F Ratios of Independent Normal Sample Variances

Why a Ratio of Sample Variances Has an Exact Distribution

A sample variance is random: a different normal sample from the same population will generally produce a different value. When two independent normal samples are available, their sample variances can be compared through a ratio. The exact distribution of that ratio depends on the two sample sizes and on whether the population variances are equal. Keeping these facts separate is essential: the ratio has a standard F distribution after it is scaled by the population variances, and it has that standard distribution without further scaling under equality of the population variances.

The central result follows from two facts about normal samples. Each sample variance, when scaled by its population variance, is a chi-square random variable divided by its degrees of freedom. Independent samples give independent chi-square variables. Taking their ratio produces the F distribution.

The order in this definition matters: the variable from the numerator supplies \(\nu_1\), and the one from the denominator supplies \(\nu_2\). A useful density formula makes both roles explicit. Write \(a=\nu_1/2\), \(b=\nu_2/2\), and define the beta function by \(B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)\).

Deriving the F Density

The chi-square density with \(\nu\) degrees of freedom is \(u^{\nu/2-1}e^{-u/2}/(2^{\nu/2}\Gamma(\nu/2))\), for \(u>0\). To derive the F density, start with independent \(U\) and \(V\) as in the definition, and transform from \((U,V)\) to \((X,V)\), where \(X=(U/\nu_1)/(V/\nu_2)\). The inverse transformation is \(U=(\nu_1/\nu_2)XV\), \(V=V\); its Jacobian determinant in absolute value is \((\nu_1/\nu_2)V\). Independence permits us to multiply the two chi-square densities.

Put \(a=\nu_1/2\), \(b=\nu_2/2\), and \(c=\nu_1/\nu_2\). After substituting \(u=cxv\) and multiplying by the Jacobian, the joint density of \((X,V)\) is

Integrate over \(v>0\) to obtain the marginal density of \(X\). The gamma integral \(\int_0^\infty v^{a+b-1}e^{-qv}\,dv=\Gamma(a+b)/q^{a+b}\), valid for \(q>0\), applies with \(q=(1+cx)/2\). Therefore

Substituting \(a=\nu_1/2\), \(b=\nu_2/2\), and \(c=\nu_1/\nu_2\) gives the stated density. Each density and transformation here is defined on positive values, so the resulting F variable is also positive. The density integrates to one because it was obtained by integrating the joint density of a valid random pair.

From Normal Samples to an F Ratio

Let \(X_1,\ldots,X_{n_1}\) and \(Y_1,\ldots,Y_{n_2}\) be independent samples, with \(n_1,n_2\geq 2\). Suppose the first sample is independent and identically distributed as \(N(\mu_1,\sigma_1^2)\), and the second is independent and identically distributed as \(N(\mu_2,\sigma_2^2)\), with \(\sigma_1^2,\sigma_2^2>0\). Use the usual unbiased sample variances

For a normal sample, the established sample-variance result is \((n-1)S^2/\sigma^2\sim\chi^2_{n-1}\). Applying it separately to these two samples gives chi-square variables with \(\nu_1=n_1-1\) and \(\nu_2=n_2-1\) degrees of freedom. The samples are independent, so their sample variances, and hence these scaled variables, are independent.

Proof. Set \(U=(n_1-1)S_1^2/\sigma_1^2\) and \(V=(n_2-1)S_2^2/\sigma_2^2\). By the normal-sample variance result and independence, \(U\sim\chi^2_{n_1-1}\), \(V\sim\chi^2_{n_2-1}\), and \(U,V\) are independent. Thus

Multiplying this identity by the positive constant \(\sigma_1^2/\sigma_2^2\) gives the distribution of the raw variance ratio. \(\square\)

Equality of Variances and F Probabilities

If the population variances are equal, \(\sigma_1^2=\sigma_2^2\), their ratio is 1. The theorem then says that the raw sample-variance ratio \(S_1^2/S_2^2\) itself has an \(F_{n_1-1,n_2-1}\) distribution. This equality is what permits an unscaled sample-variance ratio to be compared with an F distribution in exact calculations under the equal-variance model.

The beta function also gives a convenient way to express F probabilities. If \(X\sim F_{\nu_1,\nu_2}\), define \(Z=\nu_1X/(\nu_1X+\nu_2)\). Substituting \(x=\nu_2z/(\nu_1(1-z))\) into the F density, with derivative \(dx/dz=\nu_2/(\nu_1(1-z)^2)\), gives the beta density

Thus \(Z\) has a beta distribution with parameters \(\nu_1/2,\nu_2/2\). In particular, for \(x>0\), the event \(X\leq x\) is equivalent to \(Z\leq \nu_1x/(\nu_1x+\nu_2)\). This transformation can simplify exact probability calculations.

Reversing the Ratio Changes the Degrees of Freedom

For positive independent \(U\sim\chi^2_{\nu_1}\) and \(V\sim\chi^2_{\nu_2}\), the reciprocal of \((U/\nu_1)/(V/\nu_2)\) is \((V/\nu_2)/(U/\nu_1)\). The numerator and denominator have exchanged places, so the reciprocal has distribution \(F_{\nu_2,\nu_1}\). The same fact applies to standardized sample-variance ratios. It does not mean that an F variable and its reciprocal have the same distribution in general.

Common Mistakes and Why the F Law Matters

• Using the wrong degrees of freedom. The sample sizes themselves are not the degrees of freedom for the sample variances. With the unbiased sample-variance convention, they are \(n_1-1\) and \(n_2-1\).
• Reversing the ratio without reversing the degrees of freedom. If the variance ratio is inverted, the numerator and denominator degrees of freedom must also be exchanged.
• Calling an unscaled ratio F when the population variances differ. The standardized ratio has the F law under the stated assumptions. The raw ratio is scaled by \(\sigma_1^2/\sigma_2^2\); it is unscaled only when these population variances are equal.
• Ignoring independence. The derivation requires independent samples. Separate normal marginal distributions alone do not establish independence of the two sample variances.

The F distribution provides an exact reference law for comparing independent normal-sample variances. Its two degrees of freedom record the separate information in the numerator and denominator; they are not interchangeable labels. When using a variance ratio, first identify which sample is in the numerator, then check whether standardization by the population variance ratio is required.