F Distribution Calculator

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Parameters

Degrees of Freedom (Numerator)

Degrees of Freedom (Denominator)

Calculation Type

Lower bound (x1)

Upper bound (x2)

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

F Distribution Calculator

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Parameters

Distribution Chart

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F Distribution: Definition, Formula, and Applications

F Distribution

Properties

Key Statistics:

Key Properties:

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