Duncan’s Multiple Range Test: Formula, Interpretation, SPSS, Python, R and Excel Guide

Quick Answer: Duncan’s Multiple Range Test Result

The Duncan’s Multiple Range Test result separates the four studytime groups into three homogeneous subsets. Studytime groups 3 and 4 share subset A, meaning they are treated as not significantly different from each other. Studytime group 2 forms subset B. Studytime group 1 forms subset C.

The ranked mean pattern is clear: group 1 has the lowest mean G3 at about 10.84, group 2 is higher at about 12.09, group 4 is about 13.06, and group 3 is the highest at about 13.23. Duncan’s test indicates that group 1 differs from groups 2, 3 and 4; group 2 differs from group 3; group 2 does not clearly differ from group 4; and groups 3 and 4 do not differ from each other.

What Is Duncan’s Multiple Range Test?

Duncan’s Multiple Range Test, often shortened to the Duncan test or DMRT, is a post hoc mean-separation procedure used after ANOVA. It ranks group means from smallest to largest, calculates critical ranges for different mean separations, and decides which groups are statistically different.

Unlike a single omnibus ANOVA test, Duncan’s test tells which means form different groups. It is often displayed with homogeneous subset letters. Groups that share a letter are treated as not significantly different in that output. Groups with different letters are interpreted as significantly separated by the Duncan procedure.

In this example, Duncan’s test is used after comparing G3 final grade across four studytime groups. The test ranks the group means and shows a clear separation: group 1 is lowest, group 2 is middle, and groups 3 and 4 form the top subset.

This guide connects naturally with One Way ANOVA, ANOVA in SPSS, ANOVA in R, ANOVA in Python, ANOVA Assumptions, P Value, Confidence Interval, Effect Size, Eta Squared and Omega Squared.

Duncan’s Multiple Range Test Formula

Duncan’s test starts by ordering the group means. The absolute difference between two ordered means is compared with a Duncan critical range. A pair is treated as significant when the observed mean difference exceeds the critical range for that span.

Here, |X̄_i − X̄_j| is the observed absolute mean difference, and R_p is the Duncan critical range for the number of ordered means covered by the comparison.

A simplified critical range form is:

When group sizes are unequal, software uses an adjusted standard error or harmonic-style sample-size handling. The exact calculation depends on the implementation, but the interpretation remains the same: if the observed difference is larger than the Duncan critical range, the pair is separated.

Key Terms in the Formula

Duncan Test Hypotheses

Duncan’s Multiple Range Test is applied pair by pair after ANOVA. Each pairwise comparison has a null hypothesis that the two group means are equal and an alternative hypothesis that the two group means are different.

Dataset and Variables Used

The worked example uses student performance data. The dependent variable is G3 final grade. The grouping factor is studytime, which has four groups. Duncan’s Multiple Range Test is used as the post hoc mean-separation method after the ANOVA workflow.

Group Mean Pattern

The group mean pattern shows why Duncan’s test forms three subsets. Group 1 is clearly lower. Group 2 is higher than group 1 but lower than group 3. Groups 3 and 4 are close together at the top and share the same letter.

For supporting concepts, review Descriptive Statistics, Mean Median and Mode, Standard Deviation, Variance, Standard Error, Confidence Interval, Five Number Summary, Box Plot Interpretation and Histogram Interpretation.

Duncan Pairwise Decision Summary

The pairwise mean-difference heatmap and the difference-versus-critical-range chart show the same decision pattern. A pair is significant when the observed mean difference is larger than the Duncan critical range. In this output, four comparisons are separated and two comparisons are not separated.

Assumptions Before Duncan’s Multiple Range Test

Duncan’s Multiple Range Test is usually applied after an ANOVA model. That means the main ANOVA assumptions still matter. The outcome should be numeric, observations should be independent, the factor should define meaningful groups, and the ANOVA error term should be appropriate for post hoc comparison.

For assumption support, use ANOVA Assumptions, Levene Test, Bartlett’s Test, Brown-Forsythe Test, Hartley F Max Test, Cochran C Test, Q-Q Plot Normality Check, P-P Plot Normality Check, Shapiro-Wilk Test and Outlier Detection.

SPSS Output Interpretation for Duncan’s Multiple Range Test

The SPSS output for Duncan’s Multiple Range Test should be read in this order: first the descriptive means, then the ANOVA context, then the Duncan homogeneous subsets table. The homogeneous subsets table is usually the easiest SPSS output section to report because it directly assigns group letters or subset columns.

SPSS Reading Order

SPSS Homogeneous Subset Interpretation

Python Chart-by-Chart Interpretation

The Python chart sequence explains Duncan’s Multiple Range Test through group means, ranked means, pairwise mean differences, critical range comparison, homogeneous subsets and distribution boxplots.

Python Chart 1: Group Means with Confidence Intervals

The group mean chart shows that G3 increases from studytime group 1 to group 3, while group 4 remains high. Group 1 has the lowest average final grade, group 2 is higher, and groups 3 and 4 form the top region of the chart.

This chart prepares the reader for Duncan’s subset result. The means are not equally spaced, but the overall pattern supports a clear low-middle-high separation.

Python Chart 2: Ranked Group Means

The ranked mean chart orders the groups from lowest to highest: group 1, group 2, group 4 and group 3. The labels show group 1 as subset C, group 2 as subset B, and groups 4 and 3 as subset A.

This is the clearest visual summary of the Duncan result. Groups with different letters should be interpreted as separated by the Duncan procedure. Groups 3 and 4 share A, so they are not separated from each other.

Python Chart 3: Pairwise Mean Difference Heatmap

The heatmap shows the absolute mean differences between each pair of studytime groups. Starred cells indicate pairs separated by the Duncan-style critical range rule. The largest separation is between groups 1 and 3, followed by groups 1 and 4.

The heatmap also shows two non-significant comparisons: group 2 vs group 4 and group 4 vs group 3. This matches the homogeneous subset result because group 3 and group 4 share the same subset letter.

Python Chart 4: Difference vs Critical Range

This chart explains the Duncan decision rule directly. A pair is significant when the observed mean difference is longer than the Duncan critical range. The comparisons 1 vs 3, 1 vs 4, 1 vs 2 and 2 vs 3 exceed their critical ranges.

The comparisons 2 vs 4 and 4 vs 3 do not exceed their critical ranges. This is why the output does not treat group 4 as clearly separated from group 2 or group 3 in the same way that group 1 is separated from the higher groups.

Python Chart 5: Duncan Homogeneous Subsets

The homogeneous subset chart gives the final Duncan grouping. Groups 3 and 4 share letter A, group 2 has letter B, and group 1 has letter C.

This chart is the best reporting figure for readers who need a quick result. The letters show which means are treated as statistically similar and which means are separated by the Duncan test.

Python Chart 6: Group Distribution Boxplots

The boxplots show spread, medians and outliers by studytime group. Group 1 has lower central G3 values, while groups 3 and 4 have higher distributions.

This distribution chart is important because post hoc letters should be read with the raw group pattern. Duncan’s test separates group means, while the boxplots show how much overlap and spread exist inside the groups.

R Chart-by-Chart Validation

The R charts validate the same Duncan’s Multiple Range Test workflow in a second software environment. The R sequence confirms the group mean pattern, ranked means, pairwise difference decisions, critical range comparisons, homogeneous subsets and distributions.

R Chart 1: Group Means with Confidence Intervals

The R group means chart confirms the same pattern as Python. Mean G3 is lowest in group 1, higher in group 2, and highest in groups 3 and 4.

This repeated mean profile supports the final interpretation that Duncan’s test is separating a real low-middle-high pattern across studytime groups.

R Chart 2: Ranked Group Means

The R ranked means chart confirms the same order: group 1, group 2, group 4 and group 3. The same subset letters appear in the same pattern.

This chart validates that the Duncan result is not only a Python result. The ranking and subset interpretation agree across software outputs.

R Chart 3: Pairwise Mean Difference Heatmap

The R heatmap confirms that the significant differences are 1 vs 2, 1 vs 3, 1 vs 4 and 2 vs 3. The non-significant comparisons are 2 vs 4 and 3 vs 4.

This gives the same decision table as the Python heatmap and supports the same final subset interpretation.

R Chart 4: Difference vs Critical Range

The R critical range chart confirms the same significance rule. When the observed difference is larger than the critical range, the pair is separated. When it is smaller, the pair is not separated.

The most important message is that 4 vs 3 has a very small observed difference and does not exceed the critical range. That explains why groups 3 and 4 share subset A.

R Chart 5: Duncan Homogeneous Subsets

The R homogeneous subset chart confirms the final grouping: groups 3 and 4 are A, group 2 is B, and group 1 is C.

This is the final result that should appear in the report. Groups sharing the same letter are treated as not significantly different in this Duncan output.

R Chart 6: Group Distribution Boxplots

The R boxplot confirms the same distribution story as Python. The lower studytime group has lower central performance, and the higher studytime groups show higher distributions.

This final validation chart helps readers understand that Duncan’s subset letters are attached to a visible distribution pattern, not only a numerical post hoc table.

SPSS, R, Python and Excel Workflows for Duncan’s Multiple Range Test

The same Duncan’s Multiple Range Test workflow can be reproduced in SPSS, R, Python and Excel. SPSS provides Duncan as a post hoc option in one-way ANOVA. R can run Duncan-style post hoc testing with specialist packages such as agricolae. Python can reproduce the logic by ranking means, computing pairwise mean differences and comparing those differences with critical ranges. Excel can support the ranked mean, difference and subset workflow, but SPSS or R is better for formal Duncan output.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Duncan’s Multiple Range Test

SPSS Syntax for Duncan’s Multiple Range Test

* Duncan's Multiple Range Test in SPSS.
* Dependent variable: G3.
* Factor: studytime.

TITLE "Duncan's Multiple Range Test: G3 by Studytime".

ONEWAY G3 BY studytime
  /STATISTICS DESCRIPTIVES HOMOGENEITY
  /POSTHOC = DUNCAN ALPHA(.05)
  /MISSING ANALYSIS.

EXAMINE VARIABLES=G3 BY studytime
  /PLOT BOXPLOT
  /COMPARE GROUPS
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Duncans-Multiple-Range-Test-SPSS-Output.pdf".

Python Code for Duncan-Style Mean Separation

import pandas as pd
import numpy as np
from itertools import combinations
from scipy import stats
from statsmodels.formula.api import ols
from statsmodels.stats.anova import anova_lm

df = pd.read_csv("dataset.csv")

df["G3"] = pd.to_numeric(df["G3"], errors="coerce")
df["studytime"] = df["studytime"].astype("category")

data = df[["G3", "studytime"]].dropna().copy()

# ANOVA model
model = ols("G3 ~ C(studytime)", data=data).fit()
anova_table = anova_lm(model, typ=2)

mse = anova_table.loc["Residual", "sum_sq"] / anova_table.loc["Residual", "df"]
df_error = anova_table.loc["Residual", "df"]

summary = data.groupby("studytime", observed=True)["G3"].agg(
    n="count",
    mean="mean",
    sd="std"
).reset_index()

summary = summary.sort_values("mean")
print(summary)
print(anova_table)

# Pairwise observed differences
rows = []
groups = list(summary["studytime"])

for g1, g2 in combinations(groups, 2):
    mean1 = summary.loc[summary["studytime"] == g1, "mean"].iloc[0]
    mean2 = summary.loc[summary["studytime"] == g2, "mean"].iloc[0]
    n1 = summary.loc[summary["studytime"] == g1, "n"].iloc[0]
    n2 = summary.loc[summary["studytime"] == g2, "n"].iloc[0]

    observed_difference = abs(mean2 - mean1)
    se = np.sqrt(mse * 0.5 * (1 / n1 + 1 / n2))

    # A conservative placeholder critical rule is shown here.
    # For formal Duncan critical ranges, use a Duncan-specific implementation
    # or validate against SPSS/R output.
    t_stat = observed_difference / se
    p_value = 2 * (1 - stats.t.cdf(abs(t_stat), df_error))

    rows.append({
        "group_1": g1,
        "group_2": g2,
        "mean_1": mean1,
        "mean_2": mean2,
        "observed_difference": observed_difference,
        "standard_error": se,
        "t_stat": t_stat,
        "p_value_reference": p_value
    })

pairwise_table = pd.DataFrame(rows)
print(pairwise_table)

# Final subset letters should be reported from the validated Duncan output:
# Group 3 = A, Group 4 = A, Group 2 = B, Group 1 = C.

R Code for Duncan’s Multiple Range Test

# Duncan's Multiple Range Test in R

library(tidyverse)
library(car)
library(agricolae)

df <- read.csv("dataset.csv")

df$G3 <- as.numeric(df$G3)
df$studytime <- as.factor(df$studytime)

data <- df %>%
  select(G3, studytime) %>%
  drop_na()

# Descriptive statistics
data %>%
  group_by(studytime) %>%
  summarise(
    n = n(),
    mean_G3 = mean(G3),
    sd_G3 = sd(G3),
    .groups = "drop"
  )

# ANOVA model
model <- aov(G3 ~ studytime, data = data)
summary(model)

# Assumption context
leveneTest(G3 ~ studytime, data = data)

# Duncan's Multiple Range Test
duncan_result <- agricolae::duncan.test(
  y = model,
  trt = "studytime",
  alpha = 0.05,
  group = TRUE
)

print(duncan_result$groups)
print(duncan_result$comparison)

Excel Notes for Duncan’s Multiple Range Test

Excel support workflow:

1. Arrange the data:
   G3 | studytime

2. Create group summaries:
   n, mean, standard deviation, standard error.

3. Rank group means:
   Lowest to highest:
   Group 1 = 10.84
   Group 2 = 12.09
   Group 4 = 13.06
   Group 3 = 13.23

4. Calculate observed pairwise differences:
   =ABS(mean_i - mean_j)

5. Compare with validated Duncan critical ranges:
   Significant if observed difference > critical range.

6. Assign homogeneous subset letters:
   Group 3 = A
   Group 4 = A
   Group 2 = B
   Group 1 = C

7. Report:
   Groups sharing a letter are treated as not significantly different.

APA Reporting Wording

When reporting Duncan’s Multiple Range Test, report the ANOVA context first, then the post hoc subset result. The easiest reader-friendly format is to report the group means with subset letters.

Common Mistakes

When to Use Duncan’s Multiple Range Test

Use Duncan’s Multiple Range Test when your goal is to separate and rank group means after ANOVA. It is commonly used in experimental fields where homogeneous subset grouping is useful for presenting treatments or categories.

Compare this guide with One Way ANOVA, Factorial ANOVA, Balanced ANOVA, Brown Forsythe ANOVA, ANOVA Effect Size, F Distribution, Eta Squared, Omega Squared, Cohen’s F Formula and Welch’s T Test.

Downloads and Resources for Duncan’s Multiple Range Test

Use these resources to reproduce the Duncan’s Multiple Range Test workflow. The Python report, R report and SPSS output PDF are included as verification files. Script and workbook placeholders can be replaced after the final downloadable files are uploaded to the WordPress Media Library.

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