Shapiro-Wilk Test

What Is the Shapiro-Wilk Test?

Proposed by Samuel Sanford Shapiro and Martin Wilk in 1965, the test works by comparing the ordered sample observations with the values one would expect if the data truly came from a normal distribution. The closer that comparison, the higher W and the larger the p-value — meaning the data do not contradict the normality assumption.

Two points deserve emphasis from the start. First, the test assesses whether the data are consistent with normality — a non-significant result does not prove that the population is exactly normal. Second, the test's sensitivity to departures from normality depends heavily on sample size, so a significant result in a large sample may reflect a trivially small deviation that has no practical consequence for the analysis you are planning. Both points are discussed in detail in the sample size section below.

When Should You Use the Shapiro-Wilk Test?

The Shapiro-Wilk test is typically appropriate when a researcher needs to assess normality before applying a method whose assumptions involve the normal distribution. Common situations include:

• Checking whether residuals from a linear regression or ANOVA appear consistent with normality
• Assessing whether a continuous measurement (e.g., blood pressure, reaction times, exam scores) is approximately normal before using a t-test
• Exploratory data analysis where understanding the distributional shape informs subsequent modeling decisions
• Quality control settings where a production process measurement is expected to follow a normal distribution

There are also situations where running the test adds little value or may mislead. With very small samples (n < 10), the test has limited ability to detect moderate departures from normality. With very large samples (n > several hundred), the test often returns a significant p-value for departures so small that they have no practical consequence for the statistical method being considered. In both extremes, graphical diagnostics — particularly a Q-Q plot — carry substantial weight.

Should You Test the Raw Data or the Residuals?

For many statistical models — linear regression, ANOVA, and the t-test — the normality assumption applies to the model residuals or errors, not to the raw outcome variable. Testing the raw dependent variable for normality before running a regression, for example, answers the wrong question. The correct approach is to fit the model, obtain the residuals, and then apply the Shapiro-Wilk test (or a Q-Q plot) to those residuals.

This matters in practice. A skewed raw variable can produce perfectly well-behaved residuals after accounting for predictor variables, particularly when the predictors themselves explain a substantial portion of the outcome's variance. Conversely, a raw variable that looks approximately normal can produce non-normal residuals if the model is misspecified.

Shapiro-Wilk Test Hypotheses

Rejecting H₀ provides evidence against normality under the test's assumptions, but it does not identify which specific non-normal distribution the data follow. A significant result might reflect skewness, heavy tails, outliers, bimodality, or any other departure from the normal shape. The alternative hypothesis simply says "not normal" — it does not pin down the alternative distribution.

Failing to reject H₀ means the test did not find sufficient evidence against normality at the chosen significance level. This is not proof that the population is exactly normal. No finite sample can prove exact normality, and the test's ability to detect departures depends on sample size, the magnitude of the departure, and the significance level chosen.

The Shapiro-Wilk W Statistic

The test produces a single test statistic denoted W. Conceptually, W measures how well the ordered sample observations agree with the expected order statistics of a normal distribution. W is always between 0 and 1:

A W value alone, without its associated p-value and without knowing the sample size, cannot be meaningfully interpreted. A W of 0.95 might be associated with a very small p-value (strong evidence against normality) for a large sample, but with a large p-value (no significant evidence) for a small sample. Always report and interpret W alongside the p-value and n.

Shapiro-Wilk Formula

The Shapiro-Wilk statistic W is defined as:

The numerator squares the sum of weighted ordered observations, where the weights aᵢ are derived from the expected values and covariance structure of the order statistics of a standard normal distribution. The denominator is the total sum of squared deviations from the mean — proportional to the sample variance. When the data are perfectly normal, these two quantities are proportional, and W equals 1.

The aᵢ coefficients are not simple constants. They depend on the sample size n and on the expected order statistics and their covariance matrix under the normal distribution. In practice, these coefficients are tabulated (for small n, Shapiro and Wilk's original 1965 paper provided them) or computed algorithmically by statistical software. Users implementing the test from scratch should use a validated algorithm rather than attempting to derive the coefficients manually.

How the Shapiro-Wilk Test Works

Here is the step-by-step logic behind the test:

Shapiro-Wilk P-Value Interpretation

The p-value answers this question: if the null hypothesis of normality were true, how likely would we be to observe a W statistic this small or smaller by chance alone? A small p-value means such a W would be unusual under normality — evidence against H₀.

Three common misreadings of the p-value deserve explicit correction.

Misreading 1: p > 0.05 proves the data are normal. It does not. It means the test did not find sufficient evidence against normality. The absence of statistical evidence against H₀ is not the same as evidence for H₀. The data might be non-normal in a way the test lacks power to detect — especially for small samples.

Misreading 2: p < 0.05 automatically means you cannot use a parametric test. It does not mean that either. Many parametric procedures are robust to moderate departures from normality, especially with moderate-to-large samples. A significant Shapiro-Wilk result is a signal to investigate further, not an automatic prohibition on parametric methods.

Misreading 3: The p-value measures how normal the data are. It does not measure the degree of normality. It measures the strength of statistical evidence against the null hypothesis of normality, which depends on both the degree of departure and the sample size.

Shapiro-Wilk Test and Sample Size

Sample size is the single most important contextual factor when interpreting the Shapiro-Wilk test result, and it affects the interpretation in opposite directions depending on whether n is small or large.

Small Samples

With small samples — say, fewer than 20 or 30 observations — the Shapiro-Wilk test has limited statistical power. It will often fail to detect moderate departures from normality because there simply is not enough data. A non-significant result in a small sample provides relatively weak evidence about the distributional shape. In this setting, graphical diagnostics such as a Q-Q plot matter more, and the robustness of the planned statistical method deserves particular attention.

Large Samples

With large samples — several hundred or more — the Shapiro-Wilk test becomes sensitive to trivially small departures from normality. A p-value well below 0.05 might correspond to a departure so minor that it has no practical consequence for the statistical procedure being considered. Many parametric procedures remain valid under such minor deviations, particularly when sample sizes are large enough for the central limit theorem to provide adequate protection.

The practical guidance: in large samples, place more weight on graphical diagnostics and on the known robustness of the statistical procedure than on the Shapiro-Wilk p-value alone. A statistically significant normality test result in a large sample is not necessarily a reason to abandon a well-chosen parametric method.

Q-Q Plot: The Essential Visual Companion

A Q-Q plot (quantile-quantile plot) displays the ordered sample observations on one axis against the theoretical quantiles of a normal distribution on the other. If the data are consistent with normality, the points fall approximately along a straight diagonal reference line. Systematic departures from that line indicate specific types of non-normality.

Reading a Q-Q plot is a skill worth developing. When points curve upward at both ends of the reference line, this typically indicates heavier tails than a normal distribution (leptokurtosis). When the S-curve goes in the opposite direction, the tails are lighter (platykurtosis). When there is a consistent bow shape above or below the line, this suggests skewness — a right bow indicates right-skewed data. Isolated points far from the line often correspond to outliers.

The Q-Q plot answers a question the Shapiro-Wilk test cannot: where does the departure from normality occur? A significant p-value from the test tells you there is evidence of departure; the Q-Q plot tells you what kind.

Worked Examples

Example 1 — Approximately Normal Data

Example 2 — Clearly Skewed Data

Example 3 — Data with an Outlier

Statistical Significance vs. Practical Importance

A statistically significant Shapiro-Wilk test does not automatically mean a planned parametric analysis is invalid. The relevant question is whether the departure from normality is large enough to meaningfully affect the conclusions of the specific method being used.

Many parametric procedures — the two-sample t-test, one-way ANOVA — are reasonably robust to moderate departures from normality, particularly when sample sizes are equal across groups and when n is moderate or large. The robustness comes partly from the central limit theorem: for large enough samples, the sampling distribution of means is approximately normal regardless of the raw data distribution.

Conversely, a non-significant Shapiro-Wilk result does not guarantee that every parametric assumption holds. Normality is one assumption among several — independence, equal variances (for some tests), and correct model specification all matter too. A comprehensive assumption checking process combines normality assessment with checks of these other conditions.

What If the Shapiro-Wilk Test Is Significant?

A statistically significant Shapiro-Wilk result calls for further investigation — not an automatic switch to a nonparametric test. The appropriate response depends on several factors:

Common nonparametric alternatives to consider when normality is not tenable:

Shapiro-Wilk vs. Other Normality Tests

Shapiro-Wilk vs. Kolmogorov-Smirnov

The Kolmogorov-Smirnov test in its standard form requires the reference distribution to be fully specified before seeing the data — meaning exact μ and σ must be stated, not estimated from the sample. When researchers estimate μ and σ from the sample itself (as is always the case in practice), the standard K-S critical values are inappropriate, and the Lilliefors test provides the correct critical values. This distinction matters for correct application.

Shapiro-Wilk vs. Anderson-Darling

Shapiro-Wilk vs. Jarque-Bera

The Jarque-Bera test takes a different approach: it computes sample skewness and excess kurtosis, then tests whether both are simultaneously consistent with zero (as expected for a normal distribution). Unlike Shapiro-Wilk, which uses the full ordering of observations, Jarque-Bera summarizes departures from normality through just two moments. It was originally developed for large samples and econometric contexts. For small samples, it tends to be less reliable than Shapiro-Wilk. Neither test is uniformly best, and the choice depends on the research context, sample size, and what type of departure from normality is most likely.

Running the Shapiro-Wilk Test in Software

R

R's built-in shapiro.test() function implements the AS R94 algorithm (Royston, 1995), which supports n = 3 to 5000.

# Define your data vector
x <- c(62, 65, 68, 70, 71, 73, 74, 75, 78, 82)

# Run the Shapiro-Wilk test
result <- shapiro.test(x)
print(result)

# Output:
#   Shapiro-Wilk normality test
# W = 0.97303, p-value = 0.9204

# Q-Q plot for visual inspection
qqnorm(x, main = "Q-Q Plot")
qqline(x, col = "steelblue", lwd = 2)

The output provides W and the p-value directly. Combine with qqnorm() and qqline() to generate a Q-Q plot. For residuals from a linear model, apply the test to the residuals object: shapiro.test(residuals(my_model)). See the R documentation for current details.

Python

SciPy's scipy.stats.shapiro() function provides the Shapiro-Wilk test. Check the current SciPy documentation for supported sample sizes and any version-specific behavior.

from scipy.stats import shapiro
import scipy.stats as stats
import matplotlib.pyplot as plt

data = [62, 65, 68, 70, 71, 73, 74, 75, 78, 82]

# Run the Shapiro-Wilk test
statistic, p_value = shapiro(data)
print(f"W = {statistic:.4f}, p = {p_value:.4f}")

# Interpretation
alpha = 0.05
if p_value < alpha:
    print("Evidence against normality (reject H0)")
else:
    print("Insufficient evidence to reject normality")

# Q-Q plot
stats.probplot(data, dist="norm", plot=plt)
plt.title("Q-Q Plot")
plt.show()

SciPy's implementation supports a range of sample sizes; refer to the SciPy documentation for the current supported range and any warnings for very large samples. For residuals from a fitted model, pass the residuals array directly to shapiro().

SPSS

In SPSS, the Shapiro-Wilk test is available through the Explore procedure:

Verify current SPSS menu paths in the version you are using, as exact menu names may differ across releases. IBM's documentation provides current guidance for SPSS Statistics.

Excel

Microsoft Excel does not include a native Shapiro-Wilk test function. To run the test with Excel data, the most practical options are: exporting the data to R or Python and using their implementations; using an Excel add-in such as the Real Statistics Resource Pack, which adds normality testing functions; or entering the data directly into the interactive calculator on this page. Do not use Excel's built-in statistical functions to construct an approximation and label it a Shapiro-Wilk test.

How to Report Shapiro-Wilk Results

Statistical reporting should include the test statistic, sample size, p-value, and a plain-language conclusion that avoids overclaiming. Below are standard templates:

Shapiro-Wilk Test Calculator

Enter your data below — comma-separated, one value per line, or space-separated. The calculator runs a validated Shapiro-Wilk implementation in your browser. No data is sent to any server.

Common Mistakes with the Shapiro-Wilk Test

Real-World Applications

Frequently Asked Questions

Sources and References

This guide draws on the following primary and secondary sources. Formulas and algorithmic details are cross-referenced against the original publication and established computational references.

• Shapiro, S.S. & Wilk, M.B. (1965) — "An analysis of variance test for normality (complete samples)." Biometrika, 52(3–4), 591–611. The original paper introducing the W statistic and providing coefficient tables for n = 3 to 50.
• Royston, J.P. (1995) — "Remark AS R94: A remark on algorithm AS 181." Applied Statistics, 44, 547–551. The AS R94 algorithm extending the test to n = 3–5000, implemented in R's shapiro.test().
• NIST Engineering Statistics Handbook — Normality Tests. National Institute of Standards and Technology. itl.nist.gov
• Penn State STAT 415 — Introduction to Mathematical Statistics. Penn State Eberly College of Science. online.stat.psu.edu
• SciPy Documentation — scipy.stats.shapiro. docs.scipy.org
• R Documentation — stats::shapiro.test. rdocumentation.org
• IBM SPSS Statistics Documentation — Explore procedure, normality tests. ibm.com/docs