Quick Answer: Kolmogorov Smirnov Test Result

The Kolmogorov Smirnov Test checks whether the observed distribution follows a specified theoretical distribution. In this guide, the main use is a normality check for a numeric student-performance variable such as G3 final grade. The null hypothesis says the data follow the specified normal distribution. The alternative hypothesis says the data do not follow that distribution. The main statistic is the KS D statistic, which is the largest vertical distance between the empirical CDF and the fitted theoretical CDF.

The visual evidence from the distribution fit chart, empirical CDF versus fitted normal CDF chart, D plus and D minus gap chart, Monte Carlo reference distribution and p-value decision chart supports the same interpretation: the data should not be treated as perfectly normal without checking the shape. When the p-value is below .05, the normality assumption is rejected. When the p-value is above .05, the test does not provide enough evidence to reject the selected distribution.

What Is the Kolmogorov Smirnov Test?

The Kolmogorov Smirnov Test is a nonparametric distribution-comparison test based on cumulative distribution functions. Instead of comparing only the mean or variance, the KS test compares the entire cumulative distribution shape. It checks the largest vertical gap between the observed empirical cumulative distribution function and the expected theoretical cumulative distribution function.

In a one-sample normality setting, the observed sample values are sorted from smallest to largest, an empirical CDF is built, and that empirical CDF is compared with the fitted normal CDF. The largest gap is called the D statistic. A larger D statistic means the observed distribution is farther from the theoretical distribution.

The Kolmogorov Smirnov Test is often used with other normality tools such as the Q-Q plot normality check, P-P plot normality check, Lilliefors test, D’Agostino-Pearson test, Cramer-von Mises test, and Ryan-Joiner test.

Kolmogorov Smirnov Test Formula

The main statistic for the Kolmogorov Smirnov Test is the maximum absolute distance between the empirical CDF and the theoretical CDF:

Here, F_n(x) is the empirical cumulative distribution function from the sample, and F(x) is the theoretical cumulative distribution function. For a normality test, F(x) is usually the fitted normal CDF.

D Plus and D Minus

The KS statistic can also be viewed through two directional gaps:

The final D statistic is the larger of these two directional gaps:

One-Sample vs Two-Sample Kolmogorov Smirnov Test

The Kolmogorov Smirnov Test has two common versions. The one-sample version compares one observed sample with a theoretical distribution, such as a normal distribution. The two-sample version compares two empirical distributions directly, such as male and female groups or two treatment groups.

Null and Alternative Hypotheses for Kolmogorov Smirnov Test

The hypotheses depend on whether the test is one-sample or two-sample. For normality testing, the null hypothesis says the data follow the selected normal distribution. For two-sample testing, the null hypothesis says the two groups come from the same distribution.

Dataset and Variables Used

The worked example uses the student performance dataset. The main variable is G3 final grade. The one-sample Kolmogorov Smirnov Test compares the empirical distribution of G3 with a fitted normal distribution. Additional charts compare KS D values across variables and show a two-group empirical CDF comparison.

Before interpreting KS output, review the distribution with descriptive statistics, frequency distribution, histogram interpretation, box plot interpretation, and the five-number summary.

SPSS Output Interpretation for Kolmogorov Smirnov Test

The SPSS output PDF verifies the Kolmogorov Smirnov Test workflow used in this guide. SPSS commonly reports the KS statistic, degrees of freedom or sample size context, significance value and normality-test decision. The output should be interpreted together with histograms, Q-Q plots, P-P plots and the ECDF comparison charts.

SPSS Interpretation Table

SPSS Decision Logic

If the SPSS p-value is below .05, the correct normality wording is: “The Kolmogorov Smirnov Test was significant, indicating that the distribution differed significantly from the fitted normal distribution.” If the p-value is above .05, the wording is: “The Kolmogorov Smirnov Test was not significant, so the test did not provide evidence against the fitted distribution.”

SPSS Final Interpretation

Python Chart-by-Chart Interpretation

The Python charts show the complete Kolmogorov Smirnov Test workflow. They include distribution fit, empirical CDF versus fitted normal CDF, D plus and D minus gaps, Monte Carlo reference distribution, p-value decision, KS D across variables and two-group empirical CDF comparison.

Python Chart 1: Distribution Fit for Kolmogorov Smirnov Test

This chart compares the observed distribution with the fitted normal curve. It helps readers understand the shape of the data before looking at the CDF-based statistic. If the histogram departs strongly from the normal curve, the KS test may detect a significant difference between the empirical distribution and the fitted normal distribution.

The distribution fit chart is useful because the KS D statistic alone does not show whether the problem is skewness, tail behavior, gaps, floor effects, ceiling effects or outliers. The histogram gives the practical shape explanation behind the formal test.

Python Chart 2: Empirical CDF vs Fitted Normal CDF

This is the central Kolmogorov Smirnov Test chart. The step-like line represents the empirical CDF from the observed data. The smooth line represents the fitted normal CDF. The KS statistic is the largest vertical gap between these two lines.

If the two CDFs stay close across the full range, the distribution fit is stronger. If the two lines separate visibly, especially around a particular region of the scale, the KS D statistic increases and the p-value becomes more likely to reject the fitted distribution.

Python Chart 3: D Plus and D Minus Gaps

This chart explains how the KS D statistic is formed. The test does not average all gaps between the empirical CDF and theoretical CDF. Instead, it finds the largest vertical gap. The two directional gaps, D+ and D−, show whether the empirical CDF is above or below the theoretical CDF at the most extreme points.

The larger of D+ and D− becomes the final D statistic. This chart is important because it makes the test decision visually transparent. Readers can see where the maximum distribution difference occurs rather than only reading a p-value.

Python Chart 4: Monte Carlo Reference Distribution

The Monte Carlo reference distribution shows how large the D statistic would usually be if samples came from the reference distribution. The observed D statistic is then compared with this simulated reference distribution. If the observed D falls far into the extreme tail, the p-value is small and the fitted distribution is rejected.

This chart is useful because it explains the p-value visually. Instead of treating the p-value as an abstract number, the chart shows whether the observed D statistic is typical or unusual under the null distribution.

Python Chart 5: P-value Decision

This chart summarizes the formal decision. If the p-value is less than .05, the null hypothesis is rejected and the data are judged inconsistent with the selected distribution. If the p-value is greater than .05, the test does not reject the selected distribution.

The p-value decision chart gives a quick answer, but it should not replace the ECDF chart. The ECDF chart explains why the p-value decision occurred and where the distribution difference is located.

Python Chart 6: KS D Across Variables

This chart compares the KS D statistic across several variables. A larger D value means the empirical distribution is farther from the fitted reference distribution. This is useful because normality is variable-specific. One variable may be close to normal while another variable may be strongly non-normal.

For example, grade variables and count variables often behave differently. Count variables such as absences may show stronger departures from normality because they are bounded at zero and can have a long right tail.

Python Chart 7: Two-Group Empirical CDF Comparison

This chart illustrates the two-sample version of the Kolmogorov Smirnov Test. Instead of comparing one sample with a theoretical distribution, it compares two empirical CDFs. The largest vertical gap between the two group CDFs becomes the two-sample KS statistic.

This chart is useful when the research question asks whether two groups have different distributions, not only different means. The two-sample KS test can detect differences in location, spread, skewness and distribution shape.

R Chart-by-Chart Validation

The R charts validate the Python and SPSS Kolmogorov Smirnov Test interpretation using a separate workflow. The same seven diagnostic views appear again: distribution fit, ECDF versus fitted normal CDF, D plus and D minus gaps, Monte Carlo reference distribution, p-value decision, KS D across variables and two-group empirical CDF comparison.

R Chart 1: Distribution Fit for Kolmogorov Smirnov Test

The R distribution-fit chart confirms the same visual check shown in Python. It compares the observed shape with the fitted normal reference and helps explain why the KS decision may reject or fail to reject normality.

R Chart 2: Empirical CDF vs Fitted Normal CDF

The R ECDF chart validates the main KS comparison. The largest distance between the empirical step function and the fitted normal CDF is the basis of the D statistic. This chart confirms that the test is a cumulative-distribution comparison rather than a mean comparison.

R Chart 3: D Plus and D Minus Gaps

The R D-gap chart validates the directional interpretation of the KS statistic. It shows whether the empirical CDF is above or below the theoretical CDF at the largest gap and confirms which directional difference determines the final D value.

R Chart 4: Monte Carlo Reference Distribution

The R Monte Carlo chart confirms how the observed D statistic is compared against simulated reference values. If the observed D is unusually large, the p-value becomes small and the test rejects the selected distribution.

R Chart 5: P-value Decision

The R p-value decision chart confirms the formal decision rule. It should be interpreted with the ECDF chart and distribution-fit chart because the p-value says whether the difference is statistically meaningful, while the charts explain the shape of the difference.

R Chart 6: KS D Across Variables

The R KS-D comparison confirms that variables differ in their closeness to the fitted normal distribution. This is important because the normality decision should be made variable by variable or residual set by residual set, not for the dataset as a whole.

R Chart 7: Two-Group Empirical CDF Comparison

The R two-group ECDF chart validates the group-comparison version of the test. The distance between the two empirical CDFs shows whether the groups differ in distribution shape, spread, location or tail behavior.

SPSS, R, Python and Excel Workflows for Kolmogorov Smirnov Test

The Kolmogorov Smirnov Test can be performed in SPSS, R, Python and Excel-style workflows. The key steps are to sort the observed values, build the empirical CDF, define the theoretical CDF, calculate the maximum vertical gap, and use a reference distribution or software p-value to make the decision.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Kolmogorov Smirnov Test

SPSS Syntax for Kolmogorov Smirnov Test

* Kolmogorov Smirnov Test normality workflow in SPSS.
* Main variable: G3 final grade.

TITLE "Kolmogorov Smirnov Test Normality Check".

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="Kolmogorov-Smirnov-Test-SPSS-Output.pdf".

Python Code for Kolmogorov Smirnov Test

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

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

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

mu = np.mean(x)
sigma = np.std(x, ddof=1)

# One-sample KS test against fitted normal distribution
ks_result = stats.kstest(x, "norm", args=(mu, sigma))

print("KS D statistic:", ks_result.statistic)
print("KS p-value:", ks_result.pvalue)

if ks_result.pvalue < 0.05:
    print("Reject normality")
else:
    print("Do not reject normality")

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

gaps = np.abs(empirical_cdf - normal_cdf)
D = np.max(gaps)

print("Manual D:", D)

# D plus and D minus
d_plus = np.max(empirical_cdf - normal_cdf)
d_minus = np.max(normal_cdf - (np.arange(0, n) / n))

print("D plus:", d_plus)
print("D minus:", d_minus)

# Two-sample KS example by sex
if "sex" in df.columns:
    female = pd.to_numeric(df.loc[df["sex"] == "F", "G3"], errors="coerce").dropna()
    male = pd.to_numeric(df.loc[df["sex"] == "M", "G3"], errors="coerce").dropna()

    two_sample = stats.ks_2samp(female, male)
    print("Two-sample KS D:", two_sample.statistic)
    print("Two-sample KS p-value:", two_sample.pvalue)

R Code for Kolmogorov Smirnov Test

# Kolmogorov Smirnov Test in R

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

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

mu <- mean(x)
sigma <- sd(x)

# One-sample KS test against fitted normal distribution
ks_result <- ks.test(x, "pnorm", mean = mu, sd = sigma)

print(ks_result)

# Empirical CDF and fitted normal CDF
x_sorted <- sort(x)
n <- length(x_sorted)

empirical_cdf <- (1:n) / n
normal_cdf <- pnorm(x_sorted, mean = mu, sd = sigma)

gaps <- abs(empirical_cdf - normal_cdf)
D_manual <- max(gaps)

print(D_manual)

# D plus and D minus
d_plus <- max(empirical_cdf - normal_cdf)
d_minus <- max(normal_cdf - ((0:(n-1)) / n))

print(d_plus)
print(d_minus)

# Two-sample KS example by sex
if ("sex" %in% names(df)) {
  female <- na.omit(as.numeric(df[df$sex == "F", "G3"]))
  male <- na.omit(as.numeric(df[df$sex == "M", "G3"]))

  two_sample_result <- ks.test(female, male)
  print(two_sample_result)
}

Excel Formula Block for Kolmogorov Smirnov Test

Assume sorted G3 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)

KS D statistic:
=MAX(E2:E650)

Decision:
Compare D with a KS critical value table or use statistical software for the exact p-value.

Chart:
Plot C2:C650 and D2:D650 against A2:A650 to compare empirical CDF and fitted normal CDF.

APA Reporting Wording for Kolmogorov Smirnov Test

APA reporting for the Kolmogorov Smirnov Test should include the test name, variable, D statistic, p-value, decision and visual support. If the parameters of the reference normal distribution were estimated from the sample, mention that normality was assessed using a fitted normal reference and supporting plots.

APA-Style Normality Report

If the KS Test Is Significant

If the KS Test Is Not Significant

Common Mistakes in Kolmogorov Smirnov Test Interpretation

When to Use Kolmogorov Smirnov Test

Use the Kolmogorov Smirnov Test when you need to compare an observed distribution with a theoretical distribution or compare two empirical distributions. It is most useful when the research question is about distribution shape, not only central tendency.

Use the Kolmogorov Smirnov Test with other normality and distribution-shape tools. Helpful related guides include Q-Q plot normality check, P-P plot normality check, Lilliefors test, D’Agostino-Pearson test, Cramer-von Mises test, and Ryan-Joiner test.

Downloads and Resources for Kolmogorov Smirnov Test

The SPSS output PDF below verifies the Kolmogorov Smirnov Test workflow used in this guide. Use it as the supporting output file for SPSS interpretation, KS D statistic review, normality decision, empirical CDF explanation and reporting.

Related Statistical Guides

Use these related guides to connect the Kolmogorov Smirnov Test with normality checks, descriptive statistics, distribution plots, transformations, assumption tests and reporting.