Quick Answer: Lilliefors Test Result

The Lilliefors Test checks whether a sample follows a normal distribution when the normal distribution is fitted using the sample mean and sample standard deviation. This matters because in real projects, we usually do not know the true population mean and true population standard deviation. We estimate them from the data. The standard KS test assumes a fully specified reference distribution, while the Lilliefors version adjusts the normality-test logic for estimated parameters.

The null hypothesis says the sample comes from a normal distribution. The alternative hypothesis says the sample does not come from a normal distribution. If the p-value is less than .05, reject normality. If the p-value is greater than or equal to .05, do not reject normality. The test statistic is commonly written as D, the largest distance between the empirical CDF and the fitted normal CDF.

What Is the Lilliefors Test?

The Lilliefors Test is a normality test based on the Kolmogorov-Smirnov distance, but it is adjusted for the common situation where the mean and standard deviation of the normal reference distribution are estimated from the sample. That makes the Lilliefors Test very useful in student research, thesis analysis, SPSS normality tables, Python diagnostics, R normality workflows, and statistical consulting reports.

In the ordinary KS normality idea, the reference normal distribution must be fully specified in advance. In real analysis, that almost never happens. Instead, the analyst calculates the sample mean and sample standard deviation, fits a normal curve using those values, and compares the observed distribution with that fitted curve. The Lilliefors Test adjusts the p-value and decision rule for this estimated-parameter situation.

Because it is a normality-test method, the Lilliefors Test belongs with other assumption-checking tools such as the D’Agostino-Pearson Test, Cramer-von Mises Test, Ryan-Joiner Test, Kolmogorov-Smirnov Test, and Q-Q Plot Normality Check 2.

Why the Lilliefors Test Is Important

The Lilliefors Test is important because many researchers incorrectly use a basic KS normality test even though the normal distribution parameters are estimated from the same data. This can lead to inaccurate interpretation. The Lilliefors version is specifically designed for the practical normality-checking situation in which the reference mean and standard deviation are not known before analysis.

For example, if a student checks whether final grades are normally distributed, the student normally calculates the sample mean and sample standard deviation from the grade data. The fitted normal curve then depends on the sample. The Lilliefors Test accounts for that dependency. That is why it is often discussed as a corrected or adjusted KS-style normality test.

When reporting a normality assumption, use the Lilliefors Test with supporting descriptive tools such as descriptive statistics, frequency distribution, five-number summary, and coefficient of variation.

Lilliefors Test Formula

The Lilliefors Test uses the largest absolute gap between the empirical cumulative distribution function and the fitted normal cumulative distribution function.

Where:

D Plus and D Minus

The D statistic can also be explained using two directional gaps. D plus shows where the empirical CDF is above the fitted normal CDF. D minus shows where the fitted normal CDF is above the empirical CDF. The final Lilliefors D statistic is the larger absolute gap.

Null and Alternative Hypotheses for the Lilliefors Test

The Lilliefors Test is a formal normality test, so the null and alternative hypotheses should be written clearly before interpreting output.

The hypothesis decision is especially important before parametric methods such as a one-tailed t test, one-sample z test, confidence interval, regression diagnostics such as the Ramsey RESET Test, and assumption checks such as the Levene Test.

Assumptions and Decision Logic

The Lilliefors Test is itself an assumption-checking tool, but it still requires careful interpretation. The observations should be independent, the variable should be numeric, and the test should be interpreted with visual evidence rather than as an isolated p-value.

For repeated-measures or variance assumption contexts, use separate assumption checks such as Mauchly’s Test of Sphericity, Greenhouse-Geisser Correction, Cochran C Test, and Brown-Forsythe Test.

Dataset and Variables Used

The worked Lilliefors Test example uses student performance variables. The main chart set focuses on normality diagnostics for a numeric variable such as G3 final grade, while the cross-variable charts compare Lilliefors D statistics across several numeric variables. This helps show that normality is variable-specific. One variable can look close to normal while another variable can be highly skewed.

Before running the Lilliefors Test, it is good practice to examine descriptive statistics, frequency distributions, histograms, box plots, and five-number summaries.

Verified SPSS Output Interpretation

The SPSS output PDF supports the Lilliefors Test normality workflow for this post. In SPSS-style interpretation, the analyst should report the normality-test statistic, significance value, decision about the null hypothesis, and supporting graphical evidence. Because the Lilliefors Test is connected to KS-style normality testing with estimated parameters, the interpretation should be precise and not confused with a fully specified ordinary KS test.

How to Read the SPSS Output

SPSS Decision Logic

If the SPSS p-value is below .05, write that the Lilliefors Test indicated a statistically significant departure from normality. If the p-value is above .05, write that the test did not find sufficient evidence to reject normality. Do not write that the data are “perfectly normal.” A nonsignificant result only means the evidence against normality is not strong enough under this test.

Python Chart-by-Chart Interpretation

The provided Python output contains one distribution-fit chart for the Lilliefors Test. This chart is still important because it shows the practical visual comparison that motivates the test: observed data shape versus fitted normal shape.

Python Chart 1: Distribution Fit for Lilliefors Test

This Python chart shows how the sample distribution compares with the fitted normal distribution. The fitted normal curve is based on the sample mean and sample standard deviation, which is exactly why the Lilliefors Test is needed. If the histogram or density shape follows the normal reference curve closely, the normality assumption becomes more plausible. If the observed distribution is skewed, heavy-tailed, too peaked, too flat, or irregular, the Lilliefors D statistic becomes larger.

The chart is useful because the p-value alone does not explain the pattern of non-normality. A significant result may be caused by skewness, outliers, kurtosis, floor effects, ceiling effects, or distribution gaps. This chart gives the visual reason behind the test decision. In a full normality workflow, this figure should be read together with the Q-Q plot normality check and P-P plot normality check.

Decision/reporting conclusion: Use this Python chart to support the Lilliefors normality decision. If the observed distribution visibly departs from the fitted normal curve, report that the chart supports non-normality. If the fit is close, report that the visual evidence supports approximate normality, while still using the p-value for the formal decision.

R Chart-by-Chart Validation

The R charts provide a complete validation workflow for the Lilliefors Test. They show the distribution fit, empirical CDF comparison, D plus and D minus gaps, Monte Carlo reference distribution, p-value decision, D statistic across variables, and group comparison.

R Chart 1: Distribution Fit for Lilliefors Test

This R chart validates the Python distribution-fit result. It compares the observed data distribution with the fitted normal reference. If the observed bars or density shape are close to the normal curve, the Lilliefors D statistic should be smaller. If the distribution has skewness, heavy tails, or unusual clustering, the D statistic becomes larger.

Decision/reporting conclusion: This chart should be used as the first visual support for the Lilliefors decision. It tells the reader whether the formal result is visually reasonable.

R Chart 2: Empirical CDF vs Fitted Normal CDF

This is the most important conceptual chart for the Lilliefors Test. The empirical CDF shows the cumulative pattern of the observed sample. The fitted normal CDF shows the expected cumulative pattern if the sample were normally distributed with the sample’s estimated mean and standard deviation. The Lilliefors D statistic is based on the largest vertical separation between these two curves.

If the two curves stay close together, the data are closer to normal. If the curves separate strongly in the middle, lower tail, or upper tail, the evidence against normality becomes stronger. This chart explains the test more clearly than a table alone.

Decision/reporting conclusion: Use this ECDF chart to explain where the empirical distribution differs most from the fitted normal distribution. The largest visible vertical gap is the core evidence behind the Lilliefors statistic.

R Chart 3: D Plus and D Minus Gaps

This chart shows the directional gaps behind the Lilliefors statistic. D plus occurs when the empirical CDF is above the fitted normal CDF. D minus occurs when the fitted normal CDF is above the empirical CDF. The final D statistic is the larger absolute gap.

This figure is useful for teaching and reporting because it shows the exact area where the data depart most strongly from normality. Instead of only saying that D is large or small, the chart shows the location and direction of the maximum discrepancy.

Decision/reporting conclusion: Use this chart to justify the D statistic and to describe whether the strongest departure is in the lower tail, center, or upper tail of the distribution.

R Chart 4: Monte Carlo Reference Distribution

The Lilliefors Test requires special p-value logic because the normal mean and standard deviation are estimated from the data. A Monte Carlo reference distribution shows what D values are expected when samples are drawn from a normal distribution and parameters are estimated in the same way.

If the observed D statistic is far into the extreme tail of this simulated reference distribution, the p-value becomes small and the normality assumption is rejected. This chart makes the p-value visually understandable.

Decision/reporting conclusion: Use this chart to explain whether the observed D statistic is typical or unusual under the fitted-normal null model.

R Chart 5: P-value Decision

This chart presents the final test decision in a simple visual form. If the p-value is below .05, the Lilliefors result is statistically significant and normality is rejected. If the p-value is greater than or equal to .05, normality is not rejected.

The p-value decision is useful for quick reporting, but it should be interpreted with the distribution-fit chart and ECDF chart. A p-value tells whether the difference is statistically meaningful; the charts tell what the difference looks like.

Decision/reporting conclusion: Use this p-value chart as the final decision aid, but do not report it without the supporting visual evidence.

R Chart 6: Lilliefors D Across Variables

This chart compares Lilliefors D values across different numeric variables. A larger D statistic indicates a stronger departure from the fitted normal distribution. This is useful because normality is not a property of the entire dataset; it must be assessed for the specific variable or residual series relevant to the analysis.

For example, a grade variable may show a different normality pattern than a count variable such as absences. A count variable may require a transformation such as reciprocal transformation, a square-root transformation, or a nonparametric approach.

Decision/reporting conclusion: Use this chart to identify which variables need closer normality review, transformation, robust analysis, or nonparametric methods.

R Chart 7: Group Lilliefors D Comparison

This chart compares the strength of non-normality across groups. A variable may look approximately normal in one subgroup but non-normal in another. This is important when the next analysis compares groups, such as a t test, ANOVA, or group-based regression model.

Group-specific normality checks should be interpreted with variance assumptions too. For example, if a group comparison is planned, pair the Lilliefors normality decision with variance checks such as the Levene Test, Brown-Forsythe Test, or Cochran C Test.

Decision/reporting conclusion: Use this chart to decide whether normality problems are general or concentrated in particular groups. This helps choose between parametric, robust, transformed, or nonparametric analysis.

SPSS, Python, R and Excel Workflows for the Lilliefors Test

A complete Lilliefors Test workflow should not stop at one p-value. The best workflow includes descriptive statistics, fitted normal comparison, ECDF diagnostics, D statistic, p-value, and a final reporting decision. This workflow is useful before parametric tests, confidence interval estimation, regression modeling, cross tabulation reporting for categorical summaries, and broader data-quality review.

SPSS Workflow

Python Workflow

R Workflow

Excel Workflow

SPSS, Python, R and Excel Code for the Lilliefors Test

SPSS Syntax for Lilliefors Normality Workflow

* Lilliefors-style normality workflow in SPSS.
* Replace G3 with your selected numeric variable.

TITLE "Lilliefors Test Normality Workflow".

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

NPAR TESTS
  /K-S(NORMAL)=G3
  /MISSING ANALYSIS.

FREQUENCIES VARIABLES=G3
  /STATISTICS=MEAN MEDIAN STDDEV SKEWNESS SESKEW KURTOSIS SEKURT MINIMUM MAXIMUM
  /HISTOGRAM NORMAL
  /ORDER=ANALYSIS.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Lilliefors-Test-SPSS-Output.pdf".

Python Code for the Lilliefors Test

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
from statsmodels.stats.diagnostic import lilliefors

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

x = pd.to_numeric(df["G3"], errors="coerce").dropna().to_numpy()

# Lilliefors test for normality
d_stat, p_value = lilliefors(x, dist="norm")

print("Lilliefors D statistic:", d_stat)
print("p-value:", p_value)

if p_value < 0.05:
    print("Reject normality")
else:
    print("Do not reject normality")

# Fitted normal parameters
mu = np.mean(x)
sigma = np.std(x, ddof=1)

# Empirical CDF and fitted normal CDF
x_sorted = np.sort(x)
n = len(x_sorted)
ecdf = np.arange(1, n + 1) / n
fitted_cdf = norm.cdf(x_sorted, loc=mu, scale=sigma)

# D statistic manually
abs_gap = np.abs(ecdf - fitted_cdf)
manual_d = np.max(abs_gap)

print("Manual maximum gap:", manual_d)

# Plot ECDF and fitted CDF
plt.figure(figsize=(8, 5))
plt.step(x_sorted, ecdf, where="post", label="Empirical CDF")
plt.plot(x_sorted, fitted_cdf, label="Fitted Normal CDF")
plt.title("Lilliefors Test: Empirical CDF vs Fitted Normal CDF")
plt.xlabel("Observed value")
plt.ylabel("Cumulative probability")
plt.legend()
plt.tight_layout()
plt.show()

R Code for the Lilliefors Test

# Lilliefors Test in R

# install.packages("nortest")
library(nortest)

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

x <- as.numeric(df$G3)
x <- na.omit(x)

# Run Lilliefors Test
result <- lillie.test(x)
print(result)

# Estimate normal parameters
mu <- mean(x)
sigma <- sd(x)

# Create ECDF and fitted normal CDF
x_sorted <- sort(x)
n <- length(x_sorted)
ecdf_vals <- (1:n) / n
fitted_cdf <- pnorm(x_sorted, mean = mu, sd = sigma)

# Maximum gap
gap <- abs(ecdf_vals - fitted_cdf)
D_manual <- max(gap)

print(D_manual)

# Plot
plot(x_sorted, ecdf_vals, type = "s",
     main = "Lilliefors Test: ECDF vs Fitted Normal CDF",
     xlab = "Observed value",
     ylab = "Cumulative probability")
lines(x_sorted, fitted_cdf, col = "blue", lwd = 2)
legend("bottomright",
       legend = c("Empirical CDF", "Fitted Normal CDF"),
       col = c("black", "blue"),
       lwd = c(1, 2),
       bty = "n")

Excel Formulas for a Lilliefors-Style Manual Check

Assume sorted values are in A2:A650.

Sample size:
=COUNT(A2:A650)

Mean:
=AVERAGE(A2:A650)

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

Rank position in B2:
=ROW(A1)

Empirical CDF in C2:
=B2/COUNT($A$2:$A$650)

Fitted normal CDF in D2:
=NORM.DIST(A2,$Mean_Cell$,$SD_Cell$,TRUE)

Absolute gap in E2:
=ABS(C2-D2)

Lilliefors-style D statistic:
=MAX(E2:E650)

Decision:
Use SPSS, R, or Python for the formal Lilliefors p-value.
Use Excel mainly to understand and visualize the ECDF-CDF gap.

APA Reporting Wording for the Lilliefors Test

APA reporting for the Lilliefors Test should include the variable, D statistic, p-value, hypothesis decision, and a short interpretation. A strong report should also mention supporting visual evidence such as the histogram, ECDF chart, Q-Q plot, or P-P plot.

If the Lilliefors Test Is Significant

If the Lilliefors Test Is Not Significant

Student-Friendly Report Sentence

Common Mistakes in Lilliefors Test Interpretation

When to Use the Lilliefors Test

Use the Lilliefors Test when you need a normality test and the population mean and standard deviation are unknown. That is the normal situation in most real research datasets. It is especially useful before parametric tests, regression modeling, confidence interval reporting, or any analysis where normality is part of the assumption discussion.

If the Lilliefors Test rejects normality, do not panic automatically. Review sample size, plot shape, outliers, skewness, kurtosis, and the planned method. In some situations, transformation, robust methods, bootstrapping, or effect-size reporting may be more useful than forcing the data to look perfectly normal. For reporting strength, connect the normality result with effect size, confidence interval, and visual diagnostics.

Downloads and Resources for the Lilliefors Test

The SPSS output PDF below supports the Lilliefors Test workflow used in this guide. The chart URLs also provide the visual diagnostic evidence for distribution fit, ECDF comparison, D gaps, Monte Carlo reference distribution, p-value decision, variable comparison, and group comparison.

Related Internal Guides

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