
pdf = stats.chi2.pdf(x, df)

plt.figure(figsize=(10, 6))
plt.plot(x, pdf, 'blue', label='PDF')

# Add shaded area
x_shade = x[(x >= x1) & (x <= x2)]
pdf_shade = stats.chi2.pdf(x_shade, df)
plt.fill_between(x_shade, pdf_shade, alpha=0.2, color='blue')

# Customize plot
plt.title(f'Chi-Square Distribution (df = {df})')
plt.xlabel('x')
plt.ylabel('Probability Density')
plt.annotate(f'P({x1} < X < {x2}) = {prob:.4f}',
            xy=(8, max(pdf)/2),
            xytext=(8, max(pdf)/2))

plt.grid(True, alpha=0.3)
plt.show()