Statistical Tables/Shapiro-Wilk Critical Values

Hypothesis Testing

Shapiro-Wilk Critical Values

The critical W value below which your Shapiro-Wilk statistic signals non-normality — Monte Carlo estimated and clearly labeled as such.

These critical values are Monte Carlo estimates, not exact. The Shapiro-Wilk W statistic has no known closed-form null distribution, and the standard software algorithm for it (Royston's AS R94) is only published as compiled Fortran/C with no independently checkable source. Each value below comes from 3,000,000 simulated samples — the same method Shapiro & Wilk used to build their original 1965 tables — and lands within about 0.001–0.002 of the modern published reference values checked against it (see the FAQ below).

Critical W (reject normality if your W is below this value)

n	α = 0.1	α = 0.05	α = 0.02	α = 0.01
5	0.8129	0.7773	0.7322	0.7002
6	0.8273	0.7930	0.7500	0.7192
7	0.8403	0.8085	0.7679	0.7373
8	0.8518	0.8224	0.7842	0.7554
9	0.8618	0.8343	0.7981	0.7710
10	0.8704	0.8448	0.8108	0.7850
11	0.8782	0.8541	0.8220	0.7976
12	0.8849	0.8621	0.8320	0.8084
13	0.8910	0.8696	0.8407	0.8184
14	0.8964	0.8760	0.8486	0.8274
15	0.9013	0.8820	0.8560	0.8359
16	0.9058	0.8874	0.8626	0.8432
17	0.9099	0.8922	0.8686	0.8500
18	0.9135	0.8967	0.8738	0.8562
19	0.9169	0.9008	0.8788	0.8618
20	0.9199	0.9044	0.8833	0.8670
21	0.9228	0.9078	0.8875	0.8717
22	0.9254	0.9110	0.8914	0.8763
23	0.9279	0.9140	0.8954	0.8806
24	0.9303	0.9169	0.8987	0.8844
25	0.9324	0.9195	0.9020	0.8884
26	0.9345	0.9219	0.9049	0.8914
27	0.9364	0.9241	0.9077	0.8948
28	0.9382	0.9264	0.9103	0.8978
29	0.9398	0.9283	0.9126	0.9004
30	0.9414	0.9302	0.9150	0.9032
31	0.9429	0.9320	0.9172	0.9056
32	0.9443	0.9337	0.9193	0.9081
33	0.9456	0.9353	0.9213	0.9104
34	0.9469	0.9368	0.9233	0.9126
35	0.9481	0.9383	0.9250	0.9147
36	0.9493	0.9397	0.9267	0.9165
37	0.9504	0.9410	0.9284	0.9186
38	0.9515	0.9424	0.9299	0.9203
39	0.9525	0.9435	0.9314	0.9219
40	0.9534	0.9447	0.9328	0.9234
41	0.9543	0.9457	0.9341	0.9250
42	0.9553	0.9469	0.9355	0.9265
43	0.9561	0.9479	0.9368	0.9281
44	0.9569	0.9488	0.9379	0.9294
45	0.9577	0.9498	0.9390	0.9307
46	0.9585	0.9507	0.9402	0.9320
47	0.9592	0.9516	0.9414	0.9333
48	0.9599	0.9524	0.9423	0.9344
49	0.9606	0.9532	0.9433	0.9356
50	0.9612	0.9540	0.9442	0.9366
55	0.9642	0.9576	0.9486	0.9416
60	0.9667	0.9606	0.9523	0.9458
65	0.9689	0.9632	0.9554	0.9494
70	0.9708	0.9654	0.9582	0.9526
75	0.9725	0.9674	0.9607	0.9553
80	0.9739	0.9692	0.9628	0.9578
85	0.9753	0.9708	0.9647	0.9600
90	0.9764	0.9722	0.9664	0.9620
95	0.9775	0.9735	0.9680	0.9637
100	0.9785	0.9746	0.9694	0.9653

n is your sample size. Reject the null hypothesis of normality if your computed W is less than the tabled value — the Shapiro-Wilk test is one-tailed, since only small W (not large W) indicates non-normality. For n outside this range, use statistical software's built-in exact p-value computation instead.

How to use this tool

Compute your Shapiro-Wilk W statistic from your sorted data.
Choose your significance level, α.
Find your sample size, n.
Reject the null hypothesis of normality if your W is less than the tabled value.

Formula Reference

Shapiro-Wilk W statistic

W = (Σ aᵢ x₍ᵢ₎)² / Σ(xᵢ − x̄)²

Weighting coefficients

built from the expected order statistics of a standard normal sample, with a polynomial correction to the two most extreme

ArticleThe Shapiro-Wilk Test ArticleQ-Q Plots TableKolmogorov-Smirnov Critical Values GlossaryGlossary: Shapiro-Wilk Test

Frequently Asked Questions

Why does this table say the values are estimated rather than exact?+

Every other table on this site is computed exactly. This one can't be: the Shapiro-Wilk statistic's null distribution has no closed form, and the standard software transform for it (Royston's AS R94, used by R and SciPy) is only published as compiled Fortran with no independently checkable source. Rather than silently reproduce numbers we couldn't verify, each critical value here comes from 3 million simulated normal samples — the same method Shapiro and Wilk used to build their own original 1965 tables.

How accurate are these values?+

Checked against modern published reference values (Hanusz & Tarasińska, and a 2024 Wasserstein-distance-based approximation) at several sample sizes — n=20, 30, and 45 all land within 0.001 to 0.002 of both, consistently closer to them than to Shapiro & Wilk's original 1965 table, which is documented to have had small errors for n ≤ 20 later corrected by others.

Is the W statistic itself exact, or also estimated?+

Exact. The coefficients used to compute W are independently verified: computed for real and synthetic data across n = 5 to 100, they reproduce SciPy's own W statistic to within 1e-9 in every case checked. Only the critical value — the threshold W is compared against — relies on simulation.

Should I trust software's exact p-value over this table?+

For any sample size your software supports, yes — a computed p-value from a validated implementation like R's or SciPy's shapiro.test is more precise than reading a table. This table is for hand calculation, teaching, or a quick check without running software.

α = 0.1

α = 0.05

α = 0.02

α = 0.01

0.8129

0.7773

0.7322

0.7002

0.8273

0.7930

0.7500

0.7192

0.8403

0.8085

0.7679

0.7373

0.8518

0.8224

0.7842

0.7554

0.8618

0.8343

0.7981

0.7710

0.8704

0.8448

0.8108

0.7850

0.8782

0.8541

0.8220

0.7976

0.8849

0.8621

0.8320

0.8084

0.8910

0.8696

0.8407

0.8184

0.8964

0.8760

0.8486

0.8274

0.9013

0.8820

0.8560

0.8359

0.9058

0.8874

0.8626

0.8432

0.9099

0.8922

0.8686

0.8500

0.9135

0.8967

0.8738

0.8562

0.9169

0.9008

0.8788

0.8618

0.9199

0.9044

0.8833

0.8670

0.9228

0.9078

0.8875

0.8717

0.9254

0.9110

0.8914

0.8763

0.9279

0.9140

0.8954

0.8806

0.9303

0.9169

0.8987

0.8844

0.9324

0.9195

0.9020

0.8884

0.9345

0.9219

0.9049

0.8914

0.9364

0.9241

0.9077

0.8948

0.9382

0.9264

0.9103

0.8978

0.9398

0.9283

0.9126

0.9004

0.9414

0.9302

0.9150

0.9032

0.9429

0.9320

0.9172

0.9056

0.9443

0.9337

0.9193

0.9081

0.9456

0.9353

0.9213

0.9104

0.9469

0.9368

0.9233

0.9126

0.9481

0.9383

0.9250

0.9147

0.9493

0.9397

0.9267

0.9165

0.9504

0.9410

0.9284

0.9186

0.9515

0.9424

0.9299

0.9203

0.9525

0.9435

0.9314

0.9219

0.9534

0.9447

0.9328

0.9234

0.9543

0.9457

0.9341

0.9250

0.9553

0.9469

0.9355

0.9265

0.9561

0.9479

0.9368

0.9281

0.9569

0.9488

0.9379

0.9294

0.9577

0.9498

0.9390

0.9307

0.9585

0.9507

0.9402

0.9320

0.9592

0.9516

0.9414

0.9333

0.9599

0.9524

0.9423

0.9344

0.9606

0.9532

0.9433

0.9356

0.9612

0.9540

0.9442

0.9366

0.9642

0.9576

0.9486

0.9416

0.9667

0.9606

0.9523

0.9458

0.9689

0.9632

0.9554

0.9494

0.9708

0.9654

0.9582

0.9526

0.9725

0.9674

0.9607

0.9553

0.9739

0.9692

0.9628

0.9578

0.9753

0.9708

0.9647

0.9600

0.9764

0.9722

0.9664

0.9620

0.9775

0.9735

0.9680

0.9637

100

0.9785

0.9746

0.9694

0.9653

Frequently Asked Questions

Why does this table say the values are estimated rather than exact?+

How accurate are these values?+

Is the W statistic itself exact, or also estimated?+

Should I trust software's exact p-value over this table?+