Mauchly’s Test of Sphericity: Formula, Interpretation, SPSS, R, Python and Excel Improvement Guide 1

What Is Mauchly’s Test of Sphericity?

Mauchly’s Test of Sphericity is an assumption test used mainly with repeated-measures ANOVA. Repeated-measures ANOVA is used when the same subjects are measured more than two times, or when the same person contributes scores under three or more related conditions. In this worked example, each student has three related grade measurements: G1, G2 and G3.

The sphericity assumption concerns the pattern of variances among the pairwise difference scores. In simple words, repeated-measures ANOVA expects the differences between repeated measurements to have similar variances. For three grade measurements, the relevant differences are G2 − G1, G3 − G2 and G3 − G1. If these difference-score variances are too unequal, the standard repeated-measures ANOVA degrees of freedom can become too optimistic. That can make a test appear more precise than it really is.

Mauchly’s Test checks this assumption statistically. If Mauchly’s Test is not significant, the sphericity assumption is usually treated as acceptable. If Mauchly’s Test is significant, sphericity is considered violated and a correction should be used. Common corrections include the Greenhouse-Geisser Correction, the Huynh-Feldt Correction and the lower-bound correction.

When Should You Use Mauchly’s Test of Sphericity?

Use Mauchly’s Test of Sphericity when the design has a within-subject factor with at least three levels. It is not needed for a simple paired-samples t-test because with only two repeated measurements, the sphericity assumption is automatically satisfied. It is most relevant in education, psychology, medical trials, training evaluation, longitudinal measurement and experimental designs where the same people are measured repeatedly.

Mauchly’s Test of Sphericity Formula and Logic

The full mathematical calculation of Mauchly’s Test of Sphericity is based on the covariance matrix of the orthonormalized transformed repeated measures. In applied data analysis, most students and researchers do not calculate Mauchly’s W by hand. SPSS, R and Python calculate it from the repeated-measures covariance structure.

The practical idea can be understood through the pairwise difference scores. If the variances of the pairwise differences are similar, sphericity is more plausible. If those variances are quite different, sphericity becomes doubtful. In this dataset, the difference variances were not equal:

The difference variance for G3 − G1 is much larger than the variance for G3 − G2. This unequal pattern explains why Mauchly’s Test became significant in the SPSS, R and Python workflows.

Null Hypothesis and Alternative Hypothesis

Dataset and Repeated-Measures Model Used

This tutorial uses the student-por.csv student performance dataset. The repeated measures are G1, G2 and G3. These are grade measurements recorded for the same students, so the design is a within-subject repeated-measures design.

External dataset source: UCI Machine Learning Repository: Student Performance dataset.

Verified Results in SPSS, R and Python

The analysis was reproduced in SPSS, R and Python. The three workflows confirmed the same conclusion: Mauchly’s Test of Sphericity is significant, meaning the sphericity assumption is violated for the repeated grade measurements.

Descriptive Statistics for G1, G2 and G3

Mauchly’s Test of Sphericity Table

Epsilon Correction Values

Tests of Within-Subjects Effects

R Charts for Mauchly’s Test of Sphericity

The R workflow produced eight charts that explain the repeated-measures pattern, the sphericity issue and the correction values. These charts are useful for teaching because Mauchly’s Test is easier to understand when the repeated grade pattern and pairwise difference variances are visible.

1. Mean Grade Profile Across G1, G2 and G3

The mean profile shows a steady increase from G1 = 11.40 to G2 = 11.57 and then to G3 = 11.91. This upward trend explains why the repeated-measures ANOVA found a significant within-subject effect. Mauchly’s Test does not test whether the means are different; it checks whether the repeated-measures covariance structure is suitable for the ordinary ANOVA row.

2. Grade Distribution Boxplot

The boxplot shows that the final grade measurement has a slightly higher center and a wider spread. The lower outliers around zero are visible, especially in G3. These distributional features are not the same as sphericity, but they help explain why the repeated grade measurements are not perfectly uniform across time.

3. Pairwise Difference Variances for Sphericity Check

This is the most important explanatory chart for Mauchly’s Test of Sphericity. The variance for G3 − G1 is 3.315, while the variance for G3 − G2 is only 1.634. This unequal pattern supports the significant Mauchly result and explains why a corrected ANOVA row is needed.

4. Mauchly’s Test Summary

The summary chart highlights the main result: W = .827, χ² = 122.814 and p < .001. Since the p-value is below .05, the null hypothesis of sphericity is rejected.

5. Sphericity Correction Epsilon Values

The epsilon chart shows that Greenhouse-Geisser = .853 and Huynh-Feldt = .855. These values are very close, so both corrected rows lead to the same practical conclusion. The lower-bound correction is .500, which is the most conservative option for three repeated measures.

6. Sample Student Grade Trajectories

The trajectory chart shows that not every student follows the same pattern. Some students improve, some remain stable and some decline. The bold average line still rises across G1, G2 and G3. This is why the repeated-measures effect is significant even though individual trajectories vary.

7. Histograms of G1, G2 and G3

The histograms show that the three grade measurements have broadly similar score ranges but not identical distribution shapes. G3 has more spread and visible low-end values. The histograms help readers understand the raw score pattern before interpreting the Mauchly and ANOVA tables.

8. Pairwise Difference Score Distributions

The difference-score histograms provide a direct visual explanation of sphericity. Mauchly’s Test is related to how these difference scores vary. Since the difference distributions do not have equal variance, the test detects a sphericity violation.

Python Validation Charts for Mauchly’s Test of Sphericity

The Python workflow independently reproduced the Mauchly result, epsilon values, corrected degrees of freedom and repeated-measures ANOVA effect. This validation is useful because it shows that the result does not depend on only one software package.

1. Python Mean Grade Profile

The Python mean profile confirms the same increasing pattern found in R and SPSS. The final grade mean is highest, and the error bars show uncertainty around each mean estimate.

2. Python Grade Distribution Boxplot

The Python boxplot confirms that G3 has a higher center and wider spread than the earlier grade measurements. This supports the descriptive interpretation of the repeated-measures model.

3. Python Pairwise Difference Variances

The Python variance chart matches the R chart: 2.188, 3.315 and 1.634. This repeated confirmation is important because the pairwise difference variance pattern is the most intuitive way to understand the sphericity violation.

4. Python Mauchly Test Summary

The Python result confirms the SPSS and R conclusion: the sphericity assumption is violated. Since p < .001, corrected repeated-measures ANOVA degrees of freedom are required.

5. Python Epsilon Values

The Python epsilon chart confirms the same correction values: Greenhouse-Geisser = .853, Huynh-Feldt = .855 and Lower-bound = .500.

6. Python Sample Student Trajectories

The trajectory chart shows strong individual variation. This is normal in educational data. Repeated-measures ANOVA tests the average pattern while accounting for the fact that the same students are measured repeatedly.

7. Python Histograms of G1, G2 and G3

The Python histograms confirm that the grade distributions are concentrated around the middle of the scale but include low outlying values. These charts are useful for describing the dataset before presenting the assumption test.

8. Python Pairwise Difference Score Histograms

The pairwise difference histograms show that the differences are centered near small positive values, but their spreads are not the same. This supports the significant Mauchly’s Test result.

9. Python Corrected Degrees of Freedom Comparison

This chart explains what the correction actually does. The F statistic remains the same, but the degrees of freedom are reduced. The Huynh-Feldt effect df becomes 1.709 and the error df becomes 1107.581.

10. Python Repeated-Measures ANOVA Effect Summary

The Python ANOVA summary confirms F = 36.287. Since the corrected p-value remains below .001, the repeated grade effect is statistically significant even after the sphericity correction.

How to Run Mauchly’s Test of Sphericity in SPSS, R, Python and Excel

Mauchly’s Test of Sphericity in SPSS

SPSS reports Mauchly’s Test automatically when repeated-measures ANOVA is run through the General Linear Model repeated-measures procedure. The key output tables are Mauchly’s Test of Sphericity, Tests of Within-Subjects Effects and the epsilon correction values.

* Mauchly's Test of Sphericity in SPSS.

GET DATA
 /TYPE=TXT
 /FILE='D:\low kda score priority basis posts\first post\Mauchly Test of Sphericity\student-por.csv'
 /ENCODING='UTF8'
 /DELCASE=LINE
 /DELIMITERS=","
 /QUALIFIER='"'
 /ARRANGEMENT=DELIMITED
 /FIRSTCASE=2
 /VARIABLES=
 school A40
 sex A20
 age F8.0
 address A20
 famsize A20
 Pstatus A20
 Medu F8.0
 Fedu F8.0
 Mjob A40
 Fjob A40
 reason A40
 guardian A40
 traveltime F8.0
 studytime F8.0
 failures F8.0
 schoolsup A40
 famsup A40
 paid A40
 activities A40
 nursery A40
 higher A40
 internet A40
 romantic A40
 famrel F8.0
 freetime F8.0
 goout F8.0
 Dalc F8.0
 Walc F8.0
 health F8.0
 absences F8.0
 G1 F8.0
 G2 F8.0
 G3 F8.0.
CACHE.
EXECUTE.

GLM G1 G2 G3
 /WSFACTOR=GradeTime 3 Polynomial
 /MEASURE=Grade
 /METHOD=SSTYPE(3)
 /PRINT=DESCRIPTIVE ETASQ
 /CRITERIA=ALPHA(.05)
 /WSDESIGN=GradeTime.

Mauchly’s Test of Sphericity in R

In R, the workflow is to read the dataset, reshape the data from wide to long format and run a repeated-measures ANOVA function that reports Mauchly’s Test and sphericity corrections.

install.packages(c("tidyverse", "ez"))

library(tidyverse)
library(ez)

base_dir <- "D:/low kda score priority basis posts/first post/Mauchly Test of Sphericity"
student <- read.csv(file.path(base_dir, "student-por.csv"), sep = ",")

clean_data <- student %>%
  mutate(subject_id = row_number()) %>%
  filter(!is.na(G1), !is.na(G2), !is.na(G3))

long_data <- clean_data %>%
  select(subject_id, G1, G2, G3) %>%
  pivot_longer(cols = c(G1, G2, G3),
               names_to = "GradeTime",
               values_to = "Grade") %>%
  mutate(
    subject_id = factor(subject_id),
    GradeTime = factor(GradeTime, levels = c("G1", "G2", "G3"))
  )

mauchly_result <- ezANOVA(
  data = long_data,
  dv = Grade,
  wid = subject_id,
  within = GradeTime,
  detailed = TRUE,
  type = 3
)

print(mauchly_result)

Mauchly’s Test of Sphericity in Python

Python can reproduce the same logic by calculating the repeated-measures covariance structure and sphericity-related correction values. The validation workflow used NumPy, pandas, SciPy and Matplotlib to reproduce the Mauchly W value, chi-square approximation, epsilon values and charts.

import numpy as np
import pandas as pd
from scipy.stats import chi2

data = pd.read_csv(
    r"D:\low kda score priority basis posts\first post\Mauchly Test of Sphericity\student-por.csv"
)

X = data[["G1", "G2", "G3"]].dropna().to_numpy(dtype=float)
n, k = X.shape
p = k - 1

cov_matrix = np.cov(X, rowvar=False, ddof=1)

C = np.array([
    [1 / np.sqrt(2),  1 / np.sqrt(6)],
    [-1 / np.sqrt(2), 1 / np.sqrt(6)],
    [0,              -2 / np.sqrt(6)]
])

contrast_cov = C.T @ cov_matrix @ C

mauchly_w = np.linalg.det(contrast_cov) / ((np.trace(contrast_cov) / p) ** p)

correction_factor = 1 - ((2 * p**2 + p + 2) / (6 * p * (n - 1)))
chi_square_value = -((n - 1) * correction_factor * np.log(mauchly_w))
df = int((p * (p + 1) / 2) - 1)
p_value = chi2.sf(chi_square_value, df)

print(mauchly_w, chi_square_value, df, p_value)

Mauchly’s Test of Sphericity in Excel

Excel is useful for checking repeated-measure means and pairwise difference variances. It is not the best tool for producing the final Mauchly W table because the full calculation uses a transformed covariance matrix. For final reporting, SPSS, R or Python is preferred. Still, Excel can help students understand the assumption visually and numerically.

Mean of G1:
=AVERAGE(B2:B650)

Mean of G2:
=AVERAGE(C2:C650)

Mean of G3:
=AVERAGE(D2:D650)

Variance of G2 - G1:
=VAR.S(E2:E650)

Variance of G3 - G2:
=VAR.S(F2:F650)

Variance of G3 - G1:
=VAR.S(G2:G650)

If the pairwise difference variances are not similar, sphericity may be violated. In this dataset, Excel would show the same practical warning pattern because the difference variances are 2.188, 1.634 and 3.315.

How to Report Mauchly’s Test of Sphericity

A complete report should include Mauchly’s W, chi-square, degrees of freedom, p-value, the decision about sphericity and the corrected repeated-measures ANOVA result. If sphericity is violated, do not report only the sphericity-assumed row as the main result.

Should You Use Greenhouse-Geisser or Huynh-Feldt?

In many applied workflows, Greenhouse-Geisser is treated as more conservative and Huynh-Feldt as slightly less conservative when epsilon is relatively high. In this example, the two values are almost the same: .853 and .855. Therefore, the interpretation does not materially change. Both corrected rows remain significant at p < .001.

Common Mistakes in Mauchly’s Test of Sphericity

1. Reporting p = .000

SPSS prints .000 when the p-value is very small. This should be written as p < .001 in formal reporting.

2. Ignoring a significant Mauchly’s Test

If Mauchly’s Test is significant, the sphericity-assumed row should not be the main row. Use Greenhouse-Geisser, Huynh-Feldt or another appropriate correction.

3. Thinking Mauchly’s Test compares means

Mauchly’s Test does not test whether G1, G2 and G3 means are equal. It tests the sphericity assumption. The repeated-measures ANOVA tests whether the means differ.

4. Using the test for only two repeated measures

Mauchly’s Test is not needed when there are only two repeated measurements because sphericity is automatically satisfied.

5. Confusing sphericity with normality

Sphericity is about the covariance pattern and difference-score variances in repeated-measures ANOVA. Normality is a different assumption. If you need normality tests, see the Lilliefors Test, Kolmogorov-Smirnov Test and DAgostino Pearson Test.

Related Data Analysis Guides

After learning Mauchly’s Test of Sphericity, continue with related statistical guides on Online Internet Cafe. The Greenhouse-Geisser Correction and Huynh-Feldt Correction are the most directly related guides because they explain what to do after a significant Mauchly result. For broader assumption checking, read the Brown-Forsythe Test, Hartley F Max Test, Cochran C Test, Cramer von Mises Test and Influence Diagnostics.

Download SPSS Output and Verification Files

The SPSS output PDF verifies the dataset import, descriptive statistics, pairwise difference variances, Mauchly’s Test of Sphericity, epsilon values, corrected within-subjects effects and repeated-measures ANOVA interpretation.

External References for Mauchly’s Test of Sphericity

This guide uses verified SPSS, R and Python outputs together with standard statistical documentation and software references. These resources help readers check the background of repeated-measures ANOVA, sphericity, Mauchly’s Test and correction methods.

• UCI Machine Learning Repository: Student Performance dataset
• IBM SPSS Statistics documentation
• The Comprehensive R Archive Network
• Pingouin statistical package for Python
• R aov documentation