Normality and Goodness of Fit

Lilliefors Test Table: Critical Values of D for Normality

Critical values for the Kolmogorov-Smirnov test of normality when the mean and standard deviation are estimated from the data.

n = 4 to 100
10 million simulations per n
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Lilliefors Test Table

Rows: sample size n · Columns: α · Cells: critical D. Reject normality if D ≥ the cell.

Lilliefors Test Table (Critical Values for Normality)
α → n ↓	0.2	0.15	0.1	0.05	0.01	0.001
4	0.3028	0.3215	0.3453	0.3752	0.4130	0.4327
5	0.2894	0.3027	0.3189	0.3430	0.3968	0.4388
6	0.2687	0.2809	0.2972	0.3234	0.3705	0.4234
7	0.2523	0.2643	0.2802	0.3041	0.3507	0.4008
8	0.2388	0.2503	0.2651	0.2879	0.3328	0.3827
9	0.2272	0.2381	0.2522	0.2742	0.3171	0.3653
10	0.2171	0.2275	0.2411	0.2621	0.3034	0.3508
11	0.2081	0.2181	0.2311	0.2513	0.2914	0.3377
12	0.2002	0.2099	0.2224	0.2420	0.2806	0.3254
13	0.1932	0.2025	0.2146	0.2335	0.2710	0.3146
14	0.1869	0.1958	0.2076	0.2259	0.2622	0.3047
15	0.1811	0.1898	0.2012	0.2189	0.2543	0.2955
16	0.1759	0.1844	0.1954	0.2127	0.2472	0.2876
17	0.1710	0.1792	0.1900	0.2068	0.2403	0.2799
18	0.1666	0.1746	0.1851	0.2015	0.2342	0.2729
19	0.1625	0.1703	0.1806	0.1965	0.2285	0.2666
20	0.1587	0.1663	0.1763	0.1919	0.2232	0.2603
21	0.1551	0.1626	0.1724	0.1877	0.2183	0.2547
22	0.1518	0.1591	0.1687	0.1836	0.2136	0.2493
23	0.1487	0.1559	0.1652	0.1799	0.2093	0.2444
24	0.1458	0.1528	0.1620	0.1763	0.2052	0.2396
25	0.1430	0.1499	0.1589	0.1730	0.2013	0.2350
26	0.1404	0.1471	0.1560	0.1698	0.1977	0.2310
27	0.1379	0.1445	0.1533	0.1669	0.1943	0.2270
28	0.1356	0.1421	0.1506	0.1640	0.1910	0.2229
29	0.1334	0.1398	0.1482	0.1614	0.1879	0.2198
30	0.1312	0.1375	0.1458	0.1588	0.1848	0.2162
35	0.1220	0.1279	0.1356	0.1476	0.1719	0.2012
40	0.1145	0.1200	0.1272	0.1385	0.1614	0.1889
45	0.1082	0.1134	0.1202	0.1309	0.1525	0.1784
50	0.1029	0.1078	0.1143	0.1245	0.1450	0.1696
60	0.0943	0.0988	0.1047	0.1140	0.1329	0.1557
70	0.0875	0.0917	0.0972	0.1058	0.1233	0.1445
80	0.0820	0.0859	0.0911	0.0992	0.1156	0.1355
90	0.0775	0.0811	0.0860	0.0936	0.1091	0.1279
100	0.0736	0.0771	0.0817	0.0890	0.1037	0.1215

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

How to read this table

Standardize the data using the sample mean and SD, and compute D, the largest distance between the empirical distribution function and the normal CDF.
Find the row for n and the column for α.
Reject normality if D is greater than or equal to the table value.

Worked example

A sample of 20 gives D = 0.21. Row 20, α = 0.05 gives 0.1919, so normality is rejected at the 5% level. The ordinary one-sample Kolmogorov-Smirnov critical value for n = 20 is 0.294, which would wrongly accept normality: estimating the mean and SD from the data makes the fit look better than it is, so the critical values must be smaller.

Click the underlined links to highlight the value in the table.

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Formula and how the values were computed

D = max |F_n(x) − Φ((x − x̄)/s)|

Critical values are percentiles of D from 10 million simulated normal samples for each n, using the sample SD with n − 1 in the denominator (as Lilliefors did). Simulation error is below 0.0002. Lilliefors' original 1967 table used 1,000 samples; these values agree closely with the later tables of Dallal and Wilkinson.

Frequently asked questions

Lilliefors or Shapiro-Wilk?

Shapiro-Wilk is usually more powerful against the departures from normality that matter in practice. Lilliefors is useful when you want a test based on the whole distribution function.

What about n above 100?

D is roughly proportional to 1/√n, so the n = 100 row multiplied by 10/√n gives a good approximation.

Lilliefors Test Table: Critical Values of D for Normality

Lilliefors Test Table

How to read this table

Worked example

Calculator

Formula and how the values were computed

Frequently asked questions

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