Calculate the Scheffé test F statistic for two group means or a mean difference using equal sample sizes, ANOVA k, MSE, and an optional upper-tail F-critical comparison.

Pairwise Scheffé comparison for a one-way ANOVA with equal sample size in every group. Assumes independent observations, approximately normal errors and common variance.

All k groups must have the same n, not just this pair.

Optional settings

From your chosen significance level with df₁ = k − 1 and df₂ = k(n − 1). Blank gives the statistic only.

Scheffe Test Formula

The Scheffe test is a conservative post hoc method used after ANOVA to evaluate differences among group means while limiting the chance of false positives across multiple comparisons. This calculator compares one pair using their means (or absolute difference), the common sample size in each of all k groups, and MSE from the full one-way ANOVA.

FS = (n(x₁ - x₂)²) / (2(k - 1)MSE)

Here n is the common sample size per group, k is the number of groups in the full ANOVA, and MSE is its mean square error. The total sample size is N = kn; the F comparison uses numerator df k − 1 and denominator df N − k.

dfwithin = N - k = k(n - 1)

Input Definitions

Input Meaning What to Enter
Means or |Δ| Two group means or their absolute difference Finite means, or a nonnegative difference
MSE Mean square within groups, often called mean square error A positive value greater than 0
n Common sample size in each of all k groups A whole number of at least 2
k Number of groups being compared A whole number of at least 2
S Pairwise Scheffé F statistic Computed by the calculator

How to Calculate the Scheffe Test Statistic

  1. Choose two group means, or calculate their absolute difference |Δ|.
  2. Take the mean square within groups or mean square error from the same ANOVA table.
  3. Enter the common sample size per group; this equal-n widget requires it to be the same in all groups.
  4. Count the number of groups included in the comparison.
  5. Compute the pairwise statistic using the formula above.
FS = (nΔ²) / (2(k - 1)MSE)

Compare F_S with the upper-tail F-critical for your chosen α, numerator df k − 1 and denominator df k(n − 1).

Reject H₀ when FS > Fα;k - 1,k(n - 1)

Example

For means 10 and 14, MSE = 80, n = 10 observations in each of k = 5 groups (N = 50), the absolute difference is 4 and:

FS = (10(14 - 10)²) / (2(5 - 1)80) = 0.25

The pairwise F statistic is 0.25. Compare it with the chosen upper-tail F-critical for df₁ = 4 and df₂ = 45; it is not the omnibus ANOVA F statistic.

How to Interpret the Result

In general, a larger statistic suggests that the group means are more separated relative to the variation inside the groups. A smaller statistic suggests that the apparent differences among group means are modest compared with the within-group noise.

For formal statistical decisions, Scheffe procedures are typically used in the broader context of ANOVA. That means the final conclusion often depends on the ANOVA model, the comparison being tested, and the relevant critical value. This calculator is most useful for quickly turning ANOVA summary values into a comparison statistic.

When the Scheffe Test Is Useful

  • After a one-way ANOVA shows evidence that not all group means are equal.
  • When you want a conservative method that controls error across many possible comparisons.
  • When broader contrasts may be relevant; the general method supports them, while this widget implements only the equal-size pairwise special case.
  • When protecting against false positives is more important than maximizing sensitivity.

Common Input Mistakes

  • Entering total sample size instead of the common per-group sample size required by this widget.
  • Using sum of squares values instead of mean square values.
  • Using the number of pairwise comparisons instead of the number of groups for k.
  • Entering an MSE of 0, which makes the ratio undefined.
  • Using unequal group sizes or n below 2. The general Scheffé contrast formula is needed for unequal group sizes.

Assumptions to Keep in Mind

  • Observations should be independent.
  • The groups should come from populations that are reasonably normal, especially for smaller samples.
  • Group variances should be fairly similar when using standard ANOVA-based procedures.
  • The ANOVA model should match how the data were collected and grouped.

Quick Summary of the Logic

The Scheffe approach asks whether the variation between groups is large compared with the variation within groups. If between-group variation grows while within-group variation stays small, the resulting statistic increases and the evidence for real mean differences becomes stronger.

Scheffe Test Calculator screenshot