Quick Answer: Newman Keuls Test Result

The worked example compares G3 final grade across four studytime groups. The sample contains 649 students. The omnibus one-way ANOVA is statistically significant, so post hoc testing is justified. The Newman Keuls Test is then used to compare ordered studytime group means and identify which groups differ from each other.

The ordered means are clear. Studytime group 3 has the highest mean G3 score, M = 13.23. Studytime group 4 follows closely with M = 13.06. Studytime group 2 has M = 12.09. Studytime group 1 has the lowest mean, M = 10.84. The main educational conclusion is that the lowest studytime group is separated from the higher studytime groups, while the highest two studytime groups are very close to each other.

What Is Newman Keuls Test?

Newman Keuls Test is a post hoc multiple comparison method used after a statistically significant one-way ANOVA. It is also called the Student-Newman-Keuls Test or SNK test. The method orders group means from lowest to highest or highest to lowest, then compares differences between means using the Studentized range statistic.

The purpose of the test is to answer the question that ANOVA cannot answer by itself. ANOVA tells whether at least one group mean differs, but it does not identify exactly which groups differ. Newman Keuls compares ordered pairs and can also produce homogeneous subset groupings, where groups in the same subset are not separated by the test.

The distinctive feature of Newman Keuls is its stepwise logic. It uses different critical values depending on the number of ordered means spanned by a comparison. Comparisons that are far apart in the ordered list are tested with a wider range, while neighboring means are tested with a smaller range. This makes SNK different from Tukey HSD, which uses a single familywise comparison logic across all pairs.

Before using Newman Keuls Test, review one-way ANOVA, ANOVA assumptions, Levene test, p-values, confidence intervals, and effect size.

When to Use Newman Keuls Test

Use Newman Keuls Test when you have one categorical factor with three or more groups, a continuous dependent variable, and a significant one-way ANOVA result. In this guide, studytime is the categorical factor and G3 final grade is the continuous outcome.

Newman Keuls Test Formula

The Newman Keuls Test starts with ordered group means. The pairwise mean difference is:

The standard error for comparing means is based on the ANOVA error term. A common balanced-design form is:

For unequal group sizes, software may use a harmonic mean sample-size approximation or pair-specific standard error logic depending on implementation. The Studentized range statistic is then:

The calculated q statistic is compared with a critical q value. The critical value depends on the error degrees of freedom and the number of ordered means spanned by the comparison.

Null and Alternative Hypotheses for Newman Keuls Test

Newman Keuls Test is interpreted pair by pair. Each comparison has its own null and alternative hypothesis, while the ordered stepwise method controls how the comparisons are tested.

Dataset and Variables Used

The worked example uses a student performance dataset. The dependent variable is G3 final grade. The factor is studytime, coded into four weekly study-time categories. The goal is to test whether average final grade differs across studytime groups and then use Newman Keuls to identify the ordered group differences.

Before interpreting Newman Keuls Test, review the group distributions, group means, sample sizes and standard deviations. Helpful related guides include descriptive statistics, box plot interpretation, standard deviation, ANOVA in SPSS and F distribution.

SPSS Output Interpretation for Newman Keuls Test

The SPSS output for a Newman Keuls Test is usually read in three parts: the one-way ANOVA table, the ordered means or homogeneous subsets table, and the post hoc decision pattern. The ANOVA table confirms whether post hoc testing is justified. The SNK table then shows which ordered group means are separated.

SPSS Interpretation Table

Ordered Mean Interpretation

The ordered means show the practical pattern before the formal decision is read. Group 3 is highest, group 4 is almost the same, group 2 is lower, and group 1 is clearly lowest. This means the Newman Keuls interpretation should not simply list p-values. It should explain the ordered pattern and how each group relates to the others.

Chart-by-Chart Interpretation

The charts below explain the Newman Keuls Test visually. They show the group means, group spread, ordered mean ranking, q statistics, p-value heatmap, decision summary and group size with standard deviation context.

Chart 1: Group Means with 95% Confidence Intervals

The group means chart shows the main result clearly. Studytime group 3 has the highest mean, group 4 is nearly the same, group 2 is lower and group 1 is the lowest. This ordered structure is exactly why Newman Keuls uses ordered means rather than treating the groups as an unordered list.

The confidence intervals help explain uncertainty around each mean. Group 4 has a smaller sample size, so its interval is wider than the larger groups. This is important because visual differences should always be interpreted with uncertainty and sample size in mind.

Chart 2: Group Boxplots

The boxplots show the spread of G3 values inside each studytime group. Group 1 is centered lower than the other groups, while groups 3 and 4 are centered higher. This supports the post hoc result by showing that the mean difference pattern also appears in the distribution view.

The chart also reminds readers that post hoc tests compare means, not entire distributions. A boxplot gives distribution context, but the final Newman Keuls decision depends on ordered mean differences, error variation and q statistics.

Chart 3: Ordered Group Means

The ordered means chart is central to Newman Keuls interpretation. It ranks the studytime groups from highest to lowest mean G3. Group 3 is highest, group 4 is close behind, group 2 follows, and group 1 is lowest.

This ordered display explains the stepwise logic. SNK first considers differences across wider ranges of ordered means and then moves toward smaller neighboring comparisons. The visual ranking helps readers understand why the lowest and highest groups are the most important comparisons.

Chart 4: SNK q Statistics

The q statistics chart shows how large each ordered comparison is relative to the standard error. Larger q values indicate stronger evidence that two ordered means are separated. Comparisons involving group 1 tend to be the strongest because group 1 is the lowest mean group.

The smallest q statistic belongs to the closest pair of means, especially the comparison between groups 3 and 4. This supports a careful interpretation: the highest two groups may look close, and they should not be overstated as clearly different unless the official SNK cutoff supports that decision.

R Validation Charts

The R validation charts repeat the same Newman Keuls visual workflow. They are useful because they confirm the same group mean order and decision logic through a separate reproducible analysis environment.

R Chart 1: Group Means with 95% Confidence Intervals

The R group means chart validates the same ordering: group 3 is highest, group 4 is close, group 2 is lower and group 1 is lowest. This agreement supports the reliability of the visual interpretation.

The confidence intervals also reinforce the role of sample size. Groups with fewer observations have less precise mean estimates, so their bars should be read with more caution.

R Chart 2: Group Boxplots

The R boxplot confirms that the lowest studytime group has a lower central distribution. The higher studytime groups are shifted upward, which supports the overall ANOVA and post hoc analysis.

The distribution view is useful for readers because it shows that the mean differences are not isolated table values. They reflect visible differences in the score distributions.

R Chart 3: Ordered Group Means

The R ordered means chart confirms the same ranking used by the SNK procedure. This is important because Newman Keuls decisions depend on ordered positions, not only on raw pair labels.

The chart also makes the practical pattern easy to report. The strongest story is the contrast between the lowest group and the higher groups, while the highest groups are close together.

R Chart 4: SNK q Statistics

The R q-statistic chart validates the same comparison pattern. Larger q values are connected with bigger ordered mean gaps. The strongest q statistics involve the lowest group compared with higher studytime groups.

The chart supports the interpretation that Newman Keuls is driven by ordered mean separation. It also shows why near-neighbor comparisons should be interpreted more carefully than the largest ordered gaps.

R Chart 5: Pairwise p-value Heatmap

The p-value heatmap gives a compact matrix view of the group comparisons. Darker or stronger cells usually represent smaller p-values or stronger decision evidence, while weaker cells represent comparisons that are not clearly separated.

For this example, the heatmap should be used to confirm the main interpretation: group 1 is the clearest low group, and the highest two groups are close. The heatmap makes it easier to avoid the mistake of claiming that every pair differs equally.

R Chart 6: SNK Decision Summary

The decision summary chart is the best visual table for final reporting. It translates q statistics and p-values into practical decisions. Significant comparisons should be reported with direction, such as “group 1 scored lower than group 3,” not only as “significant.”

This chart should be checked carefully before publishing because it is the most direct visual statement of the SNK result. The report should match this decision summary exactly.

R Chart 7: Group Size and Standard Deviation Context

The group size and standard deviation chart explains why post hoc comparisons should not be interpreted only from mean heights. Group 2 has the largest sample size, group 1 is also large, group 3 is smaller and group 4 is the smallest group.

Standard deviation context matters because post hoc decisions depend on mean differences relative to within-group variation. A small group with a wide standard deviation can make a comparison less stable than a chart of means alone suggests.

SPSS, R, Python and Excel Workflows for Newman Keuls Test

The Newman Keuls Test can be completed in SPSS, R, Python and Excel. SPSS provides a direct menu option for Student-Newman-Keuls in one-way ANOVA post hoc tests. R can reproduce SNK output through suitable post hoc packages. Python can reproduce the ANOVA, ordered mean structure, q-statistic logic and supporting charts. Excel can be used for a manual teaching version.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Excel can reproduce the teaching logic of Newman Keuls by calculating group means, sample sizes, ANOVA MSE, mean differences, standard errors and q statistics. Excel does not provide a simple built-in SNK button, so SPSS or R is better for final official output.

Code Blocks for Newman Keuls Test

SPSS Syntax

ONEWAY G3 BY studytime
  /STATISTICS DESCRIPTIVES HOMOGENEITY
  /POSTHOC = SNK ALPHA(0.05).

R Code

data <- read.csv("dataset.csv")
data$studytime <- factor(data$studytime)

# One-way ANOVA

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

# Group summary

aggregate(G3 ~ studytime, data = data, function(x) {
c(n = length(x), mean = mean(x), sd = sd(x))
})

# Student-Newman-Keuls can be run with a suitable post hoc package.

# Example package-based workflows may use SNK.test() where available,

# then report ordered groups, q statistics and grouping letters.

Python Code

import pandas as pd
import itertools
import statsmodels.api as sm
import statsmodels.formula.api as smf

df = pd.read_csv("dataset.csv")
df["studytime"] = df["studytime"].astype("category")

model = smf.ols("G3 ~ C(studytime)", data=df).fit()
anova = sm.stats.anova_lm(model, typ=2)

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

summary = (
df.groupby("studytime")["G3"]
.agg(["count", "mean", "std"])
.sort_values("mean", ascending=False)
)

rows = []
for g1, g2 in itertools.combinations(summary.index, 2):
n1 = summary.loc[g1, "count"]
n2 = summary.loc[g2, "count"]
m1 = summary.loc[g1, "mean"]
m2 = summary.loc[g2, "mean"]

```
diff = abs(m1 - m2)

# Teaching approximation for pair-specific comparison standard error.
se = (mse * 0.5 * (1/n1 + 1/n2)) ** 0.5
q_stat = diff / se

rows.append([g1, g2, n1, n2, m1, m2, diff, se, q_stat])
```

snk_table = pd.DataFrame(rows, columns=[
"group_1", "group_2", "n_1", "n_2",
"mean_1", "mean_2", "absolute_mean_difference",
"standard_error", "q_statistic"
])

print(anova)
print(summary)
print(snk_table.sort_values("q_statistic", ascending=False))

Excel Formula Pattern

Group mean:
=AVERAGEIF(group_range, group_id, value_range)

Group sample size:
=COUNTIF(group_range, group_id)

Mean difference:
=ABS(mean_i - mean_j)

Balanced-design standard error:
=SQRT(MSE/n)

Unequal-size teaching standard error:
=SQRT(MSE*0.5*(1/n_i+1/n_j))

SNK q statistic:
=Mean_Difference/Standard_Error

Decision:
If q statistic is greater than the range-specific critical q value,
the ordered pair is significant by Newman Keuls.

APA Reporting Wording for Newman Keuls Test

A one-way ANOVA was conducted to compare G3 final grade across four studytime groups. The ANOVA indicated that mean G3 differed across studytime levels, so Student-Newman-Keuls post hoc comparisons were examined. The ordered means showed that group 3 had the highest average G3 score, followed by group 4, group 2 and group 1.

Newman Keuls post hoc interpretation indicated that the lowest studytime group was separated from the higher studytime groups. The highest two means, groups 3 and 4, were very close and should be interpreted cautiously. These results suggest that students in the lowest studytime category had lower final grades than students in higher studytime categories, while not every higher studytime pair showed a strong practical difference.

Common Mistakes in Newman Keuls Test

Downloads and Resources

Use the following downloadable outputs to verify the Newman Keuls Test result and compare the SPSS and R workflows.

Related Guides

Continue with these related Salar Cafe guides to understand ANOVA, post hoc testing, assumptions and reporting.