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03. Descriptive Statistics & Distribution Lab8 min readAdvanced Lesson

Testing Normality: Shapiro-Wilk, Q-Q Plots, and CLT

How to combine formal normality tests, skewness/kurtosis indices, and visual Q-Q inspection to choose between parametric and non-parametric tests.

In this lesson you will learn

  • 1Interpret the null hypothesis of the Shapiro-Wilk test properly (p > .05 indicates no significant departure from normality).
  • 2Explain why large epidemiological samples (N > 300) require visual Q-Q plots and skewness checks alongside formal p-values.
  • 3Recognise how the Central Limit Theorem (CLT) stabilises the sampling distribution of the mean in moderate-to-large samples.
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01

The 3-Tier Normality Assessment Protocol

What this section is about

This section outlines the 3-step diagnostic check—Shapiro-Wilk test, skewness/kurtosis bounds, and Normal Q-Q plot—used before running parametric models.

Why it matters for your study

Students often misread Shapiro-Wilk p-values backwards—thinking p < .05 means 'good/normal'—which leads them to choose the wrong inferential test across their entire thesis.

Before choosing between a Student's t-test and Mann-Whitney U (or between ANOVA and Kruskal-Wallis), biostatisticians evaluate normality across three complementary lenses:

  1. Formal Hypothesis Test: Run the Shapiro-Wilk test (preferred for N < 2,000). Because H₀ states that the sample is drawn from a normal distribution, p > .05 means normality is satisfied, whereas p < .05 indicates significant non-normality.
  2. Numerical Shape Indices: Check that sample Skewness lies within [-1.0, +1.0] and Excess Kurtosis lies within [-2.0, +2.0], and verify that SD does not exceed the Mean on strictly positive variables.
  3. Visual Q-Q Inspection: Plot observed quantiles against theoretical normal quantiles. Under normality, points hug the 45-degree diagonal reference line; right-skewed data curves upward in a concave arc.
Normal Q-Q Plot (Points hug 45° line)Right-Skewed Q-Q Plot (Curves away from line)
Figure:On a Normal Q-Q plot, Gaussian data tracks the 45-degree diagonal closely, whereas skewed distributions arch away from the reference line.

3-Tier Normality Diagnostic Lab (Shapiro-Wilk + Skewness + Q-Q Plot)

Switch between three clinical variables to inspect numerical and visual normality checks

Shapiro-Wilk (W, p):W = 0.986, p = .412
Skewness / Kurtosis:+0.14 / -0.22
Mean (SD):14.1 (1.3)
Median [IQR]:14 [13.2–15.0]
Normal Q-Q Plot Inspection

Shapiro-Wilk p = .412 (> .05), |Skewness| < 1.0, and Q-Q points hug the 45° diagonal. Report Mean (SD) and use parametric tests (t-test / ANOVA).

Clinical Example

An investigator checks normality on two baseline variables in 84 cardiac patients: Adult Haemoglobin (g/dL) and Serum hs-CRP (mg/L).

Haemoglobin: W = 0.986, p = .412, Skew = +0.14 → Normal | hs-CRP: W = 0.742, p < .001, Skew = +2.38 → Right-Skewed

Interpretation: Haemoglobin passes Shapiro-Wilk (p = .412 > .05) and is summarised as Mean (SD) and compared via t-test. hs-CRP fails Shapiro-Wilk (p < .001) and is summarised as Median [IQR] and compared via Mann-Whitney U.

Common Pitfall

Interpreting a significant Shapiro-Wilk result (p < .05) as permission to use parametric tests, when in fact p < .05 rejects normality.

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Testing Normality: Shapiro-Wilk, Q-Q Plots, and CLT | Biostatistics Handbook | Analysez