Newman-Keuls q Critical Value Calculator
What Is the Newman-Keuls Table?
The Newman-Keuls table — also called the Student-Newman-Keuls (SNK) table — provides critical values of the studentized range statistic q used in the SNK post hoc procedure. After a one-way ANOVA returns a significant F-statistic, the test identifies which specific pairs of group means are significantly different from each other.
What sets the Newman-Keuls procedure apart is its stepwise approach: rather than applying one fixed critical value to every comparison (as Tukey HSD does), it uses a different qcrit for each pair depending on how many ordered positions separate the two means. This gives the test more power than Tukey HSD for pairs that are close together in the ranking, at the cost of somewhat less strict family-wise error rate control.
One-sentence definition: The Newman-Keuls table lists the minimum q value — the studentized range statistic — that a pair of means must produce to be declared significantly different, based on p (number of steps apart in the ranked order of means), dferror, and α.
Newman-Keuls Critical Value Table (Studentized Range q)
All values below are qcrit. Select the α tab that matches your significance level. Columns correspond to p = number of steps between means in ranked order. Click any cell to highlight and load it into the calculator above.
p = number of steps between compared means in ranked order; df = error degrees of freedom from ANOVA (dfwithin = N − k). Reject H₀ for that pair if q ≥ qcrit. Values based on the studentized range distribution (Keuls, 1952; Newman, 1939). For df not listed, interpolate or use the next smaller df (conservative).
How the Newman-Keuls Procedure Works
The SNK test follows a structured, stepwise algorithm. Each step builds on the previous comparison. The procedure below applies to any one-way ANOVA with equal group sizes.
Step 1 — Confirm F is Significant
The SNK test is a post hoc procedure, meaning it runs only after the overall ANOVA F-test rejects H₀. Record MSE (mean square error, or MSwithin) and dferror (dfwithin = N − k, where N is total observations and k is number of groups) from the ANOVA table — both are needed for every comparison.
Step 2 — Rank Group Means
Arrange all k group means from smallest (position 1) to largest (position k). Write them in order. The rank positions determine p for each pair of comparisons — this is the defining feature of the Newman-Keuls approach.
Step 3 — Compute p for Each Pair
For any two means at rank positions i and j (where j > i), p = j − i + 1. Two means that are adjacent in the ranking have p = 2. Means that are two ranks apart have p = 3. The extreme pair (rank 1 vs rank k) has p = k. This p value is what you use to look up qcrit in the table.
Step 4 — Compute the q Statistic
For each pair of means, compute the SNK test statistic:
MSE = mean square error from ANOVA. n = group size (common sample size per group). For unequal group sizes, use n* = 2 / (1/n_i + 1/n_j) — the harmonic mean of the two group sizes.
Step 5 — Look Up q Critical
For each pair, use the table with: (a) p for that specific pair, (b) dferror from the ANOVA table, and (c) your chosen α. The cell at the intersection of the df row and p column is qcrit. Each pair gets its own qcrit — a smaller critical value for adjacent means, a larger one for means farther apart.
Step 6 — Apply Decision Rule and Stopping Rule
If q < qcrit (at this p) → Fail to reject H₀ → Means are not significantly different
Stopping rule: The SNK procedure uses a top-down sequence. Start with the largest p (comparing the extreme means). If a comparison is not significant, all comparisons with smaller p that are completely within that non-significant range are also declared not significant without testing. This is what makes the procedure stepwise.
Newman-Keuls Formula and Notation
The core formula produces a single q value per comparison. That value is then compared against the appropriate qcrit from the Newman-Keuls table.
Test Statistic
x̄_i, x̄_j = the two group means (larger minus smaller). MSE = mean square error from ANOVA. n = per-group sample size (equal groups).
Lookup Parameters
p = number of steps between means in ranked order. N = total observations. k = number of groups. qcrit depends on p, df, and α.
Minimum Significant Difference
The minimum difference between two means required for significance, computed separately for each p. Easier to use than q when comparing many pairs: if x̄_i − x̄_j ≥ MSD_p, the pair is significant.
Worked Example: Step-by-Step Newman-Keuls Test
Scenario: An educational researcher tests three teaching methods (A, B, C) on exam performance. Each group has n = 10 students. One-way ANOVA returns F(2, 27) = 4.82, p < 0.05. From the ANOVA table: MSE = 18.40, dferror = 27. Group means: x̄_A = 72.1, x̄_B = 78.6, x̄_C = 83.4. Which pairs differ?
Step 1 — Rank Means and Determine p
| Rank | Group | Mean |
|---|---|---|
| 1 | Group A | 72.1 |
| 2 | Group B | 78.6 |
| 3 | Group C | 83.4 |
Pairs: C vs A → p = 3 − 1 + 1 = 3. C vs B → p = 3 − 2 + 1 = 2. B vs A → p = 2 − 1 + 1 = 2.
Step 2 — Compute SEw
Step 3 — Compute q and Look Up qcrit (df = 27, α = 0.05)
| Comparison | p | Mean diff | q computed | qcrit | Decision |
|---|---|---|---|---|---|
| C vs A | 3 | 11.3 | 8.33 | 3.53 | Sig. ✓ |
| C vs B | 2 | 4.8 | 3.54 | 2.90 | Sig. ✓ |
| B vs A | 2 | 6.5 | 4.79 | 2.90 | Sig. ✓ |
Interpretation
All three pairs of teaching methods produce statistically different exam means at α = 0.05. Method C (mean = 83.4) significantly outperforms both B (mean = 78.6) and A (mean = 72.1), and Method B significantly outperforms Method A. The researcher can conclude that the three teaching methods produce distinguishably different outcomes on this sample.
Newman-Keuls Test in Practice: Applied Examples
The SNK test originated in agricultural research and remains widely used wherever group comparisons follow a significant ANOVA. The examples below show how p changes with the ranking of means across different real-world contexts.
Agriculture: Fertilizer Yield Comparison
Four fertilizer treatments (Control, Low N, Medium N, High N) are applied to k = 4 groups of crops with n = 8 plots each. ANOVA gives F(3, 28) = 6.11, MSE = 4.25. Ranked means: Control = 42.1, Low N = 47.3, Medium N = 51.8, High N = 58.4. The comparison of High N vs Control spans p = 4, using qcrit(4, 28, 0.05) = 3.86. All pairwise tests start with this outermost pair and work inward. This is the classic original application for which Newman and Keuls designed the procedure.
Clinical Research: Drug Dosage Effects on Pain Scores
Three dosages of an analgesic (2 mg, 5 mg, 10 mg) and a placebo group produce pain ratings (lower = less pain). After a significant ANOVA with dferror = 36, the ranked means are: Placebo = 7.4, 2 mg = 6.1, 5 mg = 4.8, 10 mg = 3.2. The comparison of 10 mg vs Placebo uses p = 4; adjacent comparisons like 5 mg vs 2 mg use p = 2. This gives adjacent comparisons a lower critical value, making it easier to detect closely ranked group differences.
Psychology: Recall Across Learning Conditions
Five groups of participants learn a word list under different encoding strategies (rote repetition, imagery, semantic, elaborative, spaced practice). n = 12 per group, dferror = 55, MSE = 9.62. Ranked means produce p values from 2 to 5 depending on which pair is compared. Here the SNK test's power advantage over Tukey HSD matters most — with five groups, Tukey HSD would apply a single large critical value qcrit(5, 55, 0.05) = 3.98 to every pair, while SNK uses a smaller value for adjacent pairs.
Manufacturing: Defect Rates Across Production Lines
A quality engineer compares defect rates across six production lines (n = 15 batches each). After ANOVA confirms a significant difference (dferror = 84), the SNK test identifies which specific lines differ. Line pairs that are adjacent in ranked defect rate use p = 2 and a smaller qcrit, allowing detection of even modest differences between neighboring lines in the ranking.
Newman-Keuls vs Other Post Hoc Tests
Choosing the right post hoc test depends on how much you prioritize statistical power versus strict control of the family-wise error rate. The table below compares the main options.
| Test | Critical Value | FWER Control | Power | Best Use |
|---|---|---|---|---|
| Newman-Keuls (SNK) | Varies by p (stepwise) | Partial | High | Exploratory research; agriculture |
| Tukey HSD | Fixed at k groups | Strict (≤ α) | Moderate | All pairwise; equal n required |
| Bonferroni | α / m (per comparison) | Strict (≤ α) | Lower | Few planned comparisons |
| Scheffé | Based on F distribution | Strict (≤ α) | Lowest | Complex contrasts; unequal n |
| Duncan's MRT | Varies by p (less strict) | Weak | Highest | Agriculture; when power matters most |
| Fisher LSD | t critical value | None | Highest | Only 3 groups (protected LSD) |
FWER = family-wise error rate. Power ranking assumes equal group sizes and a fixed number of comparisons. The rankings are relative — the best choice depends on your research context, not solely on power or FWER.
When Newman-Keuls Lacks Strict FWER Control
Statisticians have shown that for k > 3 groups, the Newman-Keuls procedure does not guarantee that the probability of at least one false rejection stays at or below α. The family-wise error rate can exceed α, particularly when many groups are compared. For confirmatory research where false positives are costly — clinical trials, for example — Tukey HSD is usually the safer choice. For exploratory work or agricultural field trials where finding real differences matters more than strict error control, Newman-Keuls remains a reasonable and widely accepted option.
Which Post Hoc Test Should You Use?
The answer depends on your research goal, group structure, and how you weigh statistical power against family-wise error. This guide works through the most common decision points.
Use Newman-Keuls (SNK) when:
You want more statistical power than Tukey HSD for adjacent pairs. You have equal or near-equal group sizes. Your research is exploratory, not confirmatory. The field norm (agriculture, ecology, some psychology) accepts SNK. You understand the limited FWER guarantee and accept it for your context.
Use Tukey HSD when:
You need strict family-wise error rate control at α. You are comparing all pairwise means with equal n. The research is confirmatory (clinical, regulatory, or publication-sensitive). Reviewers or your field expect strict error control. See the Tukey's q table for critical values.
Use Bonferroni when:
You have a small number of pre-planned comparisons (fewer than all pairwise). Group sizes are unequal. You want a straightforward and defensible correction. You are willing to accept reduced power in exchange for conservative error control. See Bonferroni correction for the procedure.
Assumptions, Limitations, and Common Mistakes
The Newman-Keuls test relies on specific statistical conditions. When these assumptions are violated, the test may produce unreliable results.
Required Assumptions
Common Mistakes to Avoid
Newman-Keuls Test in R, SPSS, and SAS
Most statistical software implements the SNK procedure. The examples below show how to run the test and how to locate the values you need for manual table lookup.
R: agricolae Package
library(agricolae) # Fit one-way ANOVA model <- aov(yield ~ treatment, data = mydata) summary(model) # Check MSE and df_error # Run Newman-Keuls post hoc test snk_result <- SNK.test(model, "treatment", alpha = 0.05) print(snk_result$groups) # Shows mean groups with letter codes
The output includes group means, the q critical value used, and letter-based grouping (groups sharing a letter are not significantly different).
SPSS: One-Way ANOVA Dialog
Analyze → Compare Means → One-Way ANOVA → Post Hoc → check "S-N-K" (Student-Newman-Keuls) → Continue → OK Output: Post Hoc Tests → Homogeneous Subsets table (Groups in the same column are not significantly different)
SAS: PROC ANOVA / GLM
PROC ANOVA data=mydata; CLASS treatment; MODEL response = treatment; MEANS treatment / SNK; /* SNK keyword for Newman-Keuls */ RUN; /* Output includes: MSE, df_error, and pairwise comparisons with significance groupings */
Symbol and Concept Glossary
The table below defines every symbol and concept involved in the Newman-Keuls procedure, with the most common misinterpretation noted for each.
| Symbol | Name | Definition | Common Error |
|---|---|---|---|
| q | Studentized range statistic | Computed test value: (x̄_i − x̄_j) / √(MSE/n) | Using SD instead of √(MSE/n) in denominator |
| qcrit | Critical value | Table value at (p, df, α); reject H₀ if q ≥ qcrit | Using same qcrit for all pairs (ignoring p) |
| p | Number of steps | Rank difference + 1 between compared means | Treating p as total number of groups k for all pairs |
| MSE | Mean square error | MSwithin from ANOVA table; estimates σ² | Using pooled variance of just two groups |
| df | Error degrees of freedom | dfwithin = N − k; used for table lookup | Using total df (N−1) instead of error df (N−k) |
| k | Number of groups | Total number of treatment/factor levels | Confusing k with p (p ≤ k always) |
| SEw | Standard error of within | √(MSE/n); the denominator of the q formula | Using group-specific standard errors instead |
| FWER | Family-wise error rate | Probability of ≥1 false rejection among all comparisons | Assuming SNK fully controls FWER at α (it does not for k > 3) |
Quick Reference: Most Used q Critical Values
The values below cover the most commonly encountered combinations of p and df in published research. All at α = 0.05.
| df | p = 2 | p = 3 | p = 4 | p = 5 | p = 6 |
|---|---|---|---|---|---|
| 10 | 3.15 | 3.88 | 4.33 | 4.65 | 4.91 |
| 20 | 2.95 | 3.58 | 3.96 | 4.23 | 4.45 |
| 30 | 2.89 | 3.49 | 3.85 | 4.10 | 4.30 |
| 60 | 2.83 | 3.40 | 3.74 | 3.98 | 4.16 |
| ∞ | 2.77 | 3.31 | 3.63 | 3.86 | 4.03 |
All values at α = 0.05. qcrit increases with p and decreases with larger df. For α = 0.01, see the full table tabs above.
Frequently Asked Questions
Can I use the Newman-Keuls test without a significant ANOVA?
No. The SNK test is a post hoc procedure conditioned on a significant omnibus F-test. Running it after a non-significant ANOVA inflates Type I error because the protection provided by the ANOVA gate is gone. If ANOVA is not significant, no post hoc test should be run, including Newman-Keuls.
What happens if my group sizes are unequal?
For unequal n, replace n in the formula with the harmonic mean of the two group sizes: n* = 2 / (1/n_i + 1/n_j). This approximation is less accurate than the equal-n version. With substantially unequal group sizes, Tukey's HSD with the Kramer modification or the Games-Howell test (for unequal variances too) are more appropriate alternatives.
Why does p = 2 have the smallest critical value?
Two adjacent means in the ranked order span the narrowest range of the studentized range distribution. The critical value for p = 2 equals the critical value from a two-tailed t-test at the same α, because comparing two adjacent means is equivalent to a pairwise t-test. As p increases, the critical value increases because we are looking at a wider span of the distribution, requiring a larger difference to be considered significant.
How do I handle ties in mean rankings?
When two group means are exactly equal (or very close), their rank positions are averaged, and p is calculated using those average ranks. In practice, exactly equal group means are rare with real data. If several means are very close together, consider reporting them as a homogeneous subset.
Where do I find df_error if I only have the ANOVA summary table?
dferror is reported as dfwithin or dfresidual in the ANOVA table. It equals N − k, where N = total number of observations across all groups and k = number of groups. In software output, look for the row labeled "Within Groups," "Error," or "Residual" — the degrees of freedom in that row is what you use for the Newman-Keuls table lookup.
Does Newman-Keuls work for two-way ANOVA?
The Newman-Keuls procedure was designed for one-way ANOVA. It can be adapted for simple main effects within a two-way ANOVA if run separately within each level of the other factor. For the interaction term, most researchers use Tukey HSD or simple effects analysis rather than the SNK procedure. The MSE used in any case should come from the full two-way ANOVA model.
Sources and Further Reading
Newman, D. (1939). The distribution of range in samples from a normal population, expressed in terms of an independent estimate of standard deviation. Biometrika, 31(1–2), 20–30. doi:10.2307/2334671 — Original paper introducing the stepwise multiple range test.
Keuls, M. (1952). The use of the studentized range in connection with an analysis of variance. Euphytica, 1(2), 112–122. doi:10.1007/BF01908269 — The companion paper completing the SNK procedure; standard citation for the test.
Maxwell, S. E., & Delaney, H. D. (2004). Designing Experiments and Analyzing Data: A Model Comparison Perspective (2nd ed.). Lawrence Erlbaum. — Chapter 5 provides a rigorous treatment of post hoc tests including the FWER limitations of the SNK procedure compared to Tukey HSD.
NIST/SEMATECH e-Handbook of Statistical Methods (2013). Section 7.4.3.3: Multiple Comparisons. National Institute of Standards and Technology. itl.nist.gov — U.S. government statistical reference covering the Newman-Keuls procedure and comparison with other post hoc tests.
Penn State STAT 502: Analysis of Variance and Design of Experiments. Pennsylvania State University. online.stat.psu.edu/stat502/lesson/8/8.3 — Open-access course notes on multiple comparison procedures including Newman-Keuls, Tukey, Bonferroni, and Duncan's test.
Related Statistical Tables & Resources
Understanding What the Newman-Keuls Table Tells You
Why qcrit Increases with p
When comparing means that span many steps in the ranking, you are looking at a wider range of the studentized range distribution. The probability that the widest range in a set of means exceeds a given threshold just by chance is higher than for adjacent means. A larger critical value corrects for this, requiring a bigger difference to declare significance when the span is wide.
Why qcrit Decreases with Larger df
Larger dferror means the MSE is based on more observations and is a more precise estimate of the population variance σ². With a more accurate error estimate, smaller mean differences become detectable. This is the same logic as in t-distribution critical values: more df lowers the threshold for significance.
The Stepwise Logic of Newman-Keuls
The SNK procedure mirrors how a scientist might naturally think: start by asking whether the most extreme groups differ. If they do not, there is no reason to expect anything in between them to differ either. If they do differ, work inward, testing progressively closer comparisons with appropriate (smaller) critical values. This sequential logic is what distinguishes SNK from Tukey HSD. See one-way ANOVA for the foundation this test builds on.