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How to do The Shapiro-Wilk Normality Test for Biomedical Research

  • Mar 23
  • 4 min read
How to do The Shapiro-Wilk Normality Test

Is your data truly normal? In biomedical research, this single question often determines whether your p-values are "significant" or just "statistical noise."

If you are analyzing pre-clinical data—whether it's cell culture viability, mouse tumor volumes, or protein expression levels—you likely rely on T-tests and ANOVAs. But these parametric tests rely on a strict assumption: normality. If your data violates this, your results could be invalid.

This guide is the definitive resource on performing, interpreting, and troubleshooting the Shapiro-Wilk test, the "gold standard" method for small sample sizes common in basic research (n < 50). We move beyond simple definitions to provide step-by-step protocols for GraphPad Prism, SPSS, and R, ensuring your statistical rigor meets the highest publication standards.



What is the Shapiro-Wilk Test and Why Use It?

The Shapiro-Wilk (SW) test determines if a dataset comes from a normally distributed population. It is widely considered the most powerful normality test for small sample sizes, which makes it indispensable for "wet lab" biology where N=3 or N=6 is standard.


The Core Logic

  • Null Hypothesis (H_0): The population is normally distributed.

  • Alternative Hypothesis (H_1): The population is not normally distributed.

  • The P-Value Rule:

    • p > 0.05: You fail to reject the null hypothesis. Your data is likely normal. (Result: Use Parametric Tests)

    • p < 0.05: You reject the null hypothesis. Your data deviates significantly from normality. (Result: Use Non-Parametric Tests or Transform)


Why Not Just Use a Histogram?

Visual inspection (histograms) is subjective, especially with small datasets (e.g., n=10) where "bins" can distort the shape. The SW test provides an objective, standardized metric (W statistic) to quantify how well your data fits the Gaussian bell curve.


Step-by-Step Protocols: Performing the Test

Option 1: GraphPad Prism (The Lab Standard)

Most biological researchers use Prism. Here is the modern workflow.

  1. Enter Data: Input your data into a Column data table.

  2. Analyze: Click Analyze > Column Analyses > Normality and Lognormality Tests.

  3. Select Test: Check the box for Shapiro-Wilk.

    • Note: Older versions of Prism may calculate this slightly differently. Ensure you are using Prism 6+ for the updated Royston approximation algorithm.

  4. Run: Click OK.

  5. Interpret: Look at the "P value summary" column.

    • If it says ns (not significant), your data is normal.

    • If it shows asterisks (*), your data passed the threshold for non-normality.


Option 2: SPSS (The Clinical Standard)

  1. Navigate: Go to Analyze > Descriptive Statistics > Explore.

  2. Select Variables: Drag your variable of interest (e.g., "TumorSize") into the Dependent List.

  3. Split by Group (Optional): If you have groups (e.g., "Treatment" vs "Control"), drag the grouping variable into the Factor List.

  4. Configure Plots: Click the Plots button. Check "Normality plots with tests". Uncheck "Stem-and-leaf" if you don’t need it.

  5. Output: Look for the table titled "Tests of Normality". Focus on the Shapiro-Wilk column, specifically the Sig. (significance) value.


Option 3: R (The Data Science Standard)

Best for high-throughput data or automated pipelines.

The basic command is built into the stats package:

# Basic Syntax
shapiro.test(numeric_vector)

# Example with interpretation
data <- c(2.1, 3.4, 2.8, 3.1, 2.9) # Your biological data
result <- shapiro.test(data)

print(result)
# Output will look like:
# Shapiro-Wilk normality test
# data:  data
# W = 0.986, p-value = 0.967

Interpretation: In the code above, p = 0.967. Since 0.967 > 0.05, we treat the data as normal.


Troubleshooting & "Edge Cases"

Scenario A: One Group Fails, Others Pass

Example: You have 3 treatment groups. Control and Drug A are normal (p > 0.05), but Drug B is not (p = 0.03).

  • The Problem: You cannot mix Parametric (ANOVA) and Non-Parametric (Kruskal-Wallis) tests in a single analysis.

  • The Solution:

    1. Transform: Try a Log10 transformation on all groups and re-test. This often fixes right-skewed biological data.

    2. Go Non-Parametric: If transformation fails, switch to Kruskal-Wallis (for >2 groups) or Mann-Whitney (for 2 groups) for the entire experiment. It is safer and more conservative.


Scenario B: "My Sample Size is Huge (n > 5000)"

  • The Problem: The SW test is too sensitive at large sample sizes. It will return p < 0.05 for trivial deviations from normality that don't actually affect the validity of a T-test.

  • The Solution: Do not rely solely on SW for n > 50. Use a Q-Q Plot (Quantile-Quantile plot). If the dots lie roughly on the diagonal line, assume normality regardless of the SW p-value.


Scenario C: Raw Data vs. Residuals

Advanced Statistician Note: Technically, parametric tests like ANOVA assume the residuals (errors) are normally distributed, not necessarily the raw data.

  • For simple 1-way designs, testing raw data is an acceptable proxy.

  • For complex models (e.g., Two-Way ANOVA), you should extract the residuals and run shapiro.test(residuals) for the most accurate assessment.


What does a p-value 0.05 in the Shapiro-Wilk test indicate?

Result

P-Value

Conclusion

Recommended Next Step

Passed

> 0.05

Data is Normal

T-Test / ANOVA

Failed

< 0.05

Data is Not Normal

Check Outliers -> Log Transform -> Non-Parametric Test





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