Quick Answer: Cramer von Mises Test Result

The verified Cramer von Mises Test result for the main variable G3 final grade shows N = 649, mean = 11.9060, standard deviation = 3.2307, W2 = 1.1241, and Monte Carlo p ≈ 0.00002. Because the p-value is far below .05, the normality null hypothesis is rejected. This means the G3 distribution does not follow the fitted normal distribution closely enough under the Cramer von Mises goodness-of-fit criterion.

Hypothesis-style interpretation: The null hypothesis says that the sample follows the fitted normal distribution. The alternative hypothesis says that the sample does not follow the fitted normal distribution. Since p ≈ 0.00002, the decision is to reject H₀. The Cramer von Mises Test therefore provides strong evidence that the G3 final grade distribution departs from normality.

What Is the Cramer von Mises Test?

The Cramer von Mises Test is a goodness-of-fit test used to compare a sample distribution with a theoretical distribution. In many applied statistics tasks, it is used as a normality test. The test checks whether the empirical cumulative distribution function, also called the empirical CDF, stays close to the fitted theoretical CDF across the full range of observed values.

The Cramer von Mises Test is based on squared differences. This means it does not focus only on the largest single distance. Instead, it accumulates the squared gaps between the empirical distribution and the theoretical distribution. If these gaps are small across the distribution, the W2 statistic is small. If the gaps are large or spread across the distribution, W2 becomes larger and the p-value becomes smaller.

This makes the test different from the Kolmogorov-Smirnov test, which focuses on the maximum distance between the empirical CDF and theoretical CDF. It is also related to other normality and goodness-of-fit tools such as the D’Agostino-Pearson test, Lilliefors test, Ryan-Joiner test, Q-Q plot normality check, Q-Q plot normality check 2, and P-P plot normality check.

Why the Cramer von Mises Test Matters

The Cramer von Mises Test matters because many statistical methods assume that a variable, residual, or model error distribution is approximately normal. When the assumption is not reasonable, test results, confidence intervals, p-values and model interpretation may become less reliable. The Cramer von Mises Test provides a formal way to compare the observed distribution with the fitted normal distribution.

In the worked example, the main variable is G3 final grade. The Cramer von Mises statistic is W2 = 1.1241, and the Monte Carlo p-value is approximately 0.00002. This is very strong evidence that the empirical distribution of G3 does not match the fitted normal distribution. The charts then show where and how the distribution departs from normality.

For a broader assumption-checking workflow, use the Cramer von Mises Test with descriptive and graphical tools such as descriptive statistics, frequency distribution, histogram interpretation, box plot interpretation, five-number summary, and coefficient of variation.

Cramer von Mises Test Formula

The Cramer von Mises Test compares the empirical CDF with the fitted theoretical CDF. If the observations are sorted from smallest to largest as x₍₁₎, x₍₂₎, …, x_(n), and F(x) is the fitted theoretical CDF, the common one-sample statistic is:

In this formula, n is the sample size, F(x_(i)) is the fitted theoretical CDF value for the ordered observation, and (2i − 1)/(2n) is the expected empirical plotting position. The statistic becomes larger when the fitted CDF and empirical CDF are far apart.

How to Interpret W2

Null and Alternative Hypothesis for the Cramer von Mises Test

The Cramer von Mises Test is a formal goodness-of-fit test. In this post, it is used as a normality test for the G3 final grade variable.

Dataset and Variables Used

The worked example uses a student performance-style dataset. The main tested variable is G3 final grade, which is a numeric outcome variable. The Cramer von Mises Test evaluates whether the observed G3 distribution follows a fitted normal distribution based on the sample mean and standard deviation.

Before interpreting a formal normality result, it is useful to summarize the data with descriptive statistics, frequency distribution, histogram interpretation, and five-number summary. If the variable has extreme values, also review box plot interpretation.

Verified SPSS Output Interpretation

The verified SPSS output supports the Cramer von Mises Test interpretation for the G3 final grade variable. Because SPSS does not always provide a direct one-click Cramer von Mises normality workflow in the same way it provides common descriptive output, the analysis can be supported by computed empirical CDF values, fitted normal CDF values, squared gap contributions and Monte Carlo reference simulation.

The SPSS output PDF for this guide is available here:

SPSS Result Summary

SPSS Decision Summary

How This SPSS Output Relates to Other Normality Tests

The Cramer von Mises Test is one part of a larger normality-checking toolkit. If you want to compare results, use the Kolmogorov-Smirnov test, Lilliefors test, D’Agostino-Pearson test, and Ryan-Joiner test. If your analysis later uses regression, also examine residual diagnostics and model form using the Ramsey RESET test.

Python Chart-by-Chart Interpretation

The Python charts show the Cramer von Mises Test visually. They explain the distribution fit, empirical CDF gap, Q-Q plot behavior, squared CDF contribution, Monte Carlo reference distribution, W2 comparison across variables and group-level W2 comparison.

Python Chart 1: Distribution Fit for Cramer von Mises Test

This chart shows the observed distribution of G3 final grades compared with a fitted normal distribution. The fitted normal curve is based on the sample mean and standard deviation. If the G3 data were normal, the histogram pattern would closely follow the fitted curve across the center and tails.

The Cramer von Mises result indicates that the visual fit is not close enough across the full distribution. Some grade values may cluster more heavily than the fitted normal curve expects, while other areas may show gaps or tail differences. These distribution-wide differences contribute to the significant W2 statistic.

Decision/reporting conclusion: Report that the distribution fit chart supports the formal test result. The observed G3 distribution does not align perfectly with the fitted normal curve, and the Cramer von Mises Test rejects normality.

Python Chart 2: Empirical CDF vs Fitted Normal CDF

This empirical CDF chart is central to understanding the Cramer von Mises Test. The empirical CDF shows the cumulative proportion of observed G3 values, while the fitted normal CDF shows what those cumulative proportions would look like if the data followed the fitted normal distribution.

When the two CDF lines are close, the W2 statistic is small. When the empirical CDF repeatedly departs from the fitted normal CDF, the squared gap accumulates. The verified result W2 = 1.1241 indicates that the total squared gap is large enough to reject the normality null hypothesis.

Decision/reporting conclusion: Report that the empirical CDF differs meaningfully from the fitted normal CDF. This chart visually explains why the Cramer von Mises Test is significant.

Python Chart 3: Normal Q-Q Plot

The normal Q-Q plot compares observed quantiles with expected normal quantiles. If the data are approximately normal, the points should fall close to the reference line. Systematic curvature, tail departures or grouped steps indicate departure from normality.

This Q-Q plot supports the Cramer von Mises result by showing where the G3 distribution deviates from normality. Because G3 is a grade variable, it may have discreteness, ceiling/floor behavior or clustered values. These features can create visible Q-Q departures and contribute to a significant goodness-of-fit test.

Decision/reporting conclusion: Report that the Q-Q plot provides visual confirmation of the formal Cramer von Mises decision. The distribution is not perfectly normal, especially where points depart from the reference line.

Python Chart 4: Squared CDF Gap Contributions

This chart explains the mechanics of the Cramer von Mises statistic. Each ordered observation contributes a squared gap between the fitted normal CDF and the empirical plotting position. Larger bars or higher contribution regions show where the empirical distribution differs most from the fitted normal distribution.

This plot is very useful for interpretation because it shows that W2 is not just an abstract number. It is built from many small squared differences across the distribution. If some sections of the distribution contribute much more than others, those sections explain the main source of non-normality.

Decision/reporting conclusion: Report that the squared CDF gap chart identifies where the fitted normal distribution fails most strongly. The accumulated gap produces W2 = 1.1241, supporting rejection of normality.

Python Chart 5: Monte Carlo Reference Distribution

The Monte Carlo reference distribution chart compares the observed W2 statistic with the W2 values expected under normality. Simulated samples are generated from the fitted normal model, W2 is calculated for each simulated sample, and the observed statistic is placed against that reference distribution.

Because the observed result has Monte Carlo p ≈ 0.00002, the observed W2 is far into the extreme tail of the reference distribution. This means that a W2 value as large as the observed result would be very unlikely if the G3 data truly followed the fitted normal distribution.

Decision/reporting conclusion: Report that the Monte Carlo reference distribution strongly supports rejection of normality. The observed W2 is extreme relative to the simulated normal reference distribution.

Python Chart 6: W2 Across Variables

This chart compares W2 values across variables. It helps answer a broader diagnostic question: which variable is closest to normal, and which variable departs most strongly? A higher W2 indicates a larger distribution-wide gap from the fitted normal distribution.

For student performance data, grade variables and count variables may behave differently. A count variable such as absences may show strong right-skewness, while a grade variable may show clustering, ceiling/floor effects or discrete score patterns. This chart helps decide which variables need transformation, nonparametric alternatives or robust methods.

Decision/reporting conclusion: Report the relative W2 pattern across variables. Variables with higher W2 values show stronger normality departures and deserve closer visual and statistical review.

Python Chart 7: Group W2 Comparison

The group W2 comparison chart shows whether distributional fit differs across groups. For example, normality may be closer in one school, sex category or study-time group than another. This is useful because a full-sample normality test can hide subgroup differences.

If one group has a much higher W2 statistic, that group contributes more strongly to the overall normality departure. The analyst should inspect that group with histograms, Q-Q plots and descriptive statistics. Group-level distribution differences may also affect comparisons, confidence intervals and regression assumptions.

Decision/reporting conclusion: Report whether normality departure is similar across groups or concentrated in specific groups. If one group has a much larger W2, discuss that subgroup in the diagnostic interpretation.

R Chart-by-Chart Validation

The R charts validate the Python Cramer von Mises Test workflow using a separate statistical environment. This is useful for reproducibility because the same conclusion is supported by both Python and R outputs.

R Chart 1: Distribution Fit for Cramer von Mises Test

The R distribution fit chart validates the Python histogram and fitted normal curve. It shows whether the observed grade distribution visually follows the theoretical normal shape. The significant test result indicates that the visual departures are large enough to reject normality.

Decision/reporting conclusion: Report that the R distribution fit chart confirms the Python interpretation: the observed distribution does not match the fitted normal distribution closely enough.

R Chart 2: Empirical CDF vs Fitted Normal CDF

The R empirical CDF chart confirms the distribution-wide gap between the observed cumulative distribution and the fitted normal cumulative distribution. This chart directly supports the logic of the Cramer von Mises W2 statistic.

Decision/reporting conclusion: Report that the R empirical CDF plot validates the CDF-gap interpretation. The empirical distribution deviates enough from the fitted normal distribution to support rejecting H₀.

R Chart 3: Normal Q-Q Plot

The R Q-Q plot confirms whether deviations from normality appear in the same way as the Python Q-Q plot. Points departing from the line indicate that the observed distribution does not follow the expected normal quantile pattern.

Decision/reporting conclusion: Report that the R Q-Q plot confirms the visual non-normality pattern. This supports the statistically significant Cramer von Mises decision.

R Chart 4: Squared CDF Gap Contributions

The R squared CDF gap chart validates the Python contribution plot. It shows where the empirical distribution and fitted normal distribution differ most. These squared differences are the building blocks of the W2 statistic.

Decision/reporting conclusion: Report that the R gap-contribution chart supports the same explanation of W2: the rejection of normality comes from accumulated distribution-wide CDF differences.

R Chart 5: Monte Carlo Reference Distribution

The R Monte Carlo reference distribution confirms the p-value logic. If the observed W2 falls far in the tail of simulated W2 values, the normality assumption is unlikely. This matches the verified Monte Carlo p-value of approximately 0.00002.

Decision/reporting conclusion: Report that R confirms the observed W2 is extreme under the simulated normal reference distribution. The normality null hypothesis should be rejected.

R Chart 6: W2 Across Variables

The R W2-across-variables chart validates the Python comparison of distributional fit across variables. This is helpful when the analyst needs to decide which variables require transformation, robust methods or nonparametric alternatives.

Decision/reporting conclusion: Report that the R variable comparison confirms which variables show the largest Cramer von Mises departures from fitted normality.

R Chart 7: Group W2 Comparison

The R group W2 chart validates the subgroup diagnostic comparison. If one group shows a larger W2 value, the group may have stronger distributional departure from normality. This can affect group comparison methods, model assumptions and interpretation.

Decision/reporting conclusion: Report whether R confirms the same group-level pattern as Python. If some groups depart more strongly from normality, discuss those groups in the diagnostic conclusion.

SPSS, R, Python and Excel Workflows for the Cramer von Mises Test

The Cramer von Mises workflow is based on sorting the data, fitting the target distribution, calculating theoretical CDF values, comparing them with empirical plotting positions, and estimating the p-value. Python and R are the strongest tools for direct implementation, SPSS can support the workflow with computed variables and exported output, and Excel can be used for a transparent manual calculation.

SPSS Workflow

Python Workflow

R Workflow

Excel Workflow

Code Blocks for Cramer von Mises Test

SPSS Syntax for Cramer von Mises Test Support

* Cramer von Mises Test support in SPSS.
* Main variable: G3 final grade.

TITLE "Cramer von Mises Test: Normality and empirical CDF comparison".

DATASET ACTIVATE DataSet1.

* Inspect descriptive statistics and normality plots.
EXAMINE VARIABLES=G3
  /PLOT BOXPLOT HISTOGRAM NPPLOT
  /COMPARE GROUPS
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

* Sort cases by G3 for empirical CDF workflow.
SORT CASES BY G3 (A).

* Create case index after sorting.
COMPUTE cvm_i = $CASENUM.
EXECUTE.

* Replace 649 with the valid sample size if your dataset changes.
COMPUTE cvm_n = 649.

* Use verified sample mean and standard deviation for fitted normal CDF.
COMPUTE cvm_mean = 11.9060.
COMPUTE cvm_sd = 3.2307.

* Empirical plotting position: (2i - 1)/(2n).
COMPUTE empirical_pos = ((2*cvm_i)-1)/(2*cvm_n).

* Fitted normal CDF.
COMPUTE fitted_norm_cdf = CDF.NORMAL(G3,cvm_mean,cvm_sd).

* Squared gap contribution.
COMPUTE squared_cdf_gap = (fitted_norm_cdf - empirical_pos)**2.
EXECUTE.

DESCRIPTIVES VARIABLES=G3 fitted_norm_cdf empirical_pos squared_cdf_gap
  /STATISTICS=MEAN STDDEV MIN MAX SUM.

* The W2 statistic is:
* W2 = 1/(12*n) + SUM(squared_cdf_gap).
* Verified result for this guide: W2 = 1.1241, Monte Carlo p approximately .00002.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Cramer-von-Mises-Test-SPSS-Output.pdf".

Python Code for Cramer von Mises Test

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import stats

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

# Main variable
x = pd.to_numeric(df["G3"], errors="coerce").dropna().to_numpy()
x = np.sort(x)

n = len(x)
mu = np.mean(x)
sd = np.std(x, ddof=1)

# Fitted normal CDF values
fitted_cdf = stats.norm.cdf(x, loc=mu, scale=sd)

# Empirical plotting positions
i = np.arange(1, n + 1)
empirical_pos = (2 * i - 1) / (2 * n)

# Cramer von Mises W2 statistic
squared_gaps = (fitted_cdf - empirical_pos) ** 2
w2 = (1 / (12 * n)) + np.sum(squared_gaps)

print("N:", n)
print("Mean:", mu)
print("SD:", sd)
print("Cramer von Mises W2:", w2)

# Monte Carlo p-value with parameter re-estimation
rng = np.random.default_rng(12345)
B = 50000
sim_w2 = np.empty(B)

for b in range(B):
    sim = rng.normal(mu, sd, n)
    sim = np.sort(sim)
    sim_mu = np.mean(sim)
    sim_sd = np.std(sim, ddof=1)
    sim_cdf = stats.norm.cdf(sim, loc=sim_mu, scale=sim_sd)
    sim_gaps = (sim_cdf - empirical_pos) ** 2
    sim_w2[b] = (1 / (12 * n)) + np.sum(sim_gaps)

p_value = np.mean(sim_w2 >= w2)
print("Monte Carlo p-value:", p_value)

# Chart 1: Distribution fit
plt.figure(figsize=(8, 6))
plt.hist(x, bins=20, density=True, alpha=0.7)
grid = np.linspace(min(x), max(x), 300)
plt.plot(grid, stats.norm.pdf(grid, mu, sd), linewidth=2)
plt.xlabel("G3 final grade")
plt.ylabel("Density")
plt.title("Distribution Fit: Cramer von Mises Test")
plt.tight_layout()
plt.savefig("cramer-von-mises-test-python-01-chart_01_distribution_fit_cramer_von_mises.png", dpi=300)
plt.close()

# Chart 2: Empirical CDF vs fitted normal CDF
ecdf = np.arange(1, n + 1) / n
plt.figure(figsize=(8, 6))
plt.step(x, ecdf, where="post", label="Empirical CDF")
plt.plot(x, fitted_cdf, label="Fitted normal CDF")
plt.xlabel("G3 final grade")
plt.ylabel("Cumulative probability")
plt.title("Empirical CDF vs Fitted Normal CDF")
plt.legend()
plt.tight_layout()
plt.savefig("cramer-von-mises-test-python-02-chart_02_empirical_cdf_vs_fitted_normal_cdf.png", dpi=300)
plt.close()

# Chart 3: Normal Q-Q plot
plt.figure(figsize=(8, 6))
stats.probplot(x, dist="norm", plot=plt)
plt.title("Normal Q-Q Plot")
plt.tight_layout()
plt.savefig("cramer-von-mises-test-python-03-chart_03_normal_qq_plot.png", dpi=300)
plt.close()

# Chart 4: Squared CDF gap contributions
plt.figure(figsize=(9, 5))
plt.plot(i, squared_gaps)
plt.xlabel("Ordered case")
plt.ylabel("Squared CDF gap")
plt.title("Squared CDF Gap Contributions")
plt.tight_layout()
plt.savefig("cramer-von-mises-test-python-04-chart_04_squared_cdf_gap_contributions.png", dpi=300)
plt.close()

# Chart 5: Monte Carlo reference distribution
plt.figure(figsize=(8, 6))
plt.hist(sim_w2, bins=40, alpha=0.7)
plt.axvline(w2, linewidth=2)
plt.xlabel("Simulated W2")
plt.ylabel("Frequency")
plt.title("Monte Carlo Reference Distribution")
plt.tight_layout()
plt.savefig("cramer-von-mises-test-python-05-chart_05_monte_carlo_reference_distribution.png", dpi=300)
plt.close()

R Code for Cramer von Mises Test

# Cramer von Mises Test in R

library(ggplot2)

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

x <- as.numeric(df$G3)
x <- sort(x[!is.na(x)])

n <- length(x)
mu <- mean(x)
sd_x <- sd(x)

i <- 1:n
empirical_pos <- (2*i - 1)/(2*n)
fitted_cdf <- pnorm(x, mean = mu, sd = sd_x)
squared_gaps <- (fitted_cdf - empirical_pos)^2

w2 <- (1/(12*n)) + sum(squared_gaps)

cat("N:", n, "\n")
cat("Mean:", mu, "\n")
cat("SD:", sd_x, "\n")
cat("Cramer von Mises W2:", w2, "\n")

# Monte Carlo p-value with parameter re-estimation
set.seed(12345)
B <- 50000
sim_w2 <- numeric(B)

for (b in 1:B) {
  sim <- sort(rnorm(n, mean = mu, sd = sd_x))
  sim_mu <- mean(sim)
  sim_sd <- sd(sim)
  sim_cdf <- pnorm(sim, mean = sim_mu, sd = sim_sd)
  sim_gaps <- (sim_cdf - empirical_pos)^2
  sim_w2[b] <- (1/(12*n)) + sum(sim_gaps)
}

p_value <- mean(sim_w2 >= w2)
cat("Monte Carlo p-value:", p_value, "\n")

plot_df <- data.frame(
  x = x,
  ecdf = (1:n)/n,
  fitted_cdf = fitted_cdf,
  case = i,
  squared_gap = squared_gaps
)

# Chart 1: Distribution fit
ggplot(data.frame(x=x), aes(x=x)) +
  geom_histogram(aes(y=after_stat(density)), bins=20, alpha=.7) +
  stat_function(fun=dnorm, args=list(mean=mu, sd=sd_x), linewidth=1) +
  labs(title="Distribution Fit: Cramer von Mises Test",
       x="G3 final grade", y="Density")

# Chart 2: Empirical CDF vs fitted normal CDF
ggplot(plot_df, aes(x=x)) +
  geom_step(aes(y=ecdf), linewidth=1) +
  geom_line(aes(y=fitted_cdf), linewidth=1) +
  labs(title="Empirical CDF vs Fitted Normal CDF",
       x="G3 final grade", y="Cumulative probability")

# Chart 3: Normal Q-Q plot
qqnorm(x)
qqline(x)

# Chart 4: Squared CDF gap contributions
ggplot(plot_df, aes(x=case, y=squared_gap)) +
  geom_line() +
  labs(title="Squared CDF Gap Contributions",
       x="Ordered case", y="Squared CDF gap")

# Chart 5: Monte Carlo reference distribution
ggplot(data.frame(sim_w2=sim_w2), aes(x=sim_w2)) +
  geom_histogram(bins=40, alpha=.7) +
  geom_vline(xintercept=w2, linewidth=1) +
  labs(title="Monte Carlo Reference Distribution",
       x="Simulated W2", y="Frequency")

Excel Formulas for Cramer von Mises Test

Assume sorted G3 values are in A2:A650.

Mean in H1:
=AVERAGE(A2:A650)

Standard deviation in H2:
=STDEV.S(A2:A650)

Sample size in H3:
=COUNT(A2:A650)

Case index in B2:
=ROW(A1)

Empirical plotting position in C2:
=(2*B2-1)/(2*$H$3)

Fitted normal CDF in D2:
=NORM.DIST(A2,$H$1,$H$2,TRUE)

Squared CDF gap in E2:
=(D2-C2)^2

Copy B2:E2 down to the last observation.

Cramer von Mises W2 statistic:
=1/(12*$H$3)+SUM(E2:E650)

Decision:
If the Monte Carlo or reference p-value is below .05, reject normality.
If the p-value is .05 or above, do not reject normality.

APA Reporting Wording for the Cramer von Mises Test

When reporting the Cramer von Mises Test, include the tested variable, sample size, fitted distribution, W2 statistic, p-value and decision. Also include a visual interpretation based on the distribution fit chart, empirical CDF chart and Q-Q plot.

APA-Style Report

Short Report Sentence

Student-Friendly Report Example

Common Mistakes in Cramer von Mises Test Interpretation

When to Use the Cramer von Mises Test

Use the Cramer von Mises Test when you want to compare a sample distribution with a theoretical distribution across the whole cumulative distribution. It is commonly used for normality testing, but it can also be used as a broader goodness-of-fit tool.

If the Cramer von Mises Test rejects normality and the planned method requires stronger normality support, consider transformation, robust methods or nonparametric alternatives. For transformation context, see reciprocal transformation. For comparison with hypothesis testing and reporting, see one-tailed t-test, one-sample z-test, one-proportion z-test, confidence interval, and effect size.

Downloads and Resources for Cramer von Mises Test

The resources below include the verified SPSS output PDF, Python charts and R validation charts used in this guide. These files support direct reporting, teaching, WordPress publishing and normality diagnostics practice.

Related Statistical Guides

Use these related Salar Cafe guides to connect the Cramer von Mises Test with normality testing, descriptive statistics, assumption checks, transformations, hypothesis testing and reporting.