Compare sample variances with a robust F test. Paste raw data or type variances directly. Instant p values, critical bounds, and a clear verdict.
| Sample | n | Variance | Notes |
|---|---|---|---|
| A | 12 | 2.84 | Manufacturing line A measurements |
| B | 10 | 1.95 | Manufacturing line B measurements |
| Goal | Check if variances match | Use two-sided test at alpha = 0.05 | |
The classic variance test for two independent samples uses the F distribution. With sample variances sA² and sB²:
The p-value is computed from the F cumulative distribution. Two-sided p-values are calculated as:
This calculator supports classical F-based comparisons when two samples are independent and each population is approximately normal. In quality control, a variance shift of 20%–30% can signal tooling wear. In finance, volatility regimes often show variance ratios above 1.5 during stressed periods. Use the test as an assumption check before pooling variances in a t procedure. Always inspect plots and summary statistics first.
For samples with variances s1² and s2², the statistic F = s1²/s2² compares dispersion on a multiplicative scale. If s1² = 144 and s2² = 100, then F = 1.44. With df1 = n1−1 and df2 = n2−1, the calculator maps F to a p-value and critical boundaries, so you can decide whether the observed ratio is unusually large or small.
Right‑tailed tests target increases (σ1²>σ2²), left‑tailed tests target decreases, and two‑tailed tests flag either direction. For α = 0.05, the two‑tailed rule splits risk into 0.025 per tail. The report shows the relevant F critical value(s) so you can audit the decision without relying on software defaults.
Degrees of freedom control how tight the distribution becomes. With n1 = n2 = 10 (df = 9), critical thresholds are wider than with n1 = n2 = 50 (df = 49). Practically, small samples can require variance ratios above roughly 2.0 to reject at 5%, while larger samples may detect ratios near 1.3, depending on df balance.
The Plotly chart compares sample variances side‑by‑side and overlays the computed F ratio. A large gap between bars typically corresponds to an F far from 1. When the charted ratio crosses the displayed critical boundary, the decision tends to switch from “Fail to reject” to “Reject,” mirroring the numeric section.
A clear statement includes n1, n2, s1², s2², the chosen tail, α, F, df1, df2, and the p-value. Example wording: “F(19,19)=1.44, p=0.28, α=0.05; no evidence of unequal variances.” Exporting CSV or PDF preserves inputs and outputs for peer review and documentation.
A p-value is the probability, under the null hypothesis of equal variances, of observing an F ratio at least as extreme as yours. Smaller p-values indicate stronger evidence that the variances differ in the tested direction.
For a two‑tailed test, placing the larger sample variance in the numerator is common because it makes F ≥ 1 and simplifies interpretation. For one‑tailed tests, choose the numerator to match your hypothesis direction.
The classical F test is sensitive to departures from normality and to outliers. If your data are skewed or heavy‑tailed, consider Levene’s, Brown–Forsythe, or bootstrap methods, and use this tool mainly for learning or rough checks.
Sample variance s² estimates population variance σ² using n−1 in the denominator (Bessel’s correction). Population variance uses N. For inference, the F distribution is derived from sample variances computed with n−1.
In a two‑tailed test, the significance level α is split across both tails. You get a lower critical value and an upper critical value; rejecting occurs if F is below the lower bound or above the upper bound.
Yes, when you keep track of the numerator and denominator. If F = s1²/s2² is significant and F > 1 in a right‑tailed setting, the numerator group shows higher variability; for left‑tailed, the numerator is smaller.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.