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Normality

Shapiro-Wilk Test — Normal & Non-Normal Worked Examples

Two examples: systolic BP confirming normality W=0.972, p=.895; ER waiting times detecting skew W=0.825, p=.030.

📊 11a: W=0.972 ✓ | 11b: W=0.825 ✗
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📐 11 · Normality Testing (Shapiro-Wilk)

11a · Systolic BP — Confirming Normality
Normality
Research Question: Before running a t-test, verify that systolic blood pressure readings (n=10) are normally distributed.
i12345678910
BP (mmHg)118120122124124126128130133135
Shapiro-Wilk W Statistic
W = b 2 S S , b = ∑ i = 1 ⌊ n / 2 ⌋ a i ( x ( n + 1 − i ) − x ( i ) ) , S S = ∑ ( x i − x ¯ ) 2
  1. 1
    Sort data ascending; compute mean and SS:
    x ¯ = 126.0 , S S = ( 118 − 126 ) 2 + ⋯ + ( 135 − 126 ) 2 = 274
  2. 2
    Apply SW coefficients (n=10): a₁=0.5739, a₂=0.3291, a₃=0.2141, a₄=0.1224, a₅=0.0399
    x ( 10 ) − x ( 1 ) = 17 , x ( 9 ) − x ( 2 ) = 13 , x ( 8 ) − x ( 3 ) = 8 , x ( 7 ) − x ( 4 ) = 4 , x ( 6 ) − x ( 5 ) = 2 b = 0.5739 ( 17 ) + 0.3291 ( 13 ) + 0.2141 ( 8 ) + 0.1224 ( 4 ) + 0.0399 ( 2 ) = 9.756 + 4.278 + 1.713 + 0.490 + 0.080 = 16.317
  3. 3
    W = ( 16.317 ) 2 274 = 266.2 274 = 0.972
0.972
W
.895
p
10
n
⚪ Fail to Reject H₀ — Data is consistent with normality. Proceed with t-test. W(10) = 0.97, p = .895.
APA-7
Shapiro-Wilk testing confirmed that blood pressure scores were approximately normally distributed, W(10) = 0.97, p = .895.
11b · Emergency Waiting Times — Non-Normal (Right-Skewed)
Non-Normal
Research Question: Are emergency department waiting times (minutes, n=10) normally distributed?
i12345678910
Minutes581012152035527895
  1. 1
    x ¯ = 33.0 , S S = 9,106 Differences:  x ( 10 ) − x ( 1 ) = 90 , 70 , 42 , 23 , 5 b = 0.5739 ( 90 ) + 0.3291 ( 70 ) + 0.2141 ( 42 ) + 0.1224 ( 23 ) + 0.0399 ( 5 ) = 51.65 + 23.04 + 8.99 + 2.82 + 0.20 = 86.70 W = ( 86.70 ) 2 9106 = 7516.9 9106 = 0.825
0.825
W
.030
p
<.05
Violation
🔴 Reject H₀ — Waiting times are NOT normally distributed, W(10) = 0.83, p = .030. Use non-parametric alternatives (e.g., Mann-Whitney U).
💡 Decision rule: p > .05 → normality assumed (safe to proceed with parametric tests). p ≤ .05 → use non-parametric tests or apply log/square-root transformation. For large samples (n > 50), prefer Kolmogorov-Smirnov or visual Q-Q plots.

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