{
  "Title": "Quantiles of the $F$-distribution",
  "Definition": "The lower-tail quantile $F_{d_1,d_2,p}=\\inf\\{x>0:G_{d_1,d_2}(x)\\ge p\\}$ of the central $F$-distribution, where $G_{d_1,d_2}$ is its cumulative distribution function CITE{R}.",
  "Parameters": {
    "d1": {
      "type": "Z",
      "title": "numerator degrees of freedom",
      "display": "$d_1$",
      "constraints": "$d_1\\ge1$"
    },
    "d2": {
      "type": "Z",
      "title": "denominator degrees of freedom",
      "display": "$d_2$",
      "constraints": "$d_2\\ge1$"
    },
    "p": {
      "type": "Q",
      "title": "lower-tail probability",
      "display": "$p$",
      "constraints": "$0<p<1$"
    }
  },
  "Tags": [
    "probability theory",
    "special values"
  ],
  "Links": {
    "R": {
      "title": "R documentation: The F Distribution",
      "url": "https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Fdist.html"
    },
    "NISTCritical": {
      "title": "NIST/SEMATECH e-Handbook: Upper Critical Values of the F Distribution",
      "url": "https://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm"
    },
    "DLMFBeta": {
      "title": "DLMF 8.17: Incomplete beta functions",
      "url": "https://dlmf.nist.gov/8.17"
    }
  },
  "Data properties": {
    "type": "R",
    "digits": "100",
    "rigour": "proven",
    "complete": "no",
    "complete-note": "it holds every pair $1\\le d_1,d_2\\le10$ at $p\\in\\{1/100,1/40,1/20,1/2,19/20,39/40,99/100\\}$, covering small-sample degrees of freedom, the median, the one-sided 1% and 5% levels and their complements, and both cutoffs of an equal-tail 5% test",
    "rigour details": "Bisection encloses the inverse regularized beta function, evaluated by the hypergeometric representation CITE{DLMFBeta} in Sage ball arithmetic. Every bracket update uses a certified CDF comparison. The whole final bracket is transformed to an $F$ quantile and returned as a ball. The computation uses 461 bits for 100 requested significant digits, including 128 guard bits. Rational values are returned exactly from reciprocal symmetry and exact integer-root tests in CITE{formula-two}. The attached generator reproduces the values. The independent checks use elementary half-integer beta integrals and finite recurrences, without hypergeometric or gamma evaluation; they verify strict CDF signs at both endpoints of every stored decimal interval. Additional checks compare SciPy quantiles, NIST printed values, reciprocal symmetry, equal-degree medians, the special formulas and the equal-tail conversion. Across all 668 nonexact returned balls, the measured relative precision is at least 356 bits. There are 32 exact rational values. All 700 values agree with SciPy to relative error below $2.8\\cdot10^{-15}$. Thirty overlapping entries independently stored in the Student table satisfy CITE{formula-student} at 100 digits. All 100 overlapping values in CITE{NISTCritical} agree within one unit of its last printed decimal place. The printed value $238.882$ for $(d_1,d_2,p)=(8,1,19/20)$ is not correctly rounded to three decimal places: the certified value begins $238.8826948025\\ldots$, also confirmed by a 150-digit mpmath inversion.",
    "parameter selection": "The probabilities are the median and conventional critical levels, rather than a mesh for interpolation. CITE{NISTCritical} tabulates upper-tail 1% and 5% levels, giving $p=99/100$ and $19/20$; their complements give lower-tail cutoffs. The probabilities $1/40$ and $39/40$ are the two cutoffs of an equal-tail 5% test. The integer degrees $1$ through $10$ in each position cover the initial small-sample block of the NIST tables, with every pair included."
  },
  "Display properties": {
    "number-header": "$F_{d_1,d_2,p}$"
  },
  "Keywords": [
    "critical value",
    "percentage point",
    "percentile",
    "significance level",
    "p-value",
    "table",
    "Fisher-Snedecor distribution",
    "variance ratio",
    "ANOVA",
    "F test"
  ],
  "Comments": {
    "comment-ratio": "The central $F$-distribution is the law of $X=(U/d_1)/(V/d_2)$, where $U$ and $V$ are independent HREF{Quantiles_of_the_chi-squared_distribution}[chi-squared] variables with $d_1$ and $d_2$ degrees of freedom, respectively CITE{R}.",
    "comment-convention": "The table uses lower-tail probabilities. If $Q^{\\mathrm{upper}}_{d_1,d_2}(\\alpha)$ is defined by $\\Pr((U/d_1)/(V/d_2)>Q^{\\mathrm{upper}}_{d_1,d_2}(\\alpha))=\\alpha$, then $F_{d_1,d_2,p}=Q^{\\mathrm{upper}}_{d_1,d_2}(1-p)$. Thus a printed upper-tail column labelled $\\alpha=0.05$ corresponds to $p=19/20=0.95$ here CITE{NISTCritical}.",
    "comment-parameters": "The parameter $d_1$ counts numerator degrees of freedom, and $d_2$ counts denominator degrees of freedom. This table indexes positive integer degrees of freedom, although the central $F$-distribution extends to positive real degrees CITE{R}. Probability labels are exact rationals in lowest terms, so $p=39/40$ means $0.975$.",
    "comment-exact": "Equal-degree medians are exactly $1$ by reciprocal symmetry. The formulas with $d_1=2$ or $d_2=2$ also give rational quantiles whenever the displayed rational powers have rational roots; these values are stored exactly. For example, $F_{2,2,19/20}=19$ and $F_{2,4,99/100}=18$.",
    "comment-applications": "The critical values occur in analysis of variance and in tests and confidence intervals for the ratio of two normal-population variances CITE{R}. The central distribution assumes the null hypothesis; noncentral $F$ quantiles are different quantities."
  },
  "Formulas": {
    "formula-cdf": "$G_{d_1,d_2}(x)=I_{d_1x/(d_1x+d_2)}(d_1/2,d_2/2)$ for $x>0$, where $I_z(a,b)=B(z;a,b)/B(1;a,b)$ is the regularized form of HREF{Values_of_the_incomplete_beta_function}[the incomplete beta function] CITE{R}.",
    "formula-reciprocal": "$F_{d_1,d_2,p}=1/F_{d_2,d_1,1-p}$. In particular, $F_{d,d,1/2}=1$ for every positive integer $d$.",
    "formula-two-sided": "For $X=(U/d_1)/(V/d_2)$ and an equal-tail significance level $0<\\alpha<1$, the lower and upper cutoffs are $F_{d_1,d_2,\\alpha/2}$ and $F_{d_1,d_2,1-\\alpha/2}$: $\\Pr(F_{d_1,d_2,\\alpha/2}\\le X\\le F_{d_1,d_2,1-\\alpha/2})=1-\\alpha$.",
    "formula-student": "$F_{1,\\nu,p}=t_{\\nu,(1+p)/2}^{\\,2}$, where $t_{\\nu,q}$ is the lower-tail HREF{T454}[Student $t$ quantile] with $\\nu$ degrees of freedom CITE{R}.",
    "formula-one-one": "$F_{1,1,p}=\\tan^2(\\pi p/2)$.",
    "formula-two": "$F_{2,d,p}=\\frac d2\\big((1-p)^{-2/d}-1\\big)$ and $F_{d,2,p}=\\frac{2p^{2/d}}{d(1-p^{2/d})}$."
  },
  "Programs": {
    "program-r": {
      "language": "R",
      "code": "qf(19/20, df1 = 5, df2 = 10, lower.tail = TRUE)"
    },
    "program-python": {
      "language": "Python",
      "code": "import mpmath as mp\nmp.mp.dps = 110\nd1, d2 = 5, 10\np = mp.mpf(19)/20\ncdf = lambda x: mp.betainc(mp.mpf(d1)/2, mp.mpf(d2)/2,\n                          0, d1*x/(d1*x+d2), regularized=True)\nmp.findroot(lambda x: cdf(x)-p, (3, 4))"
    }
  },
  "Similar tables": [
    {
      "table": "HREF{T454}[Quantiles of Student's $t$-distribution]",
      "relation": "squaring a Student t variable gives an F variable with one numerator degree of freedom, as in CITE{formula-student}"
    },
    {
      "table": "HREF{Quantiles_of_the_chi-squared_distribution}[Quantiles of the chi-squared distribution]",
      "relation": "the F variable is a ratio of independent chi-squared variables divided by their degrees of freedom"
    },
    {
      "table": "HREF{Quantiles_of_the_standard_normal_distribution}[Quantiles of the standard normal distribution]",
      "relation": "a chi-squared variable with integer degrees of freedom is the sum of squares of that many independent standard normal variables"
    },
    {
      "table": "HREF{Values_of_the_incomplete_beta_function}[Values of the incomplete beta function $B(x;a,b)$]",
      "relation": "the regularized incomplete beta function gives the F cumulative distribution function in CITE{formula-cdf}"
    },
    {
      "table": "HREF{Values_of_the_lower_incomplete_gamma_function}[Values of the lower incomplete gamma function $\\gamma(a,x)$]",
      "relation": "regularized lower incomplete gamma values give lower-tail probabilities of the chi-squared variables in the defining ratio"
    },
    {
      "table": "HREF{Values_of_the_upper_incomplete_gamma_function}[Values of the upper incomplete gamma function $\\Gamma(a,x)$]",
      "relation": "regularized upper incomplete gamma values give upper-tail probabilities of the chi-squared variables in the defining ratio"
    },
    {
      "table": "HREF{Values_of_the_error_function}[Values of the error function $\\operatorname{erf}(x)$]",
      "relation": "the error function gives the standard normal cumulative distribution function, whose independent squared variables define the chi-squared variables in the ratio"
    },
    {
      "table": "HREF{Differential_entropies_of_continuous_probability_distributions}[Differential entropies of continuous probability distributions]",
      "relation": "includes differential entropies of central F distributions"
    },
    {
      "table": "HREF{Kullback-Leibler_divergences_between_probability_distributions}[Kullback-Leibler divergences between probability distributions]",
      "relation": "gives relative entropies for standard probability distributions rather than their quantiles"
    }
  ],
  "Numbers": []
}