Worked Lower-Tail and Two-Tail Variance Tests

Why Tail Direction Matters

A test comparing two population variances is not determined by the numerical size of the observed sample variances alone. The test statistic has an orientation: the first sample variance is in the numerator and the second is in the denominator. A lower-tailed alternative looks for evidence that this oriented ratio is too small; a two-tailed alternative looks for evidence that it is either unusually small or unusually large.

This distinction affects both the critical values and the decision. In particular, the lower critical value of an F distribution is often obtained through a reciprocal quantile with the degrees of freedom reversed. Reversing only one of those elements can change the answer materially.

Model and assumptions: Let two independent random samples come from normal populations with variances \(\sigma_1^2\) and \(\sigma_2^2\). Let their sample sizes be \(n_1\) and \(n_2\), and their unbiased sample variances be \(S_1^2\) and \(S_2^2\). The exact variance-ratio F tests below require both normal populations and independent samples. The samples need not have equal sizes or equal variances under the alternative.

The parameter of interest is the population variance ratio \(\rho=\sigma_1^2/\sigma_2^2\). To test a specified null ratio \(\rho_0\), use the oriented statistic

F_{\mathrm{obs}}=\frac{S_1^2/S_2^2}{\rho_0}.

Under the null hypothesis \(H_0:\rho=\rho_0\), this statistic has an \(F\) distribution with numerator degrees of freedom \(\nu_1=n_1-1\) and denominator degrees of freedom \(\nu_2=n_2-1\). In notation, \(F_{\mathrm{obs}}\sim F(\nu_1,\nu_2)\) under \(H_0\). The degrees of freedom follow the sample variances’ positions in the ratio; they do not depend on which sample has the larger observed variance.

Lower-Tailed Tests

For a lower-tailed test, the alternative is \(H_1:\rho<\rho_0\). At significance level \(\alpha\), reject \(H_0\) when the observed statistic falls below the lower \(\alpha\) quantile of \(F(\nu_1,\nu_2)\). Define \(q_p(\nu_1,\nu_2)\) as the \(p\) quantile, so that \(P(F\le q_p(\nu_1,\nu_2))=p\) for \(F\sim F(\nu_1,\nu_2)\).

Lower-tail rule: For \(H_1:\rho<\rho_0\), reject \(H_0\) if \(F_{\mathrm{obs}}<q_\alpha(\nu_1,\nu_2)\). The lower-tail p-value is \(P(F\le F_{\mathrm{obs}})\) under \(F\sim F(\nu_1,\nu_2)\).

Many printed F tables give upper-tail critical values rather than lower quantiles. F distributions have reciprocal symmetry: if \(F\sim F(\nu_1,\nu_2)\), then \(1/F\sim F(\nu_2,\nu_1)\). Consequently,

q_\alpha(\nu_1,\nu_2) = \frac{1}{q_{1-\alpha}(\nu_2,\nu_1)}.

The degrees of freedom reverse as well as the tail probability changing from \(\alpha\) to \(1-\alpha\). This is a quantile identity, not a rule to take the reciprocal of a quantile with the original degrees of freedom.

Worked Example: A Lower-Tailed Test With Unequal Degrees of Freedom

In a hypothetical laboratory comparison, independent normal samples of sizes \(n_1=5\) and \(n_2=3\) produce sample variances \(s_1^2=4\) and \(s_2^2=144\). Test \(H_0:\sigma_1^2/\sigma_2^2=1\) against \(H_1:\sigma_1^2/\sigma_2^2<1\) at \(\alpha=0.05\).

Method and model. The samples are stated to be independent and normal, so the exact F test applies. The null ratio is \(\rho_0=1\), and the degrees of freedom are \(\nu_1=5-1=4\) and \(\nu_2=3-1=2\). Under \(H_0\), the test statistic has distribution \(F(4,2)\).

Calculate the statistic.

F_{\mathrm{obs}} = \frac{4/144}{1} = \frac{1}{36} = 0.027777\ldots

Find the lower critical value. Use reciprocal symmetry with the degrees of freedom reversed:

q_{0.05}(4,2) = \frac{1}{q_{0.95}(2,4)}.

For \(F(2,4)\), \(q_{0.95}(2,4)\approx 6.94427\), so \(q_{0.05}(4,2)\approx 1/6.94427=0.144004\). The rejection rule is \(F_{\mathrm{obs}}<0.144004\). Since \(0.027777\ldots<0.144004\), reject \(H_0\) at the 5% level. The data provide evidence that the first population variance is less than the second.

Check and common mistake. The observed ratio is \(4/144=1/36\), a small number in the direction specified by the alternative, consistent with rejection. A common error is to use \(1/q_{0.95}(4,2)\) or to use an unrelated lower critical value. The correct conversion reverses the degrees of freedom: it is \(1/q_{0.95}(2,4)\), approximately \(0.144004\), not \(0.0620\).

Two-Tailed Tests

For a two-tailed test, the hypotheses are \(H_0:\rho=\rho_0\) and \(H_1:\rho\ne\rho_0\). An equal-tail test assigns probability \(\alpha/2\) to each tail under the null. Reject when the statistic is below the lower \(\alpha/2\) quantile or above the upper \(1-\alpha/2\) quantile.

Equal-tail two-sided rule: Reject \(H_0\) if \(F_{\mathrm{obs}}<q_{\alpha/2}(\nu_1,\nu_2)\) or \(F_{\mathrm{obs}}>q_{1-\alpha/2}(\nu_1,\nu_2)\). The equal-tail two-sided p-value is twice the smaller of the two null tail probabilities, capped at 1.

The two critical values are not generally reciprocals of one another when they use the same degrees of freedom. Reciprocal symmetry instead gives

q_{\alpha/2}(\nu_1,\nu_2) = \frac{1}{q_{1-\alpha/2}(\nu_2,\nu_1)}.

Thus, if an upper-tail table is used to find the lower critical value, both the probability and the degrees of freedom must be converted. The upper critical value for the original orientation is \(q_{1-\alpha/2}(\nu_1,\nu_2)\); it is not generally the reciprocal of the lower critical value for that same orientation.

Worked Example: A Two-Tailed Test and Its p-Value

Two independent samples from normal populations each have size \(3\). Their observed sample variances are \(s_1^2=100\) and \(s_2^2=5\). Test \(H_0:\sigma_1^2/\sigma_2^2=1\) against \(H_1:\sigma_1^2/\sigma_2^2\ne1\) at \(\alpha=0.10\), and calculate the equal-tail p-value.

Method and null distribution. The samples are independent and normal, so use the exact F test. Here \(\rho_0=1\), \(\nu_1=3-1=2\), and \(\nu_2=3-1=2\). Under \(H_0\), \(F_{\mathrm{obs}}\sim F(2,2)\).

Calculate the statistic.

F_{\mathrm{obs}} = \frac{100/5}{1} = 20.

For an \(F(2,2)\) variable, the cumulative distribution function is \(P(F\le x)=x/(1+x)\) for \(x>0\). Therefore its quantile is \(q_p(2,2)=p/(1-p)\). With \(\alpha=0.10\), the two critical values are

q_{0.05}(2,2)=\frac{0.05}{0.95}=\frac{1}{19}\approx0.05263, \qquad q_{0.95}(2,2)=\frac{0.95}{0.05}=19.

Since \(20>19\), reject \(H_0\) at the 10% level. For the p-value, the upper-tail probability is \(P(F\ge20)=1-P(F\le20)=1-20/21=1/21\). The lower-tail probability is \(P(F\le20)=20/21\), so the equal-tail p-value is

2\min\left(\frac{1}{21},\frac{20}{21}\right) = \frac{2}{21} \approx 0.0952.

Check and common mistake. The p-value is below \(0.10\), agreeing with the critical-value decision. A common error is to double the upper-tail probability without first checking which tail is smaller. Here the statistic is large, so the smaller tail is the upper tail; doubling \(20/21\) would be incorrect.

Why the Test Statistic Has This Null Distribution

The distributional basis for the procedure also explains why the null ratio must be included when it is not 1. For independent normal samples, the standardized sample variances satisfy

\frac{(n_1-1)S_1^2}{\sigma_1^2}\sim\chi^2_{\nu_1}, \qquad \frac{(n_2-1)S_2^2}{\sigma_2^2}\sim\chi^2_{\nu_2},

and these chi-square variables are independent. The F distribution is the distribution of the ratio of two independent chi-square variables after each is divided by its degrees of freedom. Hence

\frac{S_1^2/\sigma_1^2}{S_2^2/\sigma_2^2} \sim F(\nu_1,\nu_2).

Under \(H_0\), \(\sigma_1^2/\sigma_2^2=\rho_0\), so this ratio is exactly \((S_1^2/S_2^2)/\rho_0\), the test statistic defined above. This cancellation is why the statistic has the stated F distribution under the null. If the samples are not independent or the populations are not normal, the classical exact F distribution need not apply.

Worked Example: A Lower-Tailed Test With a Nonunit Null Ratio

A hypothetical calibration study takes independent normal samples of size \(3\) from two populations. The sample variances are \(s_1^2=3\) and \(s_2^2=20\). Test \(H_0:\sigma_1^2/\sigma_2^2=2\) against \(H_1:\sigma_1^2/\sigma_2^2<2\) at \(\alpha=0.10\).

Method and null distribution. Both samples are independent and normal, so the exact F test applies. The degrees of freedom are \(\nu_1=2\) and \(\nu_2=2\). Under \(H_0\), the statistic \((S_1^2/S_2^2)/2\) has distribution \(F(2,2)\).

Calculate the null-scaled statistic. The observed sample variance ratio is \(3/20=0.15\), but this is not yet the test statistic because the null ratio is 2:

F_{\mathrm{obs}} = \frac{3/20}{2} = \frac{3}{40} = 0.075.

For \(F(2,2)\), \(q_{0.10}(2,2)=0.10/0.90=1/9\approx0.1111\). Since \(0.075<0.1111\), reject \(H_0\) at the 10% level in favor of a variance ratio below 2.

Check and common mistake. The observed ratio \(0.15\) is below the hypothesized ratio 2, and scaling it by 2 gives \(F_{\mathrm{obs}}=0.075\), which lies in the lower rejection tail. A common mistake is to compare \(S_1^2/S_2^2\) directly with an F critical value. The critical value applies to the statistic divided by the null ratio.

A Reliable Decision Workflow

Before using a table or calculator, write down the hypotheses and preserve the chosen orientation. The following sequence keeps the tail, degrees of freedom, and null ratio aligned.

State the ratio and hypotheses.
Specify \(\rho=\sigma_1^2/\sigma_2^2\), the null value \(\rho_0\), and whether the alternative is lower-tailed or two-tailed.

Fix the statistic orientation.
Use \(F_{\mathrm{obs}}=(S_1^2/S_2^2)/\rho_0\), keeping sample 1 in the numerator.

Set the degrees of freedom.
Use \(\nu_1=n_1-1\) for the numerator and \(\nu_2=n_2-1\) for the denominator.

Find the appropriate tail cutoffs.
Use \(q_\alpha\) for a lower-tail test; use \(q_{\alpha/2}\) and \(q_{1-\alpha/2}\) for an equal-tail two-sided test. If converting a lower quantile by reciprocity, reverse the degrees of freedom.

Make and check the decision.
Compare the statistic with the stated rejection region, then confirm that its position is consistent with the alternative and the relevant tail probability.

Common Exam Traps

Changing the orientation after seeing the data. Putting the larger sample variance in the numerator may be convenient for a different calculation, but it changes the degrees of freedom and the alternative’s direction. Keep the orientation defined by the hypotheses.
Forgetting the null ratio. When \(\rho_0\ne1\), the statistic is \((S_1^2/S_2^2)/\rho_0\), not just \(S_1^2/S_2^2\).
Reversing only part of a reciprocal conversion. The identity \(q_\alpha(\nu_1,\nu_2)=1/q_{1-\alpha}(\nu_2,\nu_1)\) changes both the tail probability and the order of the degrees of freedom.
Using two full tails at level \(\alpha\). For an equal-tail two-sided test of overall size \(\alpha\), each tail has probability \(\alpha/2\), not \(\alpha\).
Applying the exact result outside its assumptions. The classical F test is exact for independent samples from normal populations. The result is not guaranteed to be exact for dependent samples or nonnormal populations.

Key takeaway: A variance-ratio test is controlled by three linked choices: the orientation of the ratio, the degrees of freedom in that orientation, and the tail allocation set by the alternative. Write all three before looking up a critical value.

Check Your Understanding

For each problem, use the stated exact F-test model and give the requested decision at the stated significance level.

Independent normal samples have sizes \(3\) and \(3\), with sample variances \(0.6\) and \(10\). Test \(H_0:\sigma_1^2/\sigma_2^2=1\) against a lower-tailed alternative at \(\alpha=0.10\). (Answer: \(F_{\mathrm{obs}}=0.06<1/9\approx0.1111\); reject \(H_0\).)
Independent normal samples have sizes \(3\) and \(3\), with sample variances \(4\) and \(2\). Test equality of the population variances against a two-sided alternative at \(\alpha=0.05\). (Answer: \(F_{\mathrm{obs}}=2\), and \(1/39\approx0.02564<2<39\); do not reject \(H_0\).)
Independent normal samples have sizes \(3\) and \(3\), with sample variances \(8\) and \(2\). Test \(H_0:\sigma_1^2/\sigma_2^2=2\) against a two-sided alternative at \(\alpha=0.10\). (Answer: \(F_{\mathrm{obs}}=(8/2)/2=2\), which is between \(1/19\approx0.05263\) and \(19\); do not reject \(H_0\).)
Independent normal samples have sizes \(5\) and \(3\), with sample variances \(0.10\) and \(1\). Test \(H_0:\sigma_1^2/\sigma_2^2=1\) against a lower-tailed alternative at \(\alpha=0.05\). (Answer: \(F_{\mathrm{obs}}=0.10<q_{0.05}(4,2)\approx0.1440\); reject \(H_0\).)
For an \(F(2,2)\) null distribution, an observed statistic is \(F_{\mathrm{obs}}=20\). Find the equal-tail two-sided p-value and decide at \(\alpha=0.10\). (Answer: \(2\min(1/21,20/21)=2/21\approx0.0952\); reject \(H_0\) at the 10% level.)

What You'll Learn