F Distribution Calculator

Calculator

Parameters

Degrees of Freedom (Numerator)

Degrees of Freedom (Denominator)

Calculation Type

Lower bound (x1)

Upper bound (x2)

Distribution Chart

Click Calculate to view the distribution chart

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F Distribution: Definition, Formula, and Applications

F Distribution

Definition: The F distribution (also known as Fisher-Snedecor distribution) is a continuous probability distribution used to compare variances and test hypotheses in analysis of variance (ANOVA). It is the ratio of two chi-square distributions divided by their respective degrees of freedom.

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

f(x; d_1, d_2) = \frac{1}{\text{B}(\frac{d_1}{2}, \frac{d_2}{2})} \left(\frac{d_1}{d_2}\right)^{d_1/2} x^{d_1/2-1}\left(1 + \frac{d_1}{d_2}x\right)^{-(d_1+d_2)/2}

where:

\text{B}(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}

is the beta function, and the parameters are:

$d_1$ is the degrees of freedom for numerator
$d_2$ is the degrees of freedom for denominator
$\Gamma$ is the gamma function

Properties

Key Statistics:

Mean: $E(X) = \frac{d_2}{d_2-2}$ for $d_2 > 2$
Variance: $\text{Var}(X) = \frac{2d_2^2(d_1+d_2-2)}{d_1(d_2-2)^2(d_2-4)}$ for $d_2 > 4$
Mode: $\frac{d_1-2}{d_1} \cdot \frac{d_2}{d_2+2}$ for $d_1 > 2$

Key Properties:

Always non-negative (defined for x ≥ 0)
Right-skewed distribution
Approaches normal distribution for large degrees of freedom
Related to ratio of chi-square distributions

Applications

1. Analysis of Variance (ANOVA)

The F distribution is fundamental in ANOVA for:

Testing equality of means across multiple groups
Assessing treatment effects in experimental design
Comparing nested statistical models

2. Regression Analysis

Used in regression analysis for:

Testing overall significance of regression models
Comparing nested regression models
Testing groups of coefficients

3. Variance Comparisons

Applied in comparing variances for:

Testing homogeneity of variances
Quality control processes
Measurement system analysis

4. Statistical Process Control

Used in quality control for:

Process capability analysis
Control chart construction
Variance component analysis

R Code Example

library(tidyverse)

# Parameters
df1 <- 4  # numerator degrees of freedom
df2 <- 10 # denominator degrees of freedom

# Calculate probability between two values
x1 <- 0.5
x2 <- 2.5
prob <- pf(x2, df1 = df1, df2 = df2) - pf(x1, df1 = df1, df2 = df2)
print(str_glue("P({x1} < X < {x2}) = {round(prob, 4)}"))

# Create plot
x <- seq(0, 5, length.out = 1000)
y <- df(x, df1 = df1, df2 = df2)
df <- tibble(x = x, y = y)

ggplot(df, aes(x = x, y = y)) +
  geom_line(color = "blue") +
  geom_area(data = subset(df, x >= x1 & x <= x2), 
            aes(x = x, y = y), 
            fill = "blue", 
            alpha = 0.2) +
  labs(title = str_glue("F Distribution (df1 = {df1}, df2 = {df2})"),
       x = "x",
       y = "Probability Density",
       caption = str_glue("P({x1} < X < {x2}) = {round(prob, 4)}")) +
  theme_minimal()

Python Code Example

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

# Set parameters
df1 = 4  # numerator degrees of freedom
df2 = 10 # denominator degrees of freedom

# Calculate probability between two values
x1, x2 = 0.5, 2.5
prob = stats.f.cdf(x2, df1, df2) - stats.f.cdf(x1, df1, df2)
print(f"P({x1} < X < {x2}) = {prob:.4f}")

# Create plot
x = np.linspace(0, 5, 1000)
pdf = stats.f.pdf(x, df1, df2)

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.f.pdf(x_shade, df1, df2)
plt.fill_between(x_shade, pdf_shade, alpha=0.2, color='blue')

# Customize plot
plt.title(f'F Distribution (df1 = {df1}, df2 = {df2})')
plt.xlabel('x')
plt.ylabel('Probability Density')
plt.annotate(f'P({x1} < X < {x2}) = {prob:.4f}',
            xy=(2.5, max(pdf)/2),
            xytext=(2.5, max(pdf)/2))

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