Use this calculator to find probabilities or critical values for the F-distribution.
$P(X \leq x_{1})$:
$P(X > x_1)$:
F-Score ($x_1$):
Probability Density ($f(x_1)$):
$P(X \leq x_2)$:
$P(X > x_2)$:
F-Score ($x_2$):
Probability Density ($f(x_2)$):
$P(x_1 \leq X \leq x_2)$:
Understanding the F Distribution
The F-distribution is a continuous probability distribution primarily used in analysis of variance (ANOVA) and regression analysis to test hypotheses about the variances of two populations. It is right-skewed and ranges from 0 to infinity. The shape of the distribution depends on two types of degrees of freedom: one associated with the numerator (df₁) and one with the denominator (df₂).
Key Components of the F Distribution
- Degrees of Freedom (df₁ and df₂):
- df₁ (numerator degrees of freedom): Degrees of freedom associated with the variance in the numerator.
- df₂ (denominator degrees of freedom): Degrees of freedom associated with the variance in the denominator.
- Probability (P): The probability that a random variable following the F-distribution is less than or equal to a particular value (X).
- Score (X): A value within the F-distribution that corresponds to a specific probability, used in hypothesis testing.
F Distribution Formula
The probability density function (pdf) of the F-distribution with degrees of freedom df₁ and df₂ is given by:
\[ f(x; df_1, df_2) = \frac{\sqrt{\frac{(df_1 \cdot x)^{df_1} \cdot df_2^{df_2}}{(df_1 \cdot x + df_2)^{df_1 + df_2}}}}{x \cdot B\left(\frac{df_1}{2}, \frac{df_2}{2}\right)} \]
where \( B \) is the beta function, and \( x \) represents the F-score.
Conditions for Using the F Distribution
- Independence of Samples: The two samples being compared should be independent.
- Normality: Each population should be normally distributed.
- Equal Variances: The populations should ideally have equal variances when using the F-test for variances.
Calculating F-Distribution Probabilities and F-Scores
To compute F-distribution probabilities and F-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 an F-score
const df1 = 4; // Numerator degrees of freedom
const df2 = 9; // Denominator degrees of freedom
const x = 0.85; // F-score
// CDF: P(X ≤ x)
const cumulativeProbability = jStat.centralF.cdf(x, df1, df2);
// Inverse CDF: Get the F-score for a given probability (e.g., 0.95)
const probability = 0.95;
const fScore = jStat.centralF.inv(probability, df1, df2);
console.log('Cumulative Probability:', cumulativeProbability);
console.log('F-Score for P(X ≤ x) = 0.95:', fScore);
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 f
# Define degrees of freedom
df1 = 4 # Numerator degrees of freedom
df2 = 9 # Denominator degrees of freedom
x = 0.85 # F-score
# CDF: P(X ≤ x)
cumulative_probability = f.cdf(x, df1, df2)
# Inverse CDF: Get the F-score for a given probability (e.g., 0.95)
probability = 0.95
f_score = f.ppf(probability, df1, df2)
print("Cumulative Probability:", cumulative_probability)
print("F-Score for P(X ≤ x) = 0.95:", f_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 F-distribution probabilities and F-scores.
# Define degrees of freedom
df1 <- 4 # Numerator degrees of freedom
df2 <- 9 # Denominator degrees of freedom
x <- 0.85 # F-score
# CDF: P(X ≤ x)
cumulative_probability <- pf(x, df1, df2)
# Inverse CDF: Get the F-score for a given probability (e.g., 0.95)
probability <- 0.95
f_score <- qf(probability, df1, df2)
cat("Cumulative Probability:", cumulative_probability, "\n")
cat("F-Score for P(X ≤ x) = 0.95:", f_score, "\n")
Note: The pf function calculates the cumulative probability, while qf gives the F-score for a specified probability.