F-Test for Two Variances
Test whether two populations have the same variance (equivalently, the same standard deviation). Enter raw data or Sx and n for each sample; the output matches the TI-84 2-SampFTest screen: F, p, and the degrees of freedom.
Need the critical value instead of a p-value? See the F-table. Part of the TI-84 statistics functions guide.
Related Calculators
F-Table (F-Distribution Critical Values)
Look up F critical values by numerator and denominator degrees of freedom for α = 0.10, 0.05, 0.025, and 0.01.
ANOVA Calculator
Run one-way ANOVA calculations and inspect variance across groups.
Variance Calculator
Compute population or sample variance with every deviation and squared term shown.
The F-Test Formula
F = s₁² / s₂², df₁ = n₁ − 1, df₂ = n₂ − 1
H₀: σ₁² = σ₂² vs. Hₐ: σ₁² ≠ σ₂², σ₁² < σ₂², or σ₁² > σ₂²
Two-tailed p-value = 2 × min(P(F ≤ f), P(F ≥ f))
The test is very sensitive to non-normal data: even mild skewness can produce misleading p-values. Check both samples for normality — for example with a histogram — before relying on it.
How to Run 2-SampFTest on a TI-84
- Press STAT → TESTS and choose E:2-SampFTest.
- Choose Data (lists L1 and L2) or Stats (Sx1, n1, Sx2, n2).
- Pick the alternative σ1 ≠, <, or > σ2, then highlight Calculate and press ENTER.
In Excel, =F.TEST(range1, range2) returns the two-tailed p-value from raw data.
Worked Example: Machine Consistency
Machine 1: Sx₁ = 4.2 from n₁ = 13 parts. Machine 2: Sx₂ = 2.9 from n₂ = 16 parts. Is machine 1 less consistent?
- F = 4.2² / 2.9² = 17.64 / 8.41 = 2.0975, df = (12, 15).
- Right-tailed p-value (σ₁ > σ₂): 0.088305.
- Two-tailed p-value (σ₁ ≠ σ₂): 0.17661.
Neither is below 0.05, so the samples do not show a significant difference in spread — even though machine 1's standard deviation is 45% larger. Small samples make variance tests weak.
Frequently Asked Questions
Should I use the F-test to decide whether to pool variances in a t-test?
It is no longer recommended. The F-test is unreliable when data are not normal, and the unpooled (Welch) t-test works whether or not the variances are equal — so most textbooks now simply use Welch's test.
Which sample goes in the numerator?
Sample 1, as entered. The TI-84 does the same: F = Sx1² / Sx2². Swapping the samples gives F' = 1/F and the same two-tailed p-value; for one-tailed tests make sure the alternative matches the order.
What is the difference between this F-test and the ANOVA F-test?
Both use the F-distribution, but this test compares the variances of two populations directly, while ANOVA compares group means by looking at the ratio of between-group to within-group variance.
Can I enter standard deviations instead of variances?
Yes — that is the default. The calculator squares Sx1 and Sx2 to form the variance ratio, just like the TI-84's Stats input.
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