Scheffé Critical Value Calculator
Enter the number of groups, total sample size, and significance level. The calculator returns the between-groups df, within-groups df, F critical value, and the adjusted Scheffé S threshold.
What Is the Scheffé Test?
The Scheffé test (also spelled Scheffe test or Scheffé's S method) is a post-hoc comparison procedure applied after a statistically significant one-way ANOVA. It controls the familywise error rate at α across every possible linear contrast that can be formed from the group means — not just the comparisons you happen to examine, but all that could theoretically be tested.
A linear contrast is any weighted combination of group means where the weights sum to zero: C = Σ ciȲi, with Σ ci = 0. Simple pairwise comparisons use weights +1 and −1. Complex contrasts compare weighted averages: testing whether the mean of Groups 1 and 2 combined differs from Group 3 uses weights +0.5, +0.5, and −1.0.
When to use Scheffé: Choose this test when you need to test complex, non-pairwise contrasts, when contrasts were not fully pre-specified before data collection, or when the number of groups is large. For simple pairwise-only comparisons, Tukey's HSD provides higher power. For a small number of pre-planned contrasts, Bonferroni correction is more powerful.
Scheffé Test Critical Value Table
Values below are Scrit = √[(k − 1) × Fα, k−1, df2]. Rows are df2 = N − k (within-groups). Columns are df1 = k − 1 (between-groups). A contrast is significant when |C| / SE(C) exceeds Scrit. Click any cell to highlight it.
Scrit = √[(k − 1) × Fα, k−1, N−k]. Reject H₀ for a contrast if |C| > Scrit × SE(C). F critical values sourced from the standard F-distribution; Scheffé thresholds derived from this formula and verified computationally. df2 = N − k (within-groups); df1 = k − 1 (between-groups).
The Mathematical Framework of Scheffé's S Method
For any linear contrast C = Σ ciȲi where the weights satisfy Σ ci = 0, the test works through three quantities.
Contrast Value
The contrast value C is the weighted sum of group sample means:
For a pairwise contrast of Groups 1 and 2: C = (+1)Ȳ₁ + (−1)Ȳ₂ = Ȳ₁ − Ȳ₂.
Standard Error of the Contrast
The standard error uses the Mean Square Error (MSE) from the ANOVA table and accounts for unequal group sizes ni:
MSE = SSwithin / (N − k). For equal group sizes: SE(C) = √[ MSE × Σ ci² / n ].
Scheffé Critical Threshold
The critical threshold against which the absolute contrast value is compared:
k = number of groups; N = total subjects; Fα, k−1, N−k = critical value from the standard F-table.
Decision Rule
If |C| ≤ Scrit × SE(C) → Fail to reject H₀ → Contrast is not statistically significant
The Contrast–Check–Conclude Framework
The three-stage process below organizes every Scheffé test into a repeatable structure.
1 Contrast — Define your weights
Write down ci for each group. The weights must sum to exactly zero. For a pairwise test of Group A vs. Group B: c = (+1, −1, 0, 0). For an organic vs. synthetic farming contrast across four groups: c = (+0.5, +0.5, −0.5, −0.5). Calculate C = Σ ciȲi and SE(C) = √[MSE × Σ ci²/ni].
2 Check — Locate the Scheffé threshold
Find df1 = k − 1 and df2 = N − k. Look up Fα, df1, df2 from the F-distribution table. Multiply by (k − 1) and take the square root: Scrit = √[(k − 1) × Fα]. The calculator at the top of this page automates this step.
3 Conclude — Compare and report
If |C| > Scrit × SE(C), the contrast is significant at α. Report the contrast weights, C, SE(C), Scrit, and your conclusion. If |C| ≤ Scrit × SE(C), fail to reject. A non-significant contrast is not evidence that the groups are equal — only that this sample does not provide sufficient evidence of a difference at this α.
Worked Example: Complex Agricultural Contrast
Research question: Do organic fertilizers (Groups 1 and 2) produce higher crop yield than synthetic fertilizers (Groups 3 and 4)? Four fertilizer types were applied to k = 4 groups with n = 6 plots each (N = 24 total).
ANOVA Summary Data
| Group | Fertilizer Type | ni | Ȳi (kg/plot) |
|---|---|---|---|
| Group 1 | Organic Compost | 6 | 48.2 |
| Group 2 | Manure | 6 | 45.8 |
| Group 3 | Nitrogen Synthetic | 6 | 39.5 |
| Group 4 | Phosphate Synthetic | 6 | 37.1 |
ANOVA result: F(3, 20) = 8.42, p < 0.001. MSE = 12.60.
Solution — Step by Step
| Step | Action | Result |
|---|---|---|
| 1 | Define contrast weights | c = (+0.5, +0.5, −0.5, −0.5) → Σci = 0 ✓ |
| 2 | Compute contrast C | 0.5(48.2)+0.5(45.8)−0.5(39.5)−0.5(37.1) = 8.70 |
| 3 | Compute SE(C) | √[12.60 × (4×0.25/6)] = √2.100 = 1.449 |
| 4 | Find F critical | F0.05, 3, 20 = 3.098 |
| 5 | Compute Scrit | √(3 × 3.098) = √9.294 = 3.049 |
| 6 | Decision | |8.70| > 3.049 × 1.449 = 4.418 → Significant ✓ |
Interpretation
The complex contrast comparing organic fertilizers against synthetic fertilizers is statistically significant at α = 0.05 (Scrit = 3.049, C = 8.70, SE(C) = 1.449). Organic fertilizer plots produced a mean yield approximately 8.70 kg/plot higher than synthetic plots. The Scheffé test controls familywise error for this and all other possible contrasts simultaneously.
Scheffé vs. Tukey HSD vs. Bonferroni
Choosing the wrong post-hoc procedure affects both the validity of conclusions and the ability to detect real effects. The table below identifies the defining property of each method.
| Feature | Tukey HSD | Bonferroni | Scheffé |
|---|---|---|---|
| Type of comparisons | Pairwise only | Pre-planned only | Any contrast |
| FWER control scope | All pairwise | Specified family | All possible contrasts |
| Power for pairwise tests | Highest | High (few tests) | Most conservative |
| Requires equal n | Standard form: yes | No | No |
| Handles complex contrasts | No | Yes, if pre-specified | Yes, any contrast |
| Best use case | All possible pairs | Few planned tests | Complex or exploratory contrasts |
When Scheffé is the right choice
Use Scheffé when: (1) you need to test contrasts that compare weighted averages of groups rather than single pairs; (2) contrasts emerged from looking at the data rather than pre-specification; or (3) you need rigorous protection against Type I error for an open-ended set of comparisons. For more on Type I and Type II errors, see the Statistics Fundamentals hypothesis testing guides.
Symbol and Notation Reference
Every symbol in the Scheffé procedure is defined below. Consistent notation matters when reading across textbooks — different authors use S, S*, or F-equivalent forms for the same threshold.
| Symbol | Name | Definition / Formula |
|---|---|---|
| k | Number of groups | Total number of treatment groups in the one-way ANOVA |
| N | Total sample size | Sum of all group sizes: N = n₁ + n₂ + … + nₖ |
| df₁ | Between-groups df | k − 1 (numerator degrees of freedom for F) |
| df₂ | Within-groups df | N − k (denominator degrees of freedom for F) |
| MSE | Mean Square Error | SSwithin / (N − k) — pooled within-group variance estimate |
| cᵢ | Contrast coefficient | Weights applied to group means; must sum to zero: Σ cᵢ = 0 |
| C | Contrast value | C = Σ cᵢ Ȳᵢ (weighted sum of group sample means) |
| SE(C) | Standard error of contrast | √[ MSE × Σ (cᵢ² / nᵢ) ] |
| Fα | F critical value | Fα, k−1, N−k from the standard F-distribution table |
| Scrit | Scheffé critical threshold | √[ (k − 1) × Fα, k−1, N−k] — the value in the table above |
Frequently Asked Questions About the Scheffé Test
What is a Scheffé test table used for?
A Scheffé test table provides the critical values used to evaluate all possible linear contrasts among group means after a significant F-statistic in ANOVA. It helps researchers determine whether complex combinations of groups are statistically significant while strictly controlling the familywise error rate.
How do you find the critical value for a Scheffé test?
Multiply the F critical value by (k − 1) and take the square root: Scrit = √[(k − 1) × Fα, k−1, N−k]. For example, with k = 4, N = 40, and α = 0.05: df1 = 3, df2 = 36, Fcrit ≈ 2.866, so Scrit = √(3 × 2.866) ≈ 2.932. Use the calculator at the top of this page to automate this.
What is the difference between Tukey and Scheffé post-hoc tests?
The main difference is flexibility versus statistical power. Tukey's HSD is designed exclusively for pairwise comparisons and offers higher power for those specific tests. The Scheffé test can evaluate any linear contrast or complex group combination, but it is more conservative for simple pairwise comparisons. If your only interest is pairwise differences, Tukey is preferred.
Does the Scheffé test require equal sample sizes?
No. The Scheffé test does not require equal group sizes. The standard error SE(C) = √[MSE × Σ(ci² / ni)] accounts for unequal ni directly. This is a practical advantage over Tukey's HSD, which assumes equal group sizes in its standard form.
Can I use the Scheffé test if the ANOVA was not significant?
The Scheffé critical threshold is mathematically equivalent to the omnibus F critical value. If the overall ANOVA F is not significant, no Scheffé contrast can be significant either. By convention, post-hoc contrasts are reported only after a significant omnibus F, because an insignificant F means there is no overall evidence of group differences to investigate further.
Can I apply Scheffé to contrasts formed after looking at the data?
Yes. This is one of the Scheffé test's defining advantages. Because the critical threshold controls FWER for the infinite set of all possible linear contrasts, it remains valid whether the specific contrasts were planned in advance or identified from the data. This is precisely why Scheffé is more conservative than methods that assume a pre-specified family. Researchers working in exploratory ANOVA contexts rely on it for this reason.
What if my df combination is not listed in the table?
If your exact df2 is absent, use the next smaller df2 listed — this is the conservative approach. Alternatively, use the calculator at the top of this page, which computes Scrit from the standard F critical value for any combination of k, N, and α without needing to interpolate. For precise F values at unlisted df, consult the full F-distribution table.
How is the Scheffé test related to the F-test in ANOVA?
The Scheffé critical threshold is derived directly from the F critical value: Scrit = √[(k − 1) × Fα]. This means the Scheffé procedure is mathematically consistent with the omnibus ANOVA F-test. When a contrast value equals the overall F-statistic value, the two tests produce identical decisions — by design. For more on degrees of freedom and F-distribution logic, see Statistics Fundamentals.
Sources & Further Reading
The Scheffé method was introduced in Henry Scheffé's 1953 paper and expanded in his 1959 textbook. Critical values in the table above are derived from the standard F-distribution using the formula Scrit = √[(k − 1) × Fα, k−1, N−k] and validated against the following standard references:
Scheffé, H. (1953). A method for judging all contrasts in the analysis of variance. Biometrika, 40(1–2), 87–104. — The original paper introducing the S method. Available through JSTOR.
Kirk, R. E. (2013). Experimental Design: Procedures for the Behavioral Sciences (4th ed.). SAGE Publications. — The most thorough applied treatment of planned and post-hoc contrast procedures, with full worked examples for both simple and complex contrasts.
Maxwell, S. E., Delaney, H. D., & Kelley, K. (2018). Designing Experiments and Analyzing Data (3rd ed.). Routledge. — Chapter 5 compares Scheffé, Tukey, Bonferroni, and Fisher LSD with simulation-based power comparisons. Widely used in graduate research methods courses.
NIST/SEMATECH e-Handbook of Statistical Methods. National Institute of Standards and Technology. itl.nist.gov — Free, government-maintained reference covering Scheffé, Tukey, and Bonferroni procedures with applied examples.
Penn State STAT 502: Analysis of Variance and Design of Experiments. Pennsylvania State University. online.stat.psu.edu/stat502 — Open-access course notes covering multiple comparison procedures including the Scheffé method with step-by-step examples.
Related Statistical Tables & Resources
Three Things Every Researcher Should Know About Scheffé
The threshold scales with k, not with the number of tests run
Scrit = √[(k − 1) × Fα] depends only on the number of groups k and the degrees of freedom — not on how many specific contrasts you actually examine. You pay the full conservatism cost for all possible contrasts the moment you have k groups, whether you test one contrast or one hundred. This is why Scheffé is best reserved for situations where the flexibility genuinely matters.
Significance and effect size are separate questions
A contrast that clears the Scheffé threshold is statistically significant — it is unlikely to be a chance artifact. Whether the mean difference is large enough to matter in practice is a separate question. Report a standardized effect size such as Cohen's d for the contrast or the partial η² for the ANOVA alongside the significance result. The effect size guide on Statistics Fundamentals covers this distinction in detail.
The Scheffé table is derived from the F-table
Every entry in the Scheffé table is computed as √[(df1) × Fα, df1, df2]. You can always derive the Scheffé threshold yourself from the standard F-distribution critical value table when the specific combination of df1 and df2 you need is not listed. The calculator at the top of this page automates this derivation for any combination of k, N, and α.