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F distribution

The idea in one sentence

The F distribution is the ratio of two independent chi-square variables, each divided by its degrees of freedom. It compares two sources of variation and takes only positive values.

Support and interpretation

df1 and df2 are the numerator and denominator degrees of freedom. The tail can be very heavy when the denominator has few degrees of freedom.

Its place in the family

It is a ratio of chi-squares; taking its reciprocal gives another F with degrees of freedom exchanged. A squared Student t is F with one numerator degree of freedom.

An applied reading

In ANOVA, the F statistic compares explained with residual variation after scaling by degrees of freedom. Its reference distribution depends on the design and on normality, independence, and equal-variance assumptions.

Do not fit df1 and df2 as ordinary shape parameters when they are determined by an experimental design or regression.

A comparison of variation, not just another right tail

The F distribution grew out of ratios of variability. In ANOVA and regression, its numerator summarizes explained variation while the denominator supplies a noise reference, each adjusted by its degrees of freedom. A large value suggests that the first component dominates more than expected under the null hypothesis.

It has a useful symmetry: if a variable is F with degrees d1 and d2, its reciprocal is F with the degrees exchanged. The distribution is also the square of a suitable Student’s t when its numerator has one degree of freedom. These relations help verify calculations and show why fitting a free F curve to positive data does not automatically preserve the interpretation of a statistical test.

Decision guide

A good candidate when: the variable is a ratio of two independent variances or mean squares with known degrees of freedom.

Compare it with: Beta Prime for a positive ratio without that derivation. Strong skewness at low degrees of freedom has a large effect on critical quantiles.

References

  • SciPy reference: scipy.stats.f — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. Wiley.
  • Fisher, R. A. (1924). On a distribution yielding the error functions of several well known statistics. Proceedings of the International Congress of Mathematicians, 2, 805–813.

F Distribution: equations and calculator

Distribution defintion

X∼F(df1,df2)X\sim\mathrm{F}\left(\text{df}_{1},\text{df}_{2}\right)

Distribution domain

x∈[0,∞)x\in [0,\infty)

Parameters domain and parameters constraints

df1∈R+,df2∈R+\text{df}_{1}\in\mathbb{R}^{+},\text{df}_{2}\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=Idf1x/(df1x+df2)(df12,df22)F_{X}\left(x\right)=I_{\text{df}_{1} x/(\text{df}_{1} x+\text{df}_{2})}\left (\tfrac{\text{df}_{1}}{2},\tfrac{\text{df}_{2}}{2}\right)

Probability density function

fX(x)=(df1x)df1df2df2(df1x+df2)df1+df2x ×Beta(df12,df22)f_{X}\left(x\right)=\frac{\sqrt{\frac{(\text{df}_{1} x)^{\text{df}_{1}} \text{df}_{2}^{\text{df}_{2}}}{(\text{df}_{1} x+\text{df}_{2})^{\text{df}_{1}+\text{df}_{2}}}}}{x\ \times \text{Beta}\left(\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)}

Percent point function/Sample

FX−1(u)=df2×I−1(u,df12,df22)df1×(1−I−1(u,df12,df22))F^{-1}_{X}\left(u\right)=\frac{\text{df}_{2}\times I^{-1}\left(u,\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)}{\text{df}_{1}\times \left(1-I^{-1}\left(u,\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)\right)}

Non-central parametric moments

μk′=E[Xk]=∫0∞xkfX(x)dx=(df2df1)kΓ(df12+k)Γ(df12)Γ(df22−k)Γ(df22)if df2>2k\mu'_{k}=E[X^k]=\int_{0}^{\infty}x^{k }f_{X}\left(x\right)dx=\left(\frac{\text{df}_{2}}{\text{df}_{1}}\right)^k\frac{\Gamma\left(\tfrac{\text{df}_{1}}{2}+k\right) }{\Gamma\left(\tfrac{\text{df}_{1}}{2}\right)}\frac{\Gamma\left(\tfrac{\text{df}_{2}}{2}-k\right) }{\Gamma\left(\tfrac{\text{df}_{2}}{2}\right) } \quad \text{if }\text{df}_{2} > 2k

Parametric mean

Mean(X)=μ1′=df2df2−2if df2>2\mathrm{Mean}(X)=\mu'_{1}=\frac{\text{df}_{2}}{\text{df}_{2}-2} \quad \text{if }\text{df}_{2} > 2

Parametric variance

Variance(X)=μ2′−μ1′2=2 df22 (df1+df2−2)df1(df2−2)2(df2−4)if df2>4\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{2\,\text{df}_{2}^2\,(\text{df}_{1}+\text{df}_{2}-2)}{\text{df}_{1} (\text{df}_{2}-2)^2 (\text{df}_{2}-4)} \quad \text{if }\text{df}_{2} > 4

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=(2df1+df2−2)8(df2−4)(df2−6)df1(df1+df2−2)if df2>6\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{(2 \text{df}_{1}+\text{df}_{2}-2) \sqrt{8 (\text{df}_{2}-4)}}{(\text{df}_{2}-6) \sqrt{\text{df}_{1} (\text{df}_{1}+\text{df}_{2} -2)}}\quad \text{if }\text{df}_{2} > 6

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3(8+(df2−6)×Skewness(X)2)2df2−16+3if df2>8\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=\frac{3\left(8+\left(\text{df}_{2}-6\right)\times \mathrm{Skewness}(X)^{2}\right)}{2\text{df}_{2}-16}+3\quad \text{if }\text{df}_{2} > 8

Parametric median

Median(X)=df2×I−1(12,df12,df22)df1×(1−I−1(12,df12,df22))\mathrm{Median}(X)=\frac{\text{df}_{2}\times I^{-1}\left(\frac{1}{2},\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)}{\text{df}_{1}\times \left(1-I^{-1}\left(\frac{1}{2},\frac{\text{df}_{1}}{2},\frac{\text{df}_{2}}{2}\right)\right)}

Parametric mode

Mode(X)=df2(df1−2)df1(df2+2)if df1>2\mathrm{Mode}(X)=\frac{\text{df}_{2}\left(\text{df}_{1}-2\right)}{\text{df}_{1}\left(\text{df}_{2}+2\right)} \quad \text{if }\text{df}_{1} > 2

Additional information and definitions

u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
I(x,a,b):Regularized incomplete beta functionI\left(x,a,b\right):\text{Regularized incomplete beta function}
I−1(x,a,b):Inverse of regularized incomplete beta functionI^{-1}\left(x,a,b\right):\text{Inverse of regularized incomplete beta function}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}