Normality and Goodness of Fit

Shapiro-Wilk Table: Critical Values of W and Coefficients

Critical values and coefficients for the Shapiro-Wilk test of normality, consistent with the W statistic reported by R, SAS, Stata and SciPy.

n = 3 to 50
Coefficients included
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Shapiro-Wilk Table

Two tables: critical values of W, then the coefficients aᵢ. Reject normality if W is smaller than the critical value.

Critical values of W: reject normality if W is smaller than the value

Critical values of W: reject normality if W is smaller than the value
α → n ↓	0.01	0.02	0.05	0.1
3	0.7545	0.7590	0.7723	0.7939
4	0.6923	0.7168	0.7600	0.7993
5	0.7002	0.7323	0.7773	0.8129
6	0.7190	0.7503	0.7931	0.8275
7	0.7379	0.7680	0.8087	0.8403
8	0.7549	0.7837	0.8222	0.8517
9	0.7712	0.7984	0.8345	0.8619
10	0.7844	0.8106	0.8447	0.8705
11	0.7975	0.8220	0.8540	0.8782
12	0.8087	0.8321	0.8623	0.8850
13	0.8186	0.8409	0.8696	0.8910
14	0.8277	0.8488	0.8761	0.8964
15	0.8358	0.8558	0.8819	0.9014
16	0.8432	0.8625	0.8873	0.9057
17	0.8503	0.8686	0.8922	0.9098
18	0.8562	0.8740	0.8966	0.9135
19	0.8618	0.8788	0.9006	0.9168
20	0.8671	0.8834	0.9044	0.9200
21	0.8723	0.8879	0.9080	0.9228
22	0.8765	0.8915	0.9111	0.9255
23	0.8806	0.8952	0.9140	0.9279
24	0.8846	0.8987	0.9168	0.9302
25	0.8881	0.9019	0.9194	0.9324
26	0.8915	0.9048	0.9219	0.9345
27	0.8946	0.9076	0.9242	0.9363
28	0.8977	0.9102	0.9263	0.9381
29	0.9006	0.9127	0.9283	0.9398
30	0.9031	0.9151	0.9302	0.9413
31	0.9057	0.9173	0.9320	0.9428
32	0.9082	0.9194	0.9337	0.9443
33	0.9104	0.9213	0.9353	0.9456
34	0.9124	0.9232	0.9368	0.9469
35	0.9146	0.9250	0.9383	0.9481
36	0.9165	0.9267	0.9397	0.9493
37	0.9182	0.9282	0.9410	0.9504
38	0.9202	0.9299	0.9423	0.9515
39	0.9217	0.9313	0.9435	0.9525
40	0.9235	0.9328	0.9447	0.9534
41	0.9252	0.9342	0.9458	0.9544
42	0.9266	0.9355	0.9468	0.9552
43	0.9280	0.9367	0.9479	0.9561
44	0.9295	0.9380	0.9489	0.9570
45	0.9308	0.9391	0.9498	0.9578
46	0.9319	0.9402	0.9507	0.9585
47	0.9332	0.9412	0.9516	0.9592
48	0.9345	0.9424	0.9524	0.9599
49	0.9355	0.9432	0.9532	0.9606
50	0.9367	0.9443	0.9540	0.9612

Coefficients aᵢ (i = 1 pairs the largest with the smallest observation)

Coefficients aᵢ (i = 1 pairs the largest with the smallest observation)
Coefficient index (i) → n ↓	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
2	0.7071
3	0.7071
4	0.6873	0.1663
5	0.6646	0.2414
6	0.6430	0.2807	0.0883
7	0.6231	0.3030	0.1411
8	0.6051	0.3163	0.1751	0.0565
9	0.5887	0.3243	0.1982	0.0951
10	0.5737	0.3290	0.2143	0.1228	0.0401
11	0.5600	0.3315	0.2260	0.1433	0.0698
12	0.5474	0.3326	0.2345	0.1589	0.0924	0.0304
13	0.5358	0.3327	0.2408	0.1709	0.1101	0.0540
14	0.5250	0.3320	0.2455	0.1804	0.1242	0.0729	0.0240
15	0.5150	0.3309	0.2489	0.1879	0.1356	0.0881	0.0435
16	0.5056	0.3295	0.2514	0.1939	0.1448	0.1007	0.0594	0.0196
17	0.4968	0.3277	0.2532	0.1987	0.1525	0.1111	0.0727	0.0360
18	0.4885	0.3259	0.2545	0.2026	0.1589	0.1199	0.0839	0.0497	0.0164
19	0.4807	0.3238	0.2552	0.2057	0.1642	0.1273	0.0934	0.0613	0.0304
20	0.4734	0.3217	0.2557	0.2083	0.1686	0.1336	0.1015	0.0713	0.0423	0.0140
21	0.4664	0.3196	0.2558	0.2104	0.1724	0.1390	0.1085	0.0799	0.0526	0.0261
22	0.4598	0.3174	0.2557	0.2120	0.1756	0.1436	0.1145	0.0874	0.0616	0.0366	0.0122
23	0.4535	0.3152	0.2554	0.2133	0.1783	0.1476	0.1198	0.0940	0.0694	0.0458	0.0228
24	0.4475	0.3130	0.2550	0.2143	0.1806	0.1511	0.1245	0.0997	0.0764	0.0539	0.0321	0.0107
25	0.4418	0.3108	0.2545	0.2151	0.1825	0.1542	0.1285	0.1048	0.0825	0.0611	0.0404	0.0201
26	0.4363	0.3087	0.2538	0.2157	0.1842	0.1568	0.1321	0.1093	0.0879	0.0675	0.0477	0.0285	0.0095
27	0.4311	0.3065	0.2531	0.2161	0.1856	0.1591	0.1353	0.1133	0.0928	0.0732	0.0543	0.0359	0.0179
28	0.4261	0.3044	0.2523	0.2163	0.1868	0.1611	0.1381	0.1169	0.0971	0.0783	0.0602	0.0427	0.0255	0.0085
29	0.4213	0.3023	0.2515	0.2165	0.1877	0.1629	0.1406	0.1201	0.1010	0.0829	0.0655	0.0487	0.0323	0.0161
30	0.4167	0.3003	0.2506	0.2165	0.1886	0.1644	0.1428	0.1230	0.1045	0.0871	0.0703	0.0542	0.0384	0.0229	0.0076
31	0.4122	0.2983	0.2496	0.2164	0.1892	0.1658	0.1448	0.1256	0.1077	0.0908	0.0747	0.0591	0.0440	0.0292	0.0145
32	0.4080	0.2963	0.2487	0.2163	0.1898	0.1669	0.1465	0.1279	0.1106	0.0942	0.0787	0.0636	0.0491	0.0348	0.0208	0.0069
33	0.4039	0.2943	0.2477	0.2161	0.1902	0.1679	0.1481	0.1300	0.1132	0.0973	0.0823	0.0678	0.0537	0.0400	0.0265	0.0132
34	0.3999	0.2924	0.2467	0.2158	0.1906	0.1688	0.1495	0.1319	0.1155	0.1002	0.0856	0.0715	0.0580	0.0448	0.0318	0.0190	0.0063
35	0.3960	0.2905	0.2457	0.2155	0.1908	0.1696	0.1508	0.1336	0.1177	0.1028	0.0886	0.0750	0.0619	0.0491	0.0366	0.0243	0.0121
36	0.3923	0.2887	0.2447	0.2151	0.1910	0.1703	0.1519	0.1351	0.1197	0.1051	0.0914	0.0782	0.0655	0.0531	0.0410	0.0292	0.0174	0.0058
37	0.3887	0.2869	0.2437	0.2147	0.1911	0.1708	0.1529	0.1365	0.1215	0.1073	0.0939	0.0811	0.0688	0.0568	0.0451	0.0336	0.0223	0.0111
38	0.3853	0.2851	0.2427	0.2142	0.1911	0.1713	0.1538	0.1378	0.1231	0.1093	0.0963	0.0838	0.0718	0.0602	0.0489	0.0378	0.0269	0.0161	0.0054
39	0.3819	0.2833	0.2417	0.2138	0.1911	0.1717	0.1546	0.1390	0.1246	0.1111	0.0984	0.0863	0.0746	0.0633	0.0524	0.0416	0.0311	0.0206	0.0103
40	0.3786	0.2816	0.2406	0.2133	0.1911	0.1721	0.1553	0.1400	0.1260	0.1128	0.1004	0.0886	0.0772	0.0663	0.0556	0.0452	0.0350	0.0249	0.0149	0.0050
41	0.3755	0.2800	0.2396	0.2128	0.1910	0.1723	0.1559	0.1410	0.1272	0.1144	0.1023	0.0907	0.0797	0.0690	0.0586	0.0485	0.0386	0.0288	0.0191	0.0096
42	0.3724	0.2783	0.2386	0.2122	0.1908	0.1726	0.1564	0.1418	0.1284	0.1158	0.1040	0.0927	0.0819	0.0715	0.0614	0.0516	0.0419	0.0325	0.0231	0.0138	0.0046
43	0.3694	0.2767	0.2376	0.2117	0.1907	0.1727	0.1569	0.1426	0.1294	0.1171	0.1056	0.0946	0.0840	0.0739	0.0640	0.0545	0.0451	0.0359	0.0268	0.0178	0.0089
44	0.3665	0.2751	0.2366	0.2111	0.1905	0.1729	0.1573	0.1433	0.1304	0.1184	0.1070	0.0963	0.0860	0.0761	0.0665	0.0572	0.0480	0.0391	0.0303	0.0215	0.0129	0.0043
45	0.3637	0.2736	0.2356	0.2105	0.1902	0.1730	0.1577	0.1439	0.1313	0.1195	0.1084	0.0979	0.0878	0.0781	0.0688	0.0597	0.0508	0.0421	0.0335	0.0250	0.0166	0.0083
46	0.3609	0.2720	0.2346	0.2099	0.1900	0.1730	0.1580	0.1445	0.1321	0.1205	0.1097	0.0994	0.0895	0.0801	0.0709	0.0620	0.0534	0.0449	0.0365	0.0283	0.0201	0.0121	0.0040
47	0.3582	0.2705	0.2336	0.2093	0.1897	0.1730	0.1583	0.1450	0.1328	0.1215	0.1108	0.1008	0.0911	0.0819	0.0729	0.0642	0.0558	0.0475	0.0394	0.0314	0.0234	0.0156	0.0078
48	0.3556	0.2691	0.2327	0.2087	0.1894	0.1730	0.1585	0.1455	0.1335	0.1224	0.1119	0.1021	0.0926	0.0835	0.0748	0.0663	0.0580	0.0500	0.0420	0.0342	0.0265	0.0189	0.0113	0.0038
49	0.3531	0.2676	0.2317	0.2081	0.1891	0.1730	0.1587	0.1459	0.1341	0.1232	0.1130	0.1033	0.0940	0.0851	0.0766	0.0683	0.0602	0.0523	0.0446	0.0369	0.0294	0.0220	0.0146	0.0073
50	0.3506	0.2662	0.2308	0.2075	0.1888	0.1729	0.1589	0.1463	0.1347	0.1240	0.1139	0.1044	0.0953	0.0866	0.0782	0.0701	0.0622	0.0545	0.0469	0.0395	0.0322	0.0249	0.0178	0.0106	0.0035

Tip: click any value to highlight its row and column.

How to read this table

Sort the data: x₍₁₎ ≤ x₍₂₎ ≤ … ≤ x₍ₙ₎.
Using the coefficients for your n, compute b = Σ aᵢ (x₍ₙ₊₁₋ᵢ₎ − x₍ᵢ₎), pairing the largest value with the smallest, and so on.
W = b² / Σ(xᵢ − x̄)².
Reject normality if W is smaller than the critical value for your α.

Worked example

Twenty biomarker values give W = 0.89. Row 20, α = 0.05 gives 0.9044 and α = 0.01 gives 0.8671. Because 0.8671 < 0.89 < 0.9044, normality is rejected at the 5% level but not at the 1% level (0.01 < p < 0.05). A log transformation is often the next step for skewed laboratory values.

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Formula and how the values were computed

W = (Σ aᵢ (x₍ₙ₊₁₋ᵢ₎ − x₍ᵢ₎))² / Σ (xᵢ − x̄)²

Coefficients follow Royston's algorithm (AS R94), which approximates the exact values m′V⁻¹ closely and is what modern software uses. Critical values are percentiles of W with these coefficients, estimated from 4 million simulated normal samples for each n (n = 3 is exact). For n above about 20 they differ from Shapiro and Wilk's original 1965 table, whose coefficients were approximations; use this table with the coefficients on this page or with software output.

Frequently asked questions

Why do these values differ from some printed tables?

Printed tables often reproduce Shapiro and Wilk's 1965 values, which used approximate coefficients for n > 20. Royston's coefficients are closer to the exact ones and are used by R, SAS, Stata and SciPy, so this table matches the W you get from software.

My sample is large. Should I rely on the test?

With large samples the test rejects for trivial departures from normality. Look at a normal Q-Q plot as well, and remember that t-tests and ANOVA are robust to moderate non-normality in large samples.

Shapiro-Wilk Table: Critical Values of W and Coefficients

Shapiro-Wilk Table

Critical values of W: reject normality if W is smaller than the value

Coefficients aᵢ (i = 1 pairs the largest with the smallest observation)

How to read this table

Worked example

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Formula and how the values were computed

Frequently asked questions

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