What is the Kolmogorov-Smirnov (K-S) critical value table?

The Kolmogorov-Smirnov critical value table gives the threshold value D_critical that the K-S test statistic D must exceed to reject the null hypothesis. D is the maximum absolute difference between the empirical distribution function and the theoretical CDF (one-sample) or between two empirical CDFs (two-sample). If D ≥ D_critical, you reject H₀ at the chosen significance level α.

How do you use the K-S critical value table?

To use the K-S table: (1) Collect your data and sort it in ascending order. (2) Compute the empirical distribution function F_n(x). (3) Compute D = max|F_n(x) − F₀(x)| for each data point. (4) Record your sample size n. (5) Select your significance level α (typically 0.05). (6) Find the row for n in the table and read D_critical from the α column. (7) If D ≥ D_critical, reject H₀ — the data do not fit the reference distribution.

What is the K-S test statistic D?

The K-S test statistic D is the maximum absolute vertical distance between the empirical distribution function F_n(x) and the theoretical cumulative distribution function F₀(x). Formally, D = sup_x |F_n(x) − F₀(x)|. A larger D means a greater discrepancy between the observed data and the reference distribution. D ranges from 0 (perfect fit) to 1 (complete mismatch).

What is the difference between the one-sample and two-sample K-S test?

The one-sample K-S test compares a single sample's empirical distribution function against a fully specified theoretical distribution (such as normal, uniform, or exponential) to test goodness of fit. The two-sample K-S test compares the empirical distribution functions of two independent samples to determine whether they were drawn from the same population distribution. Different critical value tables apply to each version.

Is the Kolmogorov-Smirnov test a normality test?

Yes, the K-S test can be used as a normality test by comparing sample data against a normal distribution with population parameters fully specified. However, when the mean and variance are estimated from the sample itself, the Lilliefors test (a corrected version of the K-S test with separate critical values) should be used instead. For general normality testing with estimated parameters, the Shapiro-Wilk test is usually preferred for small to medium samples (n < 50).

What are the assumptions of the Kolmogorov-Smirnov test?

The K-S test assumes: (1) The data are independently and identically distributed. (2) The reference distribution is fully specified before seeing the data (for the one-sample test) — parameters cannot be estimated from the sample. (3) The distribution is continuous — the standard K-S test is not valid for discrete distributions. (4) For the two-sample test, both samples must be independent.

What is the decision rule for the K-S test?

The K-S test decision rule is: if D ≥ D_critical (from the table at your chosen α and n), reject the null hypothesis H₀ that the data follow the reference distribution. If D < D_critical, fail to reject H₀. Equivalently, when software reports a p-value, reject H₀ if p ≤ α.

K-S test vs Shapiro-Wilk test: which should I use?

For testing normality specifically with small to medium samples (n < 50), the Shapiro-Wilk test is generally more powerful than the K-S test and is preferred in most statistical guidelines. The K-S test is more appropriate when testing against a non-normal theoretical distribution (such as exponential or uniform), when comparing two independent samples, or when a distribution-free approach is needed for any continuous distribution. For large samples, both tests become very sensitive to minor departures from normality.

What is the large-sample approximation formula for K-S critical values?

For large samples (n > 35), the K-S critical value is approximated using: D_critical = K_α / √n, where K_α is the Kolmogorov distribution constant for your significance level: K_α = 1.224 (α = 0.10), 1.358 (α = 0.05), 1.517 (α = 0.02), 1.628 (α = 0.01). For example, at n = 100 and α = 0.05: D_critical = 1.358 / √100 = 0.1358.

What is the empirical distribution function (EDF)?

The empirical distribution function (EDF) F_n(x) is the sample analogue of the theoretical CDF. For a sorted sample x₁ ≤ x₂ ≤ ... ≤ x_n, the EDF at each data point is F_n(x_i) = i/n — the proportion of observations less than or equal to x_i. The K-S test measures how far this step function deviates from the theoretical CDF at every point.

Kolmogorov-Smirnov Table: Critical Values, Decision Rules & How to Use It

K-S Critical Value Calculator

Sample Size (n)

Significance Level (α)

Test Type

D critical =

—

What Is the Kolmogorov-Smirnov (K-S) Table?

The Kolmogorov-Smirnov table is a reference for the critical values of the K-S test statistic D. This nonparametric test, developed independently by Soviet mathematicians Andrey Kolmogorov (1933) and Nikolai Smirnov (1948), measures how far a set of observations deviates from a specified theoretical distribution — or how far two samples deviate from each other.

The test statistic D is the maximum absolute vertical distance between the empirical distribution function and the reference distribution. A small D means the data match the theoretical distribution closely. A large D signals departure. The critical value from the table is the threshold: if D reaches or exceeds it, you reject the null hypothesis of good fit at your chosen significance level.

One-paragraph definition: The Kolmogorov-Smirnov test is a distribution-free goodness-of-fit test that quantifies the largest discrepancy between an observed empirical distribution function and a theoretical cumulative distribution function (one-sample), or between two empirical distribution functions (two-sample). The K-S critical value table gives the minimum D required to reject H₀ at each α level. Reject H₀ if D ≥ D_critical.

Kolmogorov-Smirnov Critical Value Table

All values below are D_critical — the minimum K-S statistic for significance at each α level. Select the tab for your test type. Click any cell to highlight and load it into the calculator above. For n > 35, use the large-sample approximation formula at the bottom of this section.

Reject H₀ if D ≥ D_critical. Exact values for n = 1–35. For n > 35, use the large-sample approximation: D_critical = K_α / √n, where K_α = 1.224 (α=0.10), 1.358 (α=0.05), 1.517 (α=0.02), 1.628 (α=0.01). Values derived from Massey (1951), Miller (1956), and the Kolmogorov distribution.

Large-Sample Approximation (n > 35)

D_critical = K_α / √n

α	K_α	n = 50	n = 100	n = 200
0.10	1.224	0.1731	0.1224	0.0866
0.05	1.358	0.1921	0.1358	0.0960
0.02	1.517	0.2145	0.1517	0.1072
0.01	1.628	0.2302	0.1628	0.1151

How the Kolmogorov-Smirnov Test Works

The K-S test operates by comparing how a dataset's cumulative pattern relates to what a theoretical distribution predicts. Every step below maps directly to a specific calculation or table lookup.

Step 1 — State Hypotheses

The null hypothesis is H₀: the data follow distribution F₀(x). The alternative is H₁: the data do not follow F₀(x). For the two-sample test: H₀: both samples come from the same population distribution; H₁: they do not. Set α before looking at the data — typically α = 0.05 in most research contexts.

Step 2 — Sort Data and Compute the EDF

Sort the n observations in ascending order: x₁ ≤ x₂ ≤ ... ≤ x_n. The empirical distribution function at each point is:

F_n(x_i) = i / n

This is the step function that rises by 1/n at each observation. It estimates the true CDF of the population from which your data were drawn.

Step 3 — Compute the K-S Statistic D

For each observation, take the absolute difference between the EDF and the theoretical CDF. D is the largest of these differences:

D = max_i { |F_n(x_i) − F₀(x_i)|, |F_n(x_i−1) − F₀(x_i)| }

Both terms are checked to capture the maximum gap on either side of each step in the EDF. D lies between 0 (perfect fit) and 1 (complete mismatch).

Step 4 — Look Up D_critical

Find the row for your sample size n in the K-S table. Read D_critical from the column for your chosen α. If your n falls between two listed values, use the smaller n (the conservative choice), or apply the large-sample formula for n > 35.

Step 5 — Apply the Decision Rule

If D ≥ D_critical → Reject H₀ → Data do not fit the reference distribution
If D < D_critical → Fail to reject H₀ → No significant departure from the reference distribution

Step 6 — Report the Result

State D, n, α, D_critical, and your conclusion. Example: D(20) = 0.183, D_critical(α=0.05) = 0.294, fail to reject H₀ — no significant departure from normality detected. In software output, compare the reported p-value directly: reject H₀ if p ≤ α.

Worked Example: K-S Test on Exam Scores

Scenario: A statistics instructor wants to know whether the exam scores of 10 students follow a uniform distribution on [50, 100]. The sorted scores are: 53, 58, 64, 67, 71, 76, 82, 85, 90, 96. Test at α = 0.05.

Computation Table

i	x_i	F_n(x_i) = i/n	F₀(x_i) = (x − 50)/50	\|F_n − F₀\|
1	53	0.100	0.060	0.040
2	58	0.200	0.160	0.040
3	64	0.300	0.280	0.020
4	67	0.400	0.340	0.060
5	71	0.500	0.420	0.080
6	76	0.600	0.520	0.080
7	82	0.700	0.640	0.060
8	85	0.800	0.700	0.100
9	90	0.900	0.800	0.100
10	96	1.000	0.920	0.080

Summary of Steps

Step	Action	Result
1	State hypotheses	H₀: Uniform[50,100]; α = 0.05
2	Record n	n = 10
3	Compute D statistic	D = max\|F_n − F₀\| = 0.100
4	Look up D_critical	n=10, α=0.05 → D_critical = 0.409
5	Apply decision rule	0.100 < 0.409 → Fail to Reject H₀

Interpretation

D = 0.100 is well below D_critical = 0.409 at α = 0.05 with n = 10. There is no statistically significant evidence that the exam scores depart from a uniform distribution on [50, 100]. Note that with only 10 observations, the K-S test has limited power — a genuine departure from uniformity might go undetected. A larger sample would provide a more sensitive test.

One-Sample vs Two-Sample K-S Test

The two versions of the K-S test address different research questions. Choosing the right one determines which critical value table to use.

One-Sample K-S Test

Compares one sample's empirical CDF against a fully specified theoretical distribution. The reference distribution — including all parameters — must be specified before seeing the data. Common uses: testing whether data follow a normal, uniform, exponential, or Poisson distribution with known parameters.

D = max_x |F_n(x) − F₀(x)|

Two-Sample K-S Test

Compares the empirical CDFs of two independent samples to test whether they were drawn from the same underlying distribution. No theoretical distribution is specified — the test is sensitive to differences in shape, spread, and location between the two groups.

D = max_x |F_m(x) − F_n(x)|

Important: Parameters Must Not Be Estimated from the Same Data

The standard K-S critical values assume the theoretical distribution is fully specified before data collection. If you estimate the mean and variance from your sample data and then test whether those same data fit a normal distribution with those estimated parameters, the standard K-S table is no longer valid — it gives p-values that are too large, making rejection too rare. Use the Lilliefors test instead, which has its own separate critical value table specifically for this situation.

K-S Test vs Other Goodness-of-Fit Tests

Choosing among goodness-of-fit tests depends on sample size, the distribution being tested, and whether parameters are estimated. The table below shows the main considerations for each method.

Test	Best For	Normality?	Two-Sample?	Parametric?	Power (small n)
Kolmogorov-Smirnov	Any continuous dist., two-sample	Known params only	Yes	No	Moderate
Shapiro-Wilk	Normality, small–medium n	Yes (optimal)	No	No	High
Anderson-Darling	Normality, sensitive to tails	Yes	Limited	No	High
Lilliefors	Normality, estimated parameters	Yes (estimated μ, σ)	No	No	Moderate
Chi-Square GOF	Discrete distributions, large n	With caution	No	No	Low

For normality testing with estimated parameters and n < 50, the Shapiro-Wilk test is generally the most reliable choice. See the normality tests guide on Statistics Fundamentals for a full comparison.

Running the K-S Test in R, Python, and SPSS

Statistical software computes D and the associated p-value automatically, so you compare p against α directly rather than using the critical value table. The table remains useful for manual computation and for verifying software output.

R — ks.test()

# One-sample K-S test against normal distribution
x <- c(53, 58, 64, 67, 71, 76, 82, 85, 90, 96)
ks.test(x, "pnorm", mean = 70, sd = 15)

# Two-sample K-S test
ks.test(group1, group2)

Python — scipy.stats.kstest()

from scipy import stats

# One-sample K-S test (normal, known params)
data = [53, 58, 64, 67, 71, 76, 82, 85, 90, 96]
result = stats.kstest(data, 'norm', args=(70, 15))
print(result.statistic, result.pvalue)

# Two-sample K-S test
stats.ks_2samp(group1, group2)

SPSS — Nonparametric Tests

In SPSS: Analyze → Nonparametric Tests → Legacy Dialogs → 1-Sample K-S. Select the variable, choose the test distribution (Normal, Uniform, Poisson, or Exponential), and run. SPSS reports the K-S statistic, degrees of freedom, and asymptotic significance (p-value). Compare p directly against your α level.

Common Mistakes When Using the K-S Table

Estimating parameters from the same sample

The most frequent error: computing the sample mean and variance, then using those estimates in the K-S test against a normal distribution with those parameters. The standard K-S table does not apply in this case. Use the Lilliefors table or the Shapiro-Wilk test instead.

Using the one-sample table for a two-sample test

The critical values for the two-sample test differ from the one-sample table, particularly for unequal sample sizes. For unequal n₁ and n₂, the effective sample size formula is n_eff = (n₁ × n₂) / (n₁ + n₂). Using the wrong table gives incorrect decisions.

Applying the K-S test to discrete data

The standard K-S test assumes a continuous distribution. Discrete data (counts, ranked categories) create ties in the empirical CDF that violate this assumption. The resulting p-values are conservative — real departures may go undetected. The chi-square goodness-of-fit test is generally more appropriate for discrete distributions.

Confusing failing to reject H₀ with proving good fit

A non-significant K-S result means only that this sample did not provide enough evidence to rule out the reference distribution. It does not confirm the data follow that distribution. With small samples, the test has limited power and may miss real departures.

Symbol and Entity Glossary

The table below defines every symbol and term used in K-S test calculations, along with the most common source of confusion for each.

Symbol	Name	Definition	Common Error
D	K-S test statistic	Maximum absolute difference between EDF and theoretical CDF; 0 ≤ D ≤ 1	Not checking both \|i/n − F₀\| and \|(i−1)/n − F₀\| at each point
F_n(x)	Empirical distribution function	Proportion of sample observations ≤ x; F_n(x_i) = i/n	Forgetting to sort data before computing EDF
F₀(x)	Theoretical CDF	Cumulative distribution function of the reference distribution, fully specified before data collection	Estimating parameters from sample data
D_critical	Critical value	Threshold from the K-S table; reject H₀ if D ≥ D_critical	Using one-sample table for two-sample test
n	Sample size	Number of observations; determines which row to use in the K-S table	Applying exact table beyond n = 35 without the large-sample formula
α	Significance level	Probability of Type I error; chooses which column of the K-S table to use	Setting α after seeing D (data dredging)
K_α	Kolmogorov constant	Constant from the Kolmogorov distribution used in the large-sample approximation D_crit = K_α/√n	Using approximation for n ≤ 35
p-value	Probability value	Probability of observing a D ≥ computed D if H₀ is true; reject H₀ if p ≤ α	Interpreting p as the probability H₀ is true

Quick Reference: Most Commonly Used K-S Critical Values

These are the most frequently referenced values for the one-sample K-S test at α = 0.05 and α = 0.01 — the standard levels in most applied and academic research.

n	D_critical (α = 0.10)	D_critical (α = 0.05)	D_critical (α = 0.02)	D_critical (α = 0.01)
5	0.565	0.627	0.669	0.669
10	0.369	0.409	0.457	0.490
15	0.304	0.338	0.376	0.404
20	0.265	0.294	0.329	0.352
25	0.238	0.264	0.295	0.317
30	0.218	0.242	0.270	0.290
35	0.202	0.224	0.251	0.269

One-sample K-S test. Values from Massey (1951). For n > 35, use D_critical = K_α/√n.

Applied Examples Across Research Fields

The following examples show how the K-S test addresses real distribution questions across several fields. Each uses a different sample size to show how n changes the critical value.

Manufacturing: Testing Component Dimensions

A quality engineer measures 20 metal rods and wants to verify that diameters follow N(10mm, 0.1mm²) based on design specifications. The computed D = 0.187. With n = 20 and α = 0.05, D_critical = 0.294. Since 0.187 < 0.294, the engineer fails to reject H₀ — dimensions are consistent with the specified normal distribution.

Finance: Testing Return Distributions

A portfolio analyst tests whether 30 monthly returns follow a specified normal distribution (based on historical long-run parameters). The computed D = 0.261. With n = 30, α = 0.05, D_critical = 0.242. Since 0.261 > 0.242, H₀ is rejected — the returns show a statistically significant departure from the specified normal model.

Healthcare: Two-Sample Clinical Comparison

Researchers compare recovery times for two treatment groups (n₁ = 25, n₂ = 25). The two-sample K-S statistic D = 0.320. With effective sample size n_eff = (25×25)/(25+25) = 12.5 ≈ 12, and α = 0.05, D_critical ≈ 0.375. Since 0.320 < 0.375, the distributions are not significantly different at this sample size.

Machine Learning: Checking Data Drift

Data scientists use the two-sample K-S test to detect distribution shift between training and production data. A large D between the training feature distribution and the current production distribution signals that model retraining may be necessary. This is implemented in libraries like SciPy, Evidently AI, and WhyLabs.

Frequently Asked Questions

Can I use the K-S test for a normal distribution with estimated mean and variance?

No — not with the standard K-S table. When parameters are estimated from the same data, use the Lilliefors test. Lilliefors critical values are smaller than standard K-S critical values because estimated parameters improve fit artificially, requiring a higher D to achieve significance. If both μ and σ are specified from theory or prior data, the standard K-S table applies. See the normality tests comparison for guidance on choosing between K-S, Lilliefors, and Shapiro-Wilk.

What sample size does the K-S test work best with?

The K-S test has adequate power for moderate samples (n ≈ 25–100) when testing against well-specified distributions. For very small samples (n < 15), the critical values are large and the test may miss real departures. For large samples (n > 200), it becomes very sensitive — even trivial, practically meaningless deviations from the reference distribution will produce significant results. In practice, graphical assessment with a Q-Q plot alongside the K-S test gives a more complete picture.

How is the K-S test different from the chi-square goodness-of-fit test?

The chi-square test groups data into bins and compares counts — it works for both discrete and continuous data, but bin choice affects results. The K-S test uses the full continuous distribution without binning, so it does not lose information. The K-S test is generally more powerful for continuous data with small samples. The chi-square test requires larger samples (expected count ≥ 5 per cell) and is more appropriate for discrete distributions. For a full comparison, see the chi-square test guide.

Does the K-S test work for all distributions?

The K-S test applies to any fully specified continuous distribution — normal, uniform, exponential, logistic, Weibull, and others. It does not work for discrete distributions (the D statistic is not distribution-free in that case). For distributions with unknown parameters, specialized tests exist: Lilliefors for normality with estimated parameters, and Anderson-Darling with its own critical values for several common continuous distributions.

What does a K-S p-value of 0.03 mean?

A p-value of 0.03 means that if the data truly followed the reference distribution, there would be only a 3% chance of observing a K-S statistic as large as the one computed. At α = 0.05, you reject H₀ — the observed difference between the sample distribution and the reference distribution is too large to attribute to sampling variation at this significance level. The p-value does not indicate how different the distributions are, only that the difference is statistically significant.

References and Further Reading

Massey, F. J., Jr. (1951). The Kolmogorov-Smirnov test for goodness of fit. Journal of the American Statistical Association, 46(253), 68–78. — The original published table of K-S critical values.

NIST/SEMATECH e-Handbook of Statistical Methods. Section 1.3.5.16: Kolmogorov-Smirnov Goodness-of-Fit Test. National Institute of Standards and Technology. itl.nist.gov — U.S. government statistical reference covering K-S test methodology, assumptions, and examples.

SciPy documentation: scipy.stats.kstest. docs.scipy.org — Implementation reference for one-sample and two-sample K-S tests in Python, with parameter definitions and examples.

Conover, W. J. (1999). Practical Nonparametric Statistics (3rd ed.). Wiley. — Comprehensive reference for the K-S test, Lilliefors correction, and related nonparametric methods including worked examples and tables.

Penn State STAT 415: Introduction to Mathematical Statistics. Lesson 14: Goodness-of-Fit Tests. online.stat.psu.edu — Open-access course covering the K-S test, chi-square GOF, and Lilliefors test with detailed theoretical grounding.

Related Statistical Tables & Resources

📋

Anderson-Darling Table Alternative goodness-of-fit test

🧪

Normality Tests Guide K-S, Shapiro-Wilk, Lilliefors compared

📐

Chi-Square GOF Test Alternative for discrete distributions

⚖️

Parametric vs Nonparametric Choosing the right test framework

🎓

Hypothesis Testing Guide Full framework for significance testing

🏠

Statistics Fundamentals Tables, calculators & complete guides

How to Use This Table

1Sort your data, compute the EDF

2Compute D = max|F_n − F₀|

3Record your sample size n

4Choose α, find row for n in table

5If D ≥ D_critical → reject H₀

Quick Reference (α = 0.05, One-Sample)

n = 5 0.627

n = 10 0.409

n = 15 0.338

n = 20 0.294

n = 25 0.264

n = 30 0.242

n = 35 0.224

Key Formulas

D = max |F_n(x) − F₀(x)|
F_n(x_i) = i / n
D_crit = K_α / √n (n > 35)
Reject H₀ if D ≥ D_critical

Statistics Fundamentals

Tables, calculators & complete guides

More Tables

→ Anderson-Darling Table → Chi-Square Table → Mann-Whitney U Table → Wilcoxon Signed-Rank Table → T-Distribution Table → Z-Table

What the K-S Table Tells You — and What It Doesn't

Why D_critical Decreases as n Increases

With more data, the empirical distribution function converges more tightly to the true population CDF. Small genuine deviations become detectable. This is why a K-S test with n = 100 can reject H₀ at D = 0.15, while the same D with n = 10 would not reach significance. The critical value encodes the precision-to-sample-size tradeoff inherent in any statistical test.

K-S Table vs p-value from Software

Statistical software (R, Python, SPSS) reports exact or approximate p-values. The table approach and the p-value approach give identical decisions: rejecting when D ≥ D_critical (table) is the same as rejecting when p ≤ α (software). The table is useful when software is unavailable, for manual verification, or for understanding exactly what threshold applies at your sample size and α level. For more on p-values, see the hypothesis testing section.

Distribution-Free Does Not Mean Assumption-Free

"Distribution-free" means the K-S test's sampling distribution under H₀ does not depend on the specific distribution being tested — it applies to any continuous distribution. It does not mean the test has no assumptions. Independence, continuity, and the critical requirement that parameters be specified without using the sample data all still apply. Violating these assumptions invalidates the critical values in the table. Review the parametric vs nonparametric tests guide for context.