Duncan Critical Value Lookup
What Is Duncan's Multiple Range Table?
Duncan's Multiple Range Table is a statistical reference table that provides critical values for Duncan's Multiple Range Test (DMRT), a post hoc comparison procedure used after a significant ANOVA F-test. The table was developed by David B. Duncan in 1955 and published in Biometrics.
Each entry in the table is a critical q value from the studentized range distribution. These q values depend on three things: the significance level α (0.05 or 0.01), the range position p (how many steps apart two means are when all group means are ranked from smallest to largest), and the error degrees of freedom from the ANOVA table.
Definition: The Least Significant Range (LSR) for range position p equals LSRp = q(α, p, dferror) × √(MSE / n). Two group means are significantly different if their absolute difference exceeds the LSR for their range position. Duncan's test uses a separate LSR for each range position — this is what distinguishes it from Tukey HSD, which uses one LSR for all comparisons.
Duncan's Multiple Range Table (Critical Values)
Select the significance level tab. Each row is an error df value; each column is a range position p (number of rank steps between compared means). Click any cell to load that value into the lookup tool above.
p = range position (number of rank steps between two means being compared). df = error degrees of freedom from ANOVA. Source: Duncan (1955), Biometrics, 11(1), 1–42; validated against Harter (1960) and SAS PROC ANOVA documentation. LSR = q × √(MSE/n). If |ȳᵢ − ȳⱼ| ≥ LSR → significantly different at α.
How Duncan's Multiple Range Test Works
Duncan's test compares every pair of group means by computing a range-specific threshold. The key idea is that comparing means that are far apart (large p) should require a higher bar for significance than comparing adjacent means (small p). This is reflected in the table: larger p values always produce larger critical q values.
Step 1 — Run One-Way ANOVA
Compute the one-way ANOVA F-test. If the F-test is not significant, stop — DMRT should not be applied. From the ANOVA summary table, record the Mean Square Error (MSE) and the error degrees of freedom (dferror = N − k, where N = total observations and k = number of groups). These two values are needed for every subsequent step.
Step 2 — Rank All Group Means
Calculate the mean for each treatment group, then arrange all means in ascending order from smallest (rank 1) to largest (rank k). This ranked order determines the range position p for every pair.
Step 3 — Determine Range Position p
For any pair of means at ranks i and j (i < j), the range position is p = j − i + 1. For example, comparing the mean at rank 2 and the mean at rank 5 gives p = 5 − 2 + 1 = 4. Larger p means the two means are further apart in rank order, and the critical value from the table will be larger.
Step 4 — Look Up q(α, p, df)
Find the row in the table corresponding to your error df, and the column corresponding to range position p. The cell gives the critical q value. Use the α = 0.05 tab for standard significance, or α = 0.01 for a more conservative threshold. If your exact df is not in the table, use the next smaller df (conservative interpolation) or linear interpolation between adjacent rows.
Step 5 — Compute the Least Significant Range
The Least Significant Range (LSR) for range position p is:
where MSE = mean square error from ANOVA, and n = number of observations per group. For unequal group sizes, replace n with the harmonic mean: nh = k / Σ(1/nᵢ).
Step 6 — Compare and Conclude
If |ȳᵢ − ȳⱼ| < LSRp → No significant difference at α
Report results by labeling groups with letters: means sharing a letter are not significantly different from each other.
Worked Example: Crop Yield Experiment
Scenario: An agronomist tests four fertilizer treatments (A, B, C, D) on wheat yield (tonnes/hectare), with n = 6 plots per treatment. One-way ANOVA is significant (F = 8.43, p < 0.05). The ANOVA table gives MSE = 1.60, dferror = 20. The four treatment means are: A = 4.2, B = 5.8, C = 5.1, D = 7.3. Apply DMRT at α = 0.05.
Step 1: Rank the Means
| Rank | Treatment | Mean Yield |
|---|---|---|
| 1 | A | 4.2 |
| 2 | C | 5.1 |
| 3 | B | 5.8 |
| 4 | D | 7.3 |
Step 2: Look Up q Values and Compute LSR
| p | q(0.05, p, 20) | LSRp = q × 0.516 |
|---|---|---|
| 2 | 2.950 | 1.523 |
| 3 | 3.097 | 1.598 |
| 4 | 3.190 | 1.646 |
Step 3: Compare All Pairs
| Pair | Difference | p | LSRp | Result |
|---|---|---|---|---|
| D vs A | |7.3 − 4.2| = 3.1 | 4 | 1.646 | Sig. ✓ |
| D vs C | |7.3 − 5.1| = 2.2 | 3 | 1.598 | Sig. ✓ |
| D vs B | |7.3 − 5.8| = 1.5 | 2 | 1.523 | Not sig. |
| B vs A | |5.8 − 4.2| = 1.6 | 3 | 1.598 | Not sig. |
| C vs A | |5.1 − 4.2| = 0.9 | 2 | 1.523 | Not sig. |
| B vs C | |5.8 − 5.1| = 0.7 | 2 | 1.523 | Not sig. |
Conclusion
Treatment D (mean = 7.3 t/ha) is significantly higher than both A and C, but not significantly different from B. Treatments A, B, and C do not differ significantly from each other. In letter notation: D (ab), B (ab), C (b), A (b) — or more precisely, D differs from A and C; B, C, and A form one non-distinct group. The agronomist can conclude fertilizer D produces the highest yield, statistically superior to treatments A and C at α = 0.05.
Duncan's Test vs Other Post Hoc Tests
Choosing the right post hoc test requires weighing statistical power against Type I error control. The table below compares DMRT with four alternatives commonly implemented in R, SAS, and SPSS.
| Test | Error Control | Power | Critical Value | Best Used When |
|---|---|---|---|---|
| Duncan's MRT | Comparisonwise (varies by p) | High | q(α, p, df) — varies by p | Agricultural / biological; power is priority |
| Tukey HSD | Familywise (FWER = α) | Moderate | q(α, k, df) — single value | All pairwise comparisons; balanced design |
| Newman-Keuls | Comparisonwise (varies by p) | High | q(α, p, df) — from Tukey table | Similar to Duncan; slightly more conservative |
| Bonferroni | Familywise (FWER ≤ α) | Low | t(α/m, df) — m = # comparisons | Few planned comparisons; very conservative |
| Scheffé | Familywise (all contrasts) | Lowest | √[(k−1)F(α, k−1, df)] | Complex contrasts beyond pairwise comparisons |
| Fisher LSD | Per-comparison only | Highest | t(α/2, df) — simple t-test | Exactly 3 groups only (otherwise inflated error) |
Which Test Should You Choose?
If controlling the familywise error rate at exactly α is required — as it often is in clinical or regulatory research — use Tukey HSD. Duncan's test is defensible in exploratory agricultural and biological work where maximizing power matters and a slightly elevated Type I error rate is acceptable. The Bonferroni correction suits planned comparisons involving only a subset of pairs. The F-test from ANOVA must be significant before applying any of these procedures.
Applied Examples Across Research Fields
Duncan's Multiple Range Test appears in published research across a wide range of disciplines. The following examples show typical applications with different numbers of treatment groups.
Biology: Drug Dosage Effects on Cell Growth
Five drug dosages (0, 5, 10, 20, 40 mg/L) each applied to n = 8 cell cultures. ANOVA: F(4, 35) = 12.4, p < 0.001, MSE = 0.98. Error df = 35. Researcher applies DMRT at α = 0.05 to identify which dosage levels produce different mean cell counts. The test computes five separate LSR values (p = 2 through 5) and labels each dosage group with significance letters.
Education: Teaching Method Comparison
Three teaching methods (lecture, flipped classroom, online) with n = 25 students per group. ANOVA: F(2, 72) = 5.7, p = 0.005, MSE = 144. Error df = 72. DMRT at α = 0.05 uses q values for p = 2 and p = 3. LSR₂ = q(0.05, 2, 72) × √(144/25) ≈ 2.83 × 2.4 = 6.79 points. Any pair of methods differing by more than 6.79 exam points is flagged as significantly different.
Manufacturing: Supplier Quality Comparison
Four raw material suppliers (W, X, Y, Z) compared on product defect rates across n = 12 production runs each. ANOVA significant (F = 9.2, p < 0.001, MSE = 3.6, dferror = 44). DMRT reveals Supplier Z produces significantly fewer defects than W and X but does not differ significantly from Y, guiding procurement decisions with statistical backing.
Psychology: Stress Reduction Interventions
Six stress interventions compared in a clinical trial with n = 10 participants each. ANOVA: F(5, 54) = 4.8, p = 0.001, MSE = 12.5, dferror = 54. DMRT at α = 0.05 requires q values for p = 2 through 6, yielding six different LSR thresholds. The highest-performing intervention is compared against all others with a range-appropriate threshold for each comparison.
Running Duncan's Multiple Range Test in Statistical Software
DMRT is available in all major statistical platforms. The table lookup above gives critical values for manual calculation; the code examples below automate the full procedure.
R (agricolae package)
install.packages("agricolae")
library(agricolae)
# Run ANOVA
model <- aov(yield ~ treatment,
data = crop_data)
# Apply DMRT
result <- duncan.test(model,
"treatment",
alpha = 0.05,
console = TRUE)
Output includes group means, SE, LSR values, and significance letters. See the agricolae documentation for full options.
SAS (PROC ANOVA / MEANS)
CLASS treatment;
MODEL yield = treatment;
MEANS treatment /
DUNCAN ALPHA=0.05;
RUN;
The MEANS statement with the DUNCAN option produces DMRT output with significance groupings. Available in SAS 9.4 and SAS OnDemand.
SPSS
→ One-Way ANOVA
Post Hoc → Check
"Duncan" → Continue
→ OK
SPSS outputs Duncan groupings in the Post Hoc Multiple Comparisons table. Available in SPSS Statistics 24 and later.
Python (scipy + pingouin)
from scipy import stats
# One-way ANOVA
aov = pg.anova(data=df,
dv='yield', between='trt')
# Pairwise (use Tukey
# or manual LSR for DMRT)
ph = pg.pairwise_tukey(data=df,
dv='yield', between='trt')
Python's pingouin does not implement DMRT natively. Use the table above for manual LSR calculation, then compare with scipy pairwise t-tests.
Symbol and Concept Glossary
The following table defines every term and symbol used in Duncan's Multiple Range Test. Understanding these is necessary to read the table correctly and interpret output from statistical software.
| Symbol / Term | Definition | Common Error |
|---|---|---|
| q(α, p, df) | Critical value from the studentized range distribution; indexed by significance level α, range position p, and error df | Using a single q for all p (that is Tukey's approach, not Duncan's) |
| p | Range position — the number of rank steps between two means being compared: p = |rank_i − rank_j| + 1 | Confusing p (range position) with p (significance level / p-value) |
| LSR | Least Significant Range — the minimum mean difference for significance: LSRp = q × √(MSE/n) | Using the same LSR for all pairs instead of computing a separate LSR per p |
| MSE | Mean Square Error from the ANOVA table — estimates the within-group variance | Using MSB (between-group) instead of MSE (within-group) |
| dferror | Error degrees of freedom = N − k, where N = total observations, k = number of groups. Used to look up q in the table. | Using k − 1 (treatment df) instead of N − k (error df) |
| n | Number of observations per group. For unequal n, use the harmonic mean nh = k / Σ(1/nᵢ) | Using total N instead of per-group n in the LSR formula |
| FWER | Family-wise error rate — the probability of making at least one Type I error across all comparisons. Duncan's test does not control FWER at α for all p. | Assuming Duncan's test controls FWER at α the same way Tukey HSD does |
| Studentized Range | The distribution of (max − min) / SE for k independent normal means, used as the basis for Duncan's, Tukey's, and Newman-Keuls critical values | Treating it as a simple t-distribution (it is not) |
Common Mistakes When Using Duncan's Multiple Range Test
Applying DMRT Without a Significant F-test
Duncan's test requires a significant ANOVA F-test as a precondition. Running DMRT on non-significant ANOVA results inflates the comparisonwise Type I error rate and can produce false positives that are not protected by the overall test.
Using the Wrong df in the Table
The df to look up is dferror = N − k (the ANOVA error term degrees of freedom), not the treatment df (k − 1) or the total df (N − 1). Using treatment df produces a critical value that is too small, making it too easy to declare significance.
Using the Same LSR for All Pairs
Duncan's test requires a different LSR for each range position p. Using a single LSR (the one for p = 2 or p = k) collapses the test into Fisher LSD or Tukey HSD, respectively. The range-specific LSR is what defines DMRT.
Ignoring Equal Variance Assumption
Duncan's test (like most ANOVA-based post hoc procedures) assumes homoscedasticity — roughly equal variances across groups. When Levene's test or Bartlett's test indicates unequal variances, a Welch ANOVA followed by Games-Howell post hoc test is more appropriate.
Frequently Asked Questions
What is Duncan's Multiple Range Table?
Duncan's Multiple Range Table provides critical q values — q(α, p, df) — drawn from the studentized range distribution. After a significant ANOVA F-test, you use these q values to compute the Least Significant Range (LSR) for each range position p. Any pair of group means whose absolute difference exceeds the LSR for its range position is significantly different at significance level α.
What does range position p mean in the table?
Range position p is the number of ranks separating two means in the sorted order, plus one. If you rank all group means from smallest to largest and compare the 1st and 4th ranked means, p = 4 − 1 + 1 = 4. Comparing the 2nd and 3rd ranked means gives p = 2. Larger p means a larger critical q, so widely separated pairs require a greater observed difference to reach significance.
How is Duncan's test different from Newman-Keuls?
Both tests use range-position-specific critical values from the studentized range distribution. The difference is in how the protection level is defined. Newman-Keuls sets the per-comparison error rate at α for each p. Duncan modifies the protection level using an equation based on the number of means being compared, resulting in slightly lower critical values (and therefore higher power) than Newman-Keuls at the same α. In practice, the two tests often give identical conclusions.
What if my error df is not listed in the table?
Use the next smaller df listed for a conservative result. For example, if dferror = 25, use the df = 20 row. Alternatively, interpolate linearly between the df = 20 and df = 30 rows. For df > 120, use the ∞ row — the asymptotic values change very little above df = 100, so the ∞ row is a good approximation.
Does Duncan's test require equal group sizes?
Equal group sizes (balanced design) are the standard assumption. When group sizes differ, substitute the harmonic mean of the group sizes for n in the LSR formula: nh = k / Σ(1/nᵢ), where k is the number of groups and nᵢ is the size of each group. This adjustment makes the test approximately valid for unbalanced designs, though Tukey-Kramer or Games-Howell may be more appropriate when imbalance is substantial.
Is Duncan's test available in Excel?
Excel does not include DMRT in its built-in Data Analysis ToolPak. You can perform manual calculations in Excel by using the LSR formula with MSE and n values from the ANOVA output, and by looking up q values from the table on this page. For automated DMRT output, use R (agricolae package) or SAS (PROC ANOVA with MEANS/DUNCAN option). See the ANOVA calculator for a starting point.
What significance level should I use — 0.05 or 0.01?
α = 0.05 is the standard choice in most biological, agricultural, and social science research. α = 0.01 is used when the cost of a false positive is high — for example, in medical research or regulatory contexts. Use the same α for the ANOVA F-test and the post hoc DMRT. Setting different α levels for each stage is inconsistent and misleads interpretation. The α level should be chosen before data collection, not after examining the results.
Sources & Further Reading
The critical values in this table are based on the original publication by David B. Duncan and subsequent tabulations validated by Harter (1960) and Pearson & Hartley (1966). The following references cover the theoretical foundations and practical application of DMRT.
Duncan, D. B. (1955). Multiple range and multiple F tests. Biometrics, 11(1), 1–42. doi:10.2307/3001478 — The original paper introducing Duncan's Multiple Range Test and the critical value table.
Harter, H. L. (1960). Tables of range and studentized range. Annals of Mathematical Statistics, 31(4), 1122–1147. doi:10.1214/aoms/1177705688 — Extended and validated critical value tables for the studentized range distribution used in DMRT.
Montgomery, D. C. (2017). Design and Analysis of Experiments (9th ed.). Wiley. Chapter 3 — Covers ANOVA, post hoc tests including Duncan's MRT, Tukey HSD, and Newman-Keuls within the context of designed experiments.
NIST/SEMATECH e-Handbook of Statistical Methods. Section 7.4.7: Multiple Comparisons. itl.nist.gov — U.S. government statistical handbook explaining multiple comparison procedures including DMRT, Tukey HSD, and Bonferroni.
Penn State STAT 503: Design of Experiments. Lesson 4: Multiple Comparisons. Pennsylvania State University. online.stat.psu.edu/stat503 — Open-access course notes comparing post hoc test procedures with worked examples in R and SAS.
Related Statistical Tables & Resources
Key Concepts Behind Duncan's Multiple Range Table
Why q Increases With p
Comparing means that are many ranks apart involves a wider range of outcomes, so by chance alone the largest mean in a large set will tend to be further from the smallest than in a small set. Duncan's table accounts for this by requiring a larger q for larger p. This is the core logic separating DMRT from a simple t-test applied to every pair.
The Studentized Range Connection
Both Duncan's table and Tukey's q table draw from the studentized range distribution. The difference is that Tukey uses q(α, k, df) — the critical value for the total number of groups k — for every pair, while Duncan uses q(α, p, df) for each specific range position p, yielding smaller critical values for adjacent means. This is why Duncan is more powerful than Tukey but less conservative.
Why ANOVA Must Come First
Duncan's "protection" against Type I error is conditional on first obtaining a significant ANOVA F-test. Without a significant overall F, there is no statistical basis for searching for specific pairwise differences — doing so without that protection would amount to running many unprotected t-tests, inflating the Type I error substantially. See the Statistics Fundamentals ANOVA guide for a full explanation of the F-test precondition.