What is the Shapiro-Wilk critical value for n = 10?

The Shapiro-Wilk critical value for n = 10 at α = 0.05 is W critical = 0.842. If your calculated W is below 0.842, reject the null hypothesis of normality at the 5% significance level.

What is the Shapiro-Wilk critical value for n = 20?

The Shapiro-Wilk critical value for n = 20 at α = 0.05 is W critical = 0.905. Reject H₀ if your calculated W is below 0.905.

What is the Shapiro-Wilk critical value for n = 50?

The Shapiro-Wilk critical value for n = 50 at α = 0.05 is W critical = 0.947. With larger samples, the critical value is closer to 1.0, making the test more sensitive to small departures from normality.

Why does a lower W statistic indicate non-normality?

The Shapiro-Wilk W statistic is bounded between 0 and 1. A W of exactly 1 would mean the data fit a perfectly normal distribution. Departures from normality cause W to decrease toward 0. This reverses the usual rejection logic: unlike most test statistics where large values lead to rejection, in the Shapiro-Wilk test a small W leads to rejection.

What does failing to reject H₀ in the Shapiro-Wilk test mean?

Failing to reject H₀ means the data do not provide sufficient evidence against normality at the chosen α level. This does not prove the data are normally distributed — it means normality cannot be ruled out based on this sample. The test's power depends heavily on sample size, so small samples may fail to detect real departures from normality.

Is the Shapiro-Wilk test better than Kolmogorov-Smirnov?

For testing normality, the Shapiro-Wilk test generally has greater power than the Kolmogorov-Smirnov test, particularly for small to moderate samples. The K-S test is more general (it can test against any specified distribution), while Shapiro-Wilk is specifically designed for normality. Both tests have limitations, and graphical diagnostics such as Q-Q plots should accompany formal tests.

Can the Shapiro-Wilk test be used for large samples?

Historical printed tables for the Shapiro-Wilk test typically covered n ≤ 50. Modern software implementations, including R's shapiro.test() and Python's scipy.stats.shapiro(), can handle larger samples. However, with very large n, the test may flag trivially small departures from normality as significant. For large samples, graphical tools such as Q-Q plots and histograms are often more informative.

What is the difference between the Shapiro-Wilk weight table and the critical value table?

The Shapiro-Wilk coefficient table (weights table) contains the aᵢ values used in the calculation of W itself. The critical value table contains the W threshold values used to make the hypothesis testing decision. These are two different tables serving two different purposes. This page provides the critical value table.

How do I report Shapiro-Wilk results in a research paper?

Report the W statistic, sample size, and p-value. Example using the p-value approach: 'The Shapiro-Wilk test indicated the data were consistent with normality, W = 0.962, p = .231.' Example using the critical value approach: 'The Shapiro-Wilk statistic was W = 0.930 (n = 20). Because W = 0.930 exceeded the critical value of 0.905 at α = 0.05, we failed to reject the null hypothesis of normality.'

What is α = 0.05 in the context of the Shapiro-Wilk test?

α = 0.05 is the significance level, meaning you accept a 5% probability of incorrectly rejecting H₀ when the data actually are normally distributed (Type I error). At α = 0.05, you use the corresponding column of W critical values. Most published Shapiro-Wilk tables focus on α = 0.05 and α = 0.01.

What should I do if my sample size is not in the table?

If your sample size falls between listed values, do not interpolate casually without understanding the methodology. The safest approach is to use modern statistical software such as R or Python/SciPy, which compute exact p-values for the Shapiro-Wilk test across a wider range of sample sizes.

Shapiro-Wilk Critical Value Table: W Critical Values, Sample Sizes, and Interpretation

What Is the Shapiro-Wilk Critical Value Table?

The Shapiro-Wilk critical value table lists threshold values of the W statistic for specific sample sizes and significance levels. A calculated W below the corresponding critical value provides evidence against the null hypothesis of normality. Modern statistical software (R, Python/SciPy, SPSS) typically reports a p-value directly, so researchers often do not need to consult printed critical-value tables — but the table remains essential for manual verification and understanding the test's logic.

The Shapiro-Wilk Decision Rule (Do Not Reverse This)
The Shapiro-Wilk test uses the opposite logic of most statistical tests — a smaller W leads to rejection:

W < W_critical

Reject H₀ — evidence against normality

W ≥ W_critical

Fail to reject H₀ — consistent with normality

H₀: The data are consistent with a normal distribution. H₁: The data are not consistent with normality. Failing to reject H₀ does not prove normality.

Shapiro-Wilk W Critical Value Lookup

Sample Size (n)

Significance Level (α)

Your W (optional)

W critical =

—

Shapiro-Wilk Critical Value Table (α = 0.05)

The table below provides W critical values at α = 0.05 for sample sizes n = 3 to 50. Select the tab to switch between significance levels. Click any row to load values into the lookup tool above.

Shapiro-Wilk W Critical Values
n	W critical	Reject H₀ if W <

Source note: Critical values are taken from Shapiro, S. S., & Wilk, M. B. (1965), Biometrika, 52(3/4), 591–611, and cross-referenced with Royston, P. (1995), Applied Statistics, 44(4), 547–551. Values for n = 3–50 at α = 0.05. Critical values can vary slightly among published tables due to rounding or interpolation method; use this table consistently with its cited source. Click any row to highlight it.

How to Use the Shapiro-Wilk Critical Value Table

Find your sample size row. Locate n in the leftmost column of the table. If your exact n is not listed, use software to get the exact p-value.

Choose your significance level. Most researchers use α = 0.05. Select the appropriate tab above to view the matching critical values.

Read W critical from the table. The value in the W critical column for your n is the threshold.

Compare W calculated to W critical. If W calculated < W critical, reject H₀. If W calculated ≥ W critical, fail to reject H₀.

Confirm with graphical checks. Always inspect a Q-Q plot or histogram alongside the formal test result.

Quick-Reference Critical Values at α = 0.05

n = 10

0.842

W_critical at α = 0.05

n = 20

0.905

W_critical at α = 0.05

n = 30

0.927

W_critical at α = 0.05

n = 50

0.947

W_critical at α = 0.05

Shapiro-Wilk Decision Examples (Hypothetical W values)

The table below shows how the decision is reached for various combinations of n and W calculated. W calculated values are illustrative examples.

n	W_critical (α = 0.05)	W_calculated *	Comparison	Decision
10	0.842	0.810	0.810 < 0.842	Reject H₀
10	0.842	0.870	0.870 ≥ 0.842	Fail to reject H₀
20	0.905	0.890	0.890 < 0.905	Reject H₀
20	0.905	0.930	0.930 ≥ 0.905	Fail to reject H₀

* W_calculated values shown here are hypothetical examples for illustration only. Failing to reject H₀ does not prove normality.

What Is the Shapiro-Wilk Test?

The Shapiro-Wilk test, introduced by Samuel Shapiro and Martin Wilk in 1965, is a formal statistical test for assessing whether a sample is consistent with having been drawn from a normally distributed population. It is widely regarded as one of the most powerful tests for this purpose, particularly for small to moderate sample sizes.

The test produces the W statistic, which measures the correspondence between the observed ordered sample values and the values expected under normality. W is bounded between 0 and 1, with values closer to 1 indicating better agreement with a normal distribution.

H₀: The data are consistent with a normally distributed population, under the assumptions of the test.
H₁: The data are not consistent with normality.

These are careful statistical statements. The test provides evidence for or against normality — not absolute proof in either direction.

Normality assumptions matter for many statistical procedures — including the t-test, ANOVA, and simple linear regression — though those assumptions typically concern model residuals rather than raw variable distributions. Sample size plays a central role: with very small n, the test has limited power to detect departures from normality, while with very large n, it may flag trivially minor deviations.

What Does the Shapiro-Wilk W Statistic Mean?

W is a ratio that compares two estimates of the population variance. In the numerator, the ordered sample values are weighted using special coefficients aᵢ derived from the expected order statistics of a normal distribution. In the denominator, the standard sum of squared deviations from the mean appears. When the data closely match the expected normal pattern, these two estimates agree, and W approaches 1. When the data deviate from normality, the ratio decreases.

W = ( Σ aᵢ · x(i) )² / Σ ( x(i) − x̄ )²

Where x(i) is the i-th ordered observation, x̄ is the sample mean, and aᵢ are the Shapiro-Wilk coefficients for sample size n. The coefficients are tabulated values — not the same table as W critical values.

W → 1

Data more consistent with normality

W → 0

Stronger evidence of departure from normality

W should not be interpreted in isolation. Sample size, the p-value or critical value, the distribution's shape, and graphical diagnostics such as Q-Q plots and histograms all contribute to a complete picture.

Why Does a Lower W Indicate Non-Normality?

Most hypothesis tests use large test statistics to indicate departures from H₀. The Shapiro-Wilk test is different because W is constructed as a goodness-of-fit measure, where a perfect fit to normality corresponds to the maximum possible value. When the sample departs from normality, the weighted sum in the numerator no longer captures the variance as efficiently, and W drops below 1.

Reject H₀ ⟺ W < W_critical
Fail to reject H₀ ⟺ W ≥ W_critical

This direction is specific to the structure of the W statistic. Confusing it with the usual "large statistic leads to rejection" logic is one of the most common errors when using Shapiro-Wilk tables manually. The lookup tool at the top of this page explicitly shows whether your W leads to rejection.

Shapiro-Wilk Critical Values at Different Alpha Levels

The significance level α controls the threshold for rejection. Lowering α (e.g., from 0.05 to 0.01) makes the test more conservative — it rejects normality less readily, requiring stronger evidence. At α = 0.10, the test is more liberal.

n	α = 0.10	α = 0.05	α = 0.01
5	0.686	0.762	0.686
10	0.815	0.842	0.781
20	0.891	0.905	0.868
50	0.935	0.947	0.930

Values at α = 0.10 and α = 0.01 are sourced from the Shapiro-Wilk (1965) original paper. Use the α = 0.05 column (tab above) for most research applications. Modern software handles any α level directly through p-values.

Shapiro-Wilk Critical Values by Sample Size

Sample size determines the critical value because the distribution of W under H₀ changes with n. As n increases, W critical also increases — meaning larger samples must produce a W closer to 1 to avoid rejection. This reflects greater statistical power: larger samples are more sensitive to departures from normality.

n = 3 (minimum)

W_critical = 0.767. With just three observations, the test has very limited power. A Q-Q plot and normal probability plot are essential companions.

Small samples (n < 15)

The test is less powerful. Real departures from normality may go undetected. Interpretation should be conservative and always supported by graphical diagnostics.

Moderate samples (n = 15–50)

The Shapiro-Wilk test performs well in this range, balancing power and stability. This is where printed critical value tables are most useful.

Large samples (n > 50)

Modern software extends Shapiro-Wilk beyond n = 50. With large n, even trivial non-normality becomes statistically significant. Effect size and graphical inspection matter more than the p-value alone.

What If My Sample Size Is Not in the Table?

Printed historical tables for the Shapiro-Wilk test typically cover n = 3 to 50. If your sample size falls outside this range or between listed values, several options exist.

Between listed n values

Some published tables provide all integers from 3 to 50; others list selected values. Do not interpolate casually without consulting the source methodology. Use statistical software for precision.

n < 3

The Shapiro-Wilk test is not defined for fewer than 3 observations. With such small samples, normality testing is not meaningful — consider substantive knowledge about the data-generating process instead.

n > 50

Modern implementations handle larger samples. R's shapiro.test() handles up to n = 5000, and Python's scipy.stats.shapiro() computes p-values across a broader range. Do not assume that the printed table's upper limit applies to all software implementations.

Shapiro-Wilk Weights vs Critical Values: Two Different Tables

"Shapiro-Wilk table" can refer to two entirely different things. If you need to check a hypothesis test result, you need the critical value table (this page). If you are computing W by hand, you need the coefficient table.

Table type	Contains	Used for
Coefficients table (aᵢ)	Weights aᵢ indexed by n and position i	Calculating the W statistic from raw ordered data
Critical value table (W_crit)	Threshold W values by n and α	Making the hypothesis testing decision after W is computed

Worked Examples

Example 1: n = 10, Reject H₀

A researcher tests whether reaction times (ms) from 10 participants follow a normal distribution. After running the Shapiro-Wilk test in SPSS, W = 0.810 is reported. Using the critical value table at α = 0.05:

Step	Value
Sample size n	10
W calculated (illustrative assumed statistic)	0.810
W critical at α = 0.05	0.842
Comparison	0.810 < 0.842 ✗
Decision	Reject H₀

Interpretation: There is evidence that the sample is not consistent with a normal distribution at the 5% significance level. The researcher should inspect a Q-Q plot and consider whether the analysis being planned requires normality, or whether robust alternatives are appropriate.

Example 2: n = 20, Fail to Reject H₀

A clinical researcher checks normality for a sample of 20 blood pressure readings before applying a one-sample t-test. W = 0.930 is obtained. Using the critical value table at α = 0.05:

Step	Value
Sample size n	20
W calculated (illustrative assumed statistic)	0.930
W critical at α = 0.05	0.905
Comparison	0.930 ≥ 0.905 ✓
Decision	Fail to reject H₀

Interpretation: The result is consistent with normality at the 5% level. This does not prove the population is normally distributed — it means the data did not provide sufficient evidence against normality. Paired with a Q-Q plot showing points following the reference line closely, this supports proceeding with the t-test.

Example 3: Research-Style Report (n = 30)

Parameter	Value
n	30
W calculated	0.952
W critical (α = 0.05)	0.927
α	0.05
Decision	Fail to reject H₀ (0.952 ≥ 0.927)

How to report: "The Shapiro-Wilk statistic was W = 0.952 (n = 30). Because W exceeded the critical value of 0.927 at α = 0.05, we failed to reject the null hypothesis of normality. This result, supported by inspection of the Q-Q plot, indicated that the normality assumption was tenable for subsequent analysis."

Example 4: Software Output (p-value approach)

When using R, Python, or SPSS, software returns a p-value alongside W. The p-value approach and the critical-value approach yield equivalent decisions, but the p-value does not equal W critical — they are different quantities.

# R output example (illustrative)
Shapiro-Wilk normality test
W = 0.9302, p-value = 0.1412

Approach	What you compare	Reject H₀ when
Critical value method	W_calc vs W_critical	W_calc < W_critical
P-value method	p-value vs α	p < α

In this example, p = 0.1412 > 0.05, so we fail to reject H₀ — the same conclusion as the critical value comparison. Both methods are mathematically equivalent. Modern software makes the p-value approach straightforward and removes the need to consult printed tables.

Shapiro-Wilk Test in R, Python, SPSS, and Other Software

Statistical software eliminates the need to look up critical values manually. Each package returns W and a p-value; the decision follows the p-value rule (reject H₀ if p < α).

R

shapiro.test(x)

Returns W and p-value. Works for n = 3 to 5000. Part of base R — no package required. See R documentation.

Python / SciPy

from scipy import stats
stats.shapiro(x)

Returns (W, p-value). See official SciPy docs.

SPSS / SAS / Stata

SPSS: Explore → Normality plots. SAS: PROC UNIVARIATE with NORMAL option. Stata: swilk command. All return W and p-value. JASP and jamovi include the test in descriptives.

Not every implementation has identical sample-size limits or p-value algorithms. When using software, verify which version of the test is applied (Shapiro-Wilk 1965 or the Royston 1992/1995 extension, which handles larger samples). Report the software and version used alongside your results.

Shapiro-Wilk Test vs Q-Q Plot

Graphical and statistical methods for assessing normality complement each other. Neither alone provides a complete picture.

Method	Type	What it shows
Shapiro-Wilk test	Formal statistical test	Whether the sample is significantly inconsistent with normality
Q-Q plot	Visual assessment	Where the distribution departs from normality (tails, skew, kurtosis)
Histogram	Visual assessment	Overall distribution shape, modality, skewness
Box plot	Visual assessment	Outliers, spread, symmetry around the median

Shapiro-Wilk vs Kolmogorov-Smirnov vs Anderson-Darling

Several normality tests exist, each with distinct properties. Choosing the appropriate test depends on sample size, the nature of the departure from normality, and whether parameters are estimated or specified.

Test	Statistic	Best for	Notes
Shapiro-Wilk	W	Small to moderate n (≤ 50 in original tables; wider in software)	Generally the most powerful test for normality
Kolmogorov-Smirnov	D	General goodness-of-fit; any specified distribution	Lower power for normality than Shapiro-Wilk; often corrected (Lilliefors)
Anderson-Darling	A²	Detecting departures in the tails of the distribution	Weights tails more heavily than K-S; competitive with Shapiro-Wilk

No single test is universally superior. The Shapiro-Wilk test is the most commonly recommended for normality checking in general practice, but the Anderson-Darling test performs competitively, particularly when tail behavior matters. Use graphical diagnostics alongside any formal test.

Shapiro-Wilk Test Assumptions and Limitations

What the test requires

Quantitative (continuous or discrete with many values) observations. Independence between observations — a design requirement, not something the test can verify. Random or representative sampling appropriate for the study design. No severe rounding or data truncation that distorts the distribution.

What the test cannot fix

Dependence among observations. Problems introduced by outliers from measurement error or data entry mistakes. Selection bias or non-representative sampling. Missing data patterns that systematically affect the distribution.

Testing residuals vs raw data

In regression, ANOVA, and related models, the normality assumption typically concerns model residuals, not the raw predictor or outcome variables. Testing raw variables for normality is often unnecessary and can mislead. The Shapiro-Wilk test applied to residuals checks the appropriate assumption. See normality tests for a broader discussion.

How Outliers Affect the Shapiro-Wilk Test

Unusual observations can substantially influence W. A single extreme value can cause rejection even when the bulk of the data is well-described by a normal distribution. Before interpreting a significant Shapiro-Wilk result, inspect the data using a histogram, Q-Q plot, and box plot to identify whether a small number of values are driving the result.

Do not remove outliers automatically because the Shapiro-Wilk test is significant. Whether an observation is a legitimate data point or an error requires substantive judgment. Removing valid observations simply to achieve a non-significant normality test is not an acceptable analytical practice.

Large-Sample and Small-Sample Interpretation

Large samples (n > 100)

Very small departures from normality produce statistically significant results. A significant Shapiro-Wilk result with large n does not mean the violation is practically important. Inspect Q-Q plots and consider whether the analytical method being used is sensitive to the type of departure detected.

Small samples (n < 10)

The test has low power. Substantial departures from normality may not produce significant results. A non-significant result with n < 10 provides weak evidence in favor of normality. Graphical inspection and substantive knowledge are especially important in this range.

Common Mistakes When Using the Shapiro-Wilk Table

Reversing the decision rule: rejecting H₀ when W ≥ W critical, or failing to reject when W < W critical.
Thinking larger W means rejection: in this test, smaller W indicates stronger evidence against normality.
Treating W critical as a p-value: they are not the same quantity and cannot be compared to each other.
Using the wrong row: looking up n = 20 when your sample size is 25.
Using α = 0.05 values when you chose α = 0.01: critical values differ between significance levels.
Treating failure to reject as proof of normality: it means evidence against normality was insufficient, not that normality is confirmed.
Ignoring sample size context: a significant result with n = 200 may reflect a trivial departure; a non-significant result with n = 5 provides limited information.
Skipping graphical diagnostics: Q-Q plots reveal where and how the distribution departs from normality, which a single W value cannot.
Automatically removing outliers: the test result alone does not justify removing observations.
Confusing the coefficient table with the critical value table: the aᵢ table is used to calculate W; the critical value table is used to make the decision.
Assuming printed tables cover all n: historical tables stop at n = 50; software handles larger samples but may use different algorithms.
Testing raw variables in regression: the normality assumption in regression applies to residuals, not necessarily to the raw outcome or predictor variables.
Reporting only "normal" or "not normal": always report W, n, α, and the p-value or critical value.

How to Report a Shapiro-Wilk Test

Critical-value style

"The Shapiro-Wilk statistic was W = 0.930. For n = 20 at α = 0.05, the critical value was 0.905. Because W exceeded the critical value, we failed to reject the null hypothesis of normality."

P-value style

"The Shapiro-Wilk test produced W = 0.930, p = .182 (n = 20). Because p > .05, the result did not provide sufficient evidence to reject the null hypothesis of normality."

APA-style

"Normality was assessed using the Shapiro-Wilk test, W(20) = .930, p = .18. The result was consistent with normality at the .05 level."

Exact APA reporting requirements may depend on the context and current style guidance. Confirm with the most current APA manual or your target journal's instructions.

Shapiro-Wilk Test Calculator

An online Shapiro-Wilk calculator accepts your dataset and returns the W statistic, p-value, sample size, and interpretation. When using any calculator, verify what significance level is applied, how the sample size range is handled, and whether missing values are addressed.

Use the Shapiro-Wilk Test page on Statistics Fundamentals for worked examples, step-by-step explanations, and additional resources on normality testing. See also our normality tests guide for a broader comparison.

Frequently Asked Questions

What is the Shapiro-Wilk critical value?

The Shapiro-Wilk critical value (W critical) is the threshold value below which the null hypothesis of normality is rejected. It depends on both sample size n and the significance level α. For n = 20 at α = 0.05, W critical = 0.905.

How do I use the Shapiro-Wilk critical value table?

Find your sample size n in the table. Read W critical from the column matching your α level. Compute or obtain W from your software. If W < W critical, reject H₀. If W ≥ W critical, fail to reject H₀. The lookup tool at the top of this page automates this process.

What is W critical for n = 3?

At α = 0.05, W critical for n = 3 is 0.767. With only three observations, the test has very limited power, and interpretation should be cautious.

What is W critical for n = 5?

At α = 0.05, W critical for n = 5 is 0.762. Small samples generally produce variable results; graphical checks are essential companions to the formal test.

What is W critical for n = 10?

At α = 0.05, W critical for n = 10 is 0.842. Reject H₀ if your calculated W is below 0.842 at this significance level.

What is W critical for n = 15?

At α = 0.05, W critical for n = 15 is 0.881. Reject H₀ if your calculated W falls below this threshold.

What is W critical for n = 20?

At α = 0.05, W critical for n = 20 is 0.905. If W calculated ≥ 0.905, fail to reject H₀. If W calculated < 0.905, reject H₀.

What is W critical for n = 25?

At α = 0.05, W critical for n = 25 is 0.918.

What is W critical for n = 30?

At α = 0.05, W critical for n = 30 is 0.927.

What is W critical for n = 40?

At α = 0.05, W critical for n = 40 is 0.940.

What is W critical for n = 50?

At α = 0.05, W critical for n = 50 is 0.947. This is the upper limit of most printed historical tables. For larger samples, use statistical software.

Is a higher W better in the Shapiro-Wilk test?

Yes, a higher W (closer to 1) indicates the data are more consistent with a normal distribution. A W of exactly 1 would correspond to perfect agreement with normality. Smaller W values indicate greater departure. This is opposite to most test statistics, where larger values lead to rejection.

What does α = 0.05 mean in the Shapiro-Wilk test?

α = 0.05 is the significance level — the acceptable probability of incorrectly rejecting H₀ when the data actually are normally distributed (Type I error rate). At α = 0.05, you use the 0.05 column of W critical values. Lowering α to 0.01 makes the test more stringent.

What if W is below the critical value?

Reject H₀. The data provide evidence against normality at the chosen significance level. This does not automatically invalidate subsequent analyses — it signals that normality cannot be assumed without further investigation, and that robust or non-parametric alternatives may be appropriate.

What if W is above the critical value?

Fail to reject H₀. The data are consistent with normality at the chosen level. This does not prove the data are normally distributed — it means the evidence against normality is insufficient at this sample size and α. Always examine Q-Q plots and histograms.

What is the Shapiro-Wilk p-value?

The p-value from the Shapiro-Wilk test is the probability of observing a W as small as or smaller than the calculated value, if the data actually came from a normal distribution. If p < α, reject H₀. The p-value is not the same as W critical — they are different quantities used in different comparison steps.

Critical value vs p-value: which should I use?

Both methods give equivalent decisions. The p-value approach (compare p to α) is more convenient with modern software. The critical-value approach (compare W to W critical from a table) is useful for manual calculations, verification, and educational contexts. Use whichever matches your reporting requirements and available resources.

What are Shapiro-Wilk coefficients?

The Shapiro-Wilk coefficients (aᵢ) are a set of weights used in the numerator of the W formula. They are derived from the expected order statistics of a normal distribution and depend on n. They appear in the coefficients table (a different table from the critical value table). Statistical software looks these up internally when computing W.

How does sample size affect the test?

Sample size affects both power and the critical value. Larger n means lower W critical and greater power — the test is more sensitive to departures from normality. Very large samples may detect trivially small departures. Very small samples may miss substantial departures. Interpret results with sample size in mind.

What if n is greater than 50?

Printed historical tables typically do not cover n > 50. Modern implementations of the test (R, Python/SciPy, SPSS) extend the range significantly. Use software and report the p-value. Note that Shapiro-Wilk is not universally limited to n ≤ 50 — that was a historical table constraint, not a fundamental statistical limit.

Shapiro-Wilk vs Kolmogorov-Smirnov: which is better?

For testing normality specifically, the Shapiro-Wilk test generally has greater power than the Kolmogorov-Smirnov test, particularly for small to moderate samples. The K-S test is more general and can test against any specified distribution. Neither test is universally superior across all situations. The Anderson-Darling test is a competitive alternative, especially when tail behavior matters.

Can I remove outliers to pass the Shapiro-Wilk test?

No — removing observations specifically to achieve a non-significant Shapiro-Wilk result is not acceptable statistical practice. Outlier treatment requires substantive justification (e.g., confirmed data entry error, known equipment malfunction). Investigate outliers through histograms and Q-Q plots and make a principled decision based on the research context.

What should I test: raw data or residuals?

In regression, ANOVA, and related analyses, the normality assumption applies to residuals, not to the raw predictor or outcome variables. Testing raw variables for normality is a common mistake. Apply the Shapiro-Wilk test to model residuals when checking parametric model assumptions.

How do I report APA-style Shapiro-Wilk results?

Report W, the sample size or degrees of freedom, and the p-value. Example: "The Shapiro-Wilk test indicated that the assumption of normality was met, W(20) = .930, p = .18." For a significant result: "The Shapiro-Wilk test indicated that the normality assumption was violated, W(20) = .846, p = .006." Confirm current APA style requirements for your publication.

Key Takeaways

The Shapiro-Wilk critical value table gives W_critical values for testing normality by sample size and significance level.
The decision rule: reject H₀ if W calculated < W critical. Lower W indicates stronger evidence against normality.
Failing to reject H₀ does not prove normality — it means the data are consistent with normality at the chosen level.
Historical printed tables cover n = 3 to 50. Modern software extends this range and computes exact p-values.
The Shapiro-Wilk coefficients table and the critical value table are two different tables with different purposes.
Always complement formal testing with graphical diagnostics: Q-Q plots, histograms, and box plots.
In regression and ANOVA, test residuals for normality — not necessarily the raw data.
With large samples, even trivial departures from normality become significant; with small samples, the test has limited power.

References and Sources

Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4), 591–611.
Royston, P. (1992). Approximating the Shapiro-Wilk W-test for non-normality. Statistics and Computing, 2(3), 117–119.
Royston, P. (1995). Remark AS R94: A remark on algorithm AS 181. Applied Statistics, 44(4), 547–551.
NIST/SEMATECH e-Handbook of Statistical Methods. Shapiro-Wilk Normality Test. National Institute of Standards and Technology.
R Core Team. shapiro.test documentation. R: A Language and Environment for Statistical Computing.
SciPy community. scipy.stats.shapiro documentation. SciPy.