A Million Random Digits with 100,000 Normal Deviates

Foreword to the Online Edition

This book was a product of RAND's computing power (and patience). The tables of random numbers in the book have become a standard reference in engineering and econometrics textbooks and have been widely used in gaming and simulations that employ Monte Carlo trials. Still the largest known source of random digits and normal deviates, the work is routinely used by statisticians, physicists, polltakers, market analysts, lottery administrators, and quality control engineers.

A humorous sidelight: The New York Public Library originally indexed this book under the heading "Psychology."

Acknowledgments

The following persons participated in the production, testing, and preparation for publication of the tables of random digits and random normal deviates: Paul Armer, E. C. Bower, Mrs. Bernice Brown, G. W. Brown, Walter Frantz, J. J. Goodpasture, W. F. Gunning, Cecil Hastings, Olaf Helmer, M. L. Juncosa, J. D. Madden, A. M. Mood, R. T. Nash, J. D. Williams. These tables were prepared in connection with analyses done for the United States Air Force.

Introduction

Early in the course of research at The RAND Corporation a demand arose for random numbers; these were needed to solve problems of various kinds by experimental probability procedures, which have come to be called Monte Carlo methods. Many of the applications required a large supply of random digits or normal deviates of high quality, and the tables presented here were produced to meet those requirements. The numbers have been used extensively by research workers at RAND, and by many others, in the solution of a wide range of problems during the past seven years.

One distinguishing feature of the digit table is its size. On numerous RAND problems the largest existing table of Kendall and Smith, 1939, would have had to be used many times over, with the consequent dangers of introducing unwanted correlations. The feasibility of working with as large a table as the present one resulted from developments in computing machinery which made possible the solving of very complicated distribution problems in a reasonable time by Monte Carlo methods. The tables were constructed primarily for use with punched card machines. With the high-speed electronic computers recently developed, the storage of such tables is usually not practical and, in fact, much larger tables than the present one are often required; these machines have caused research workers to turn to pseudo-random numbers which are computed by simple arithmetic processes directly by the machine as needed. These developments are summarized in Juncosa, 1953; Meyer, Gephart, and Rasmussen, 1954; and Moshman, 1954, where other references may be found. The Monte Carlo Method, 1951; Curtiss, 1949; Kahn and Marshall, 1953; and Kahn, 1956, discuss the uses and applications of the Monte Carlo methods and give references to other applications.

Production of the Random Digits

The random digits in this book were produced by rerandomization of a basic table generated by an electronic roulette wheel. Briefly, a random frequency pulse source, providing on the average about 100,000 pulses per second, was gated about once per second by a constant frequency pulse. Pulse standardization circuits passed the pulses through a 5-place binary counter. In principle the machine was a 32-place roulette wheel which made, on the average, about 3000 revolutions per trial and produced one number per second. A binary-to-decimal converter was used which converted 20 of the 32 numbers (the other twelve were discarded) and retained only the final digit of two-digit numbers; this final digit was fed into an IBM punch to produce finally a punched card table of random digits.

Production from the original machine showed statistically significant biases, and the engineers had to make several modifications and refinements of the circuits before production of apparently satisfactory numbers was achieved. The basic table of a million digits was then produced during May and June of 1947. This table was subjected to fairly exhaustive tests and it was found that it still contained small but statistically significant biases. For example, the following table^[1] shows the results of three tests (described later) on two blocks of 125,000 digits:

	Block 1		Block 2
	χ2	Probability	χ2	Probability
Frequency (9 d.f.a)	6.0	.74	21.0	.02
Odd-even (1 d.f)	3.0	.09	7.0	<.0l
Serial (81 d.f.)	78.7	.55	105.6	.03

^aThe letters "d.f." (degrees of freedom) identify a parameter associated with the test. A discussion of the test may be found in any textbook on statistics.

Block 1 was produced immediately after a careful tune-up of the machine; Block 2 was produced after one month of continuous operation without adjustment. Apparently the machine had been running down despite the fact that periodic electronic checks indicated it had remained in good order.

The table was regarded as reasonably satisfactory because the deviations from expectations in the various tests were all very small—the largest being less than 2 per cent—and no further effort was made to generate better numbers with the machine. However, the table was transformed by adding pairs of digits modulo 10 in order to improve the distribution of the digits. There were 20,000 punched cards with 50 digits per card; each digit on a given card was added modulo 10 to the corresponding digit of the preceding card to yield a rerandomized digit. It is this transformed table which is published here and which is the subject of the tests described below.

The transformation was expected to, and did, improve the distribution in view of a limit theorem to the effect that sums of random variables modulo 1 have the uniform distribution over the unit interval as their limiting distribution. (See Horton and Smith, 1949, for a version of this theorem for discrete variates.)

These tables were reproduced by photo-offset from pages printed by the IBM model 856 Cardatype. Because of the very nature of the tables, it did not seem necessary to proofread every page of the final manuscript in order to catch random errors of the Cardatype. All pages were scanned for systematic errors, every twentieth page was proofread (starting with page 10 for both the digits and deviates), and every fortieth page (starting with page 5 for both the digits and deviates) was summed and the totals checked against sums obtained from the cards.^[2]

Tests on the Random Digits

Frequency Tests. The table was divided into 1000 blocks of 1000 digits each and the frequency of each digit was recorded for each block. Then for each block a goodness-of-fit χ² was computed with 9 d.f. These 1000 values of χ² provided an empirical fit to the χ² distribution (with 9 d.f.); to test the fit, a goodness-of-fit χ² was computer using 50 class intervals, each of which was expected to contain 2 per cent of the values. (The number of intervals was chosen in accordance with the result of Wald and Mann, 1942.) The value of the test χ² was 54.6 which, for 49 d.f., corresponded to about the 0.45 probability level.

To examine further the frequencies, the digits were tallied in 20 blocks of 50,000 digits each. The results are shown in Table 1 together with the goodness-of-fit χ² for each block. On the total frequencies the χ² (13.316) for 9 d.f. has been partitioned into three components as follows:

	χ2	d.f.	Probability
Odd versus even digits	1.37	1	0.25
Within groups of odd digits	7.90	4	0.10
Within groups of even digits	4.04	4	0.40

Block No.	0	1	2	3	4	5	6	7	8	9	χ2
1	4923	5013	4916	4951	5109	4993	5055	5080	4986	4974	7.556
2	4870	4956	5080	5097	5066	5034	4902	4974	5012	5009	10.132
3	5065	5014	5034	5057	4902	5061	4942	4946	4960	5019	6.078
4	5009	5053	4966	4891	5031	4895	5037	5062	5170	4886	15.004
5	5033	4982	5180	5074	4892	4992	5011	5005	4959	4872	13.846
6	4976	4993	4932	5039	4965	5034	4943	4932	5116	5070	7.076
7	5011	5152	4990	5047	4974	5107	4869	4925	5023	4902	14.116
8	5003	5092	5163	4936	5020	5069	4914	4943	4914	4946	13.051
9	4860	4899	5138	4959	5089	5047	5030	5039	5002	4937	13.410
10	4998	4957	4964	5124	4909	4995	5053	4946	4995	5059	7.212
11	4948	5048	5041	5077	5051	5004	5024	4886	4917	5004	7.142
12	4958	4993	5064	4987	5041	4984	4991	4987	5113	4882	6.992
13	4968	4961	5029	5038	5022	5023	5010	4988	4936	5025	2.162
14	5110	4923	5025	4975	5095	5051	5035	4962	4942	4882	10.172
15	5094	4962	4945	4891	5014	5002	5038	5023	5179	4852	16.261
16	4957	5035	5051	5021	5036	4927	5022	4988	4910	5053	4.856
17	5088	4989	5042	4948	4999	5028	5037	4893	5004	4972	5.347
18	4970	5034	4996	5008	5049	5016	4954	4989	4970	5014	1.625
19	4998	4981	4984	5107	4874	4980	5057	5020	4978	5021	6.584
20	4963	5013	5101	5084	4956	4972	5018	4971	5021	4901	6.584
Total	99802	100050	100641	100311	100094	100214	99942	99559	100107	99280	13.316

Of the 200 frequencies recorded in Table 1, 59 (29.5 per cent) deviate from 5000 by more than σ (= 30√5 = 67.08), and 8 (4 per cent) deviate from 5000 by more than 2σ. Of the twenty χ² values in Table 1, eight exceed the 50 per cent value (8.34), two fall below the 10 per cent value (4.17), and two exceed the 90 per cent value (14.7).

Poker Tests.Sets of 5 digits in blocks of 5000 digits were taken to be poker hands and were classified as:

Class	Symbol	Expected Frequency Per Block
Busts	abcde	302.4
Pairs	aabcd	504
Two pairs	aabbc	108
Threes	aaabc	72
Full house	aaabb	9
Fours	aaaab	{4.5}
Five	aaaaa	{0.1}

There were 200 sets of 1000 poker hands in the table, and for each set a goodness-of-fit χ² was computed with 5 d.f. (the fours and fives were combined). The manner in which these 200 values fit the χ² distribution is shown in Table 2.

Probability	Values of χ2	Expected Frequency	Observed Frequency
P > .90	0 – 1.60	20	22
.90 >P > .80	1.61 – 2.35	20	19
.80 > P > .70	2.36 – 3.00	20	22
.70 > P > .60	3.01 – 3.70	20	19
.60 > P > .50	3.71 – 4.35	20	20
.50 > P > .40	4.36 – 5.20	20	29
.40 > P > .30	5.21 – 6.10	20	22
.30 > P > .20	6.11 – 7.30	20	15
.20 > P > .10	7.31 – 9.20	20	15
P < .10	9.21 or more	20	17
Total		200	200

The goodness-of-fit test gives:

χ2 = 7.7 for 9 d.f.,

P = 0.55.

The combined frequencies of poker hands in the whole table are shown in Table 3. The largest difference between expected and observed frequencies (for threes) is about 2.25 times its standard deviation, which is roughly at about the 9 or 10 per cent probability level (looking merely at the largest of five independent normal observations).

Classes	Expected Frequency	Observed Frequency
Busts (abcde)	60,480	60,479
Pairs (aabcd)	100,800	100,570
Two pairs (aabbc)	21,600	21,572
Threes (aaabc)	14,400	14,659
Full house (aaabb)	1,800	1,788
Fours (aaaab)	{900}	{914}
Fives (aaaaa)	{20}	{18}
Total	200,000	200,000

The goodness-of-fit test gives:

χ2 = 5.5 for 5 d.f.,

P = 0.35.

Also, the frequencies of poker hands were computed for each of ten blocks of 100,000 digits and the mean and standard deviation was computed from the ten values for each kind of hand. The results are shown in Table 4.

Classes	Theoretical Mean	Actual Mean	Theoretical Std. Dev.	Actual Std. Dev.
Busts	6048	6047.9	64.9	60.3
Pairs	10080	10057.0	70.7	78.4
Two pairs	2160	2157.2	43.9	45.8
Threes	1440	1465.9	36.9	26.6
Full house	180	178.8	13.4	8.9
Fours	90	91.4	9.5	11.5
Fives	2	1.8	1.4	1.9

Serial and Run Tests. Some further tests were made on the first block of 50,000 digits to look particularly for any evidence of serial association among the digits. The serial test classified every successive pair of digits by each digit of the pair in a ten-by-ten table. The frequencies of the different pairs are given in Table 5, where the first digit of the pair is shown in the left column of the table and the second digit is shown at the top. Thus there were 510 cases in which a zero followed a one. The frequency χ² for the row (or column) totals is 7.56, which is about the 0.60 probability level for 9 d.f.

	Second Digit
First Digit	0	1	2	3	4	5	6	7	8	9	Total
0	508	456	509	507	502	489	471	504	488	489	4923
1	510	514	474	514	504	481	496	486	507	527	5013
2	451	523	493	484	502	466	514	506	493	484	4916
3	500	472	476	466	513	478	540	513	530	463	4951
4	513	561	481	485	526	513	485	510	524	511	5109
5	475	490	527	507	493	481	489	512	465	554	4993
6	494	486	491	483	525	504	530	539	513	490	5055
7	508	512	454	498	550	533	516	504	485	520	5080
8	463	503	475	514	520	544	514	491	520	442	4986
9	501	496	536	493	474	504	500	515	461	494	4974
Total	4923	5013	4916	4951	5109	4993	5055	5080	4986	4974	50000

Table 5 can be tested by a criterion originally due to Kendall and Smith, 1939, and revised by Good, 1953. Assuming all pairs equally likely we get a normalized sum of squared deviations of 107.8. However, this statistic does not have a χ²-distribution. On the other hand, it is the sum of the error variation and twice the row (or column) variation, where under the assumption of perfect randomness, the error variation is asymptotically distributed like χ² with 81 degrees of freedom. We take the error variation as our test criterion. This gives a χ² of 107.8 – 2(7.56) ~ 92.7, which is about the 0.18 level for 81 degrees of freedom.

Finally, in the same block of 50,000 digits all runs were counted with the obviously satisfactory results shown in Table 6.

Length of Run	Expected Frequency	Observed Frequency
r = 1	40500	40410
r = 2	4050	4055
r = 3	405	421
r = 4	40.5	48
r ≥ 5	4.5	5

Normal Deviates

Half of the random digit table was used to produce 100,000 standard normal deviates by solving for x in Equation (1),

where D is a five-digit number from the table and

is the cumulative standard normal distribution. The Bureau of Standards tables of F(x) were used (1948).

The deviates were determined by the five-digit numbers on the left-hand half of every page of the digit table. The deviates in the first column correspond page by page with the five-figure digits in the first column of the first 200 pages of the digit table; the deviates in the second column correspond page by page with the first column of the second 200 pages of the digit table. Similarly, the third and fourth columns of deviates were derived from the second column of five-figure digits, etc.

A χ² test of the fit of the entire table of deviates to the normal distribution was performed using 400 class intervals (Mann and Wald, 1942) with roughly 250 expected in each. The χ² value was found to be 346.4, which for 399 d.f. indicates a very close fit; the probability of a larger value of χ² is about 0.97. The detailed data for this test are given in Table 7.

Class Limits	Expected Number	Observed Number		(Obs.- Exp.)2/Exp.
		Left Tail	Right Tail	Left Tail	Right Tail
.000	239.364	229	267	.449	3.191
.006	239.355	253	240	.778	.002
.012	239.338	225	232	.859	.225
.018	279.195	275	286	.063	.166
.025	239.271	243	245	.058	.137
.031	239.227	232	255	.218	1.040
.037	239.173	285	221	8.781	1.381
.043	278.958	284	277	.091	.014
.050	239.029	225	244	.823	.103
.056	238.948	225	218	.814	1.836
.062	238.860	260	231	1.871	.259
.068	278.546	225	287	10.293	.257
.075	238.638	253	211	.864	3.201
.081	238.522	224	224	.884	.884
.087	278.118	254	290	2.091	.508
.094	238.242	212	222	2.891	1.107
.100	238.098	225	250	.721	.595
.106	277.590	297	272	1.357	.113
.113	237.760	217	235	1.813	.032
.119	237.590	232	224	.132	.777
.125	237.413	230	266	.231	3.442
.131	276.744	262	272	.786	.081
.138	236.998	244	226	.207	.510
.144	236.792	235	250	.014	.737
.150	275.989	271	275	.090	.004
.157	236.320	246	273	.397	5.693
.163	275.415	267	279	.257	.047
.170	235.810	250	221	.854	.930
.176	235.561	233	233	.028	.028
.182	274.495	265	291	.328	.992
.189	234.994	230	236	.106	.004
.195	234.718	250	233	.995	.013
.201	273.481	263	283	.402	.331
.208	234.095	232	228	.019	.159
.214	272.731	302	292	3.141	1.361
.221	233.435	243	224	.392	.381
.227	233.117	223	241	.439	.267
.233	271.557	276	286	.073	.768
.240	232.401	234	248	.011	1.047
.246	270.701	292	302	1.676	3.619
.253	231.649	229	225	.030	.191
.259	269.802	274	253	.065	1.046
.266	230.859	251	227	1.757	.065
.272	268.860	269	271	.000	.017
.279	230.034	226	233	.071	.038
.285	267.876	266	274	.013	.140
.292	229.173	235	221	.148	.291
.298	266.850	248	238	1.332	3.119
.305	266.282	249	283	1.122	1.050
.312	227.779	240	216	.656	.609
.318	265.193	244	274	1.694	.292
.325	226.830	213	234	.843	.227
.331	264.065	284	284	1.505	1.505
.338	263.440	282	269	1.308	.117
.345	225.302	230	214	.098	.567
.351	262.251	299	265	5.150	.029
.358	261.596	267	279	.112	1.158
.365	223.693	229	217	.126	.200
.371	260.347	275	279	.825	1.336
.378	259.659	266	253	.155	.171
.385	222.009	224	225	.018	.040
.391	258.353	266	247	.226	.499
.398	257.633	250	244	.226	.721
.405	256.905	268	269	.479	.569
.412	256.164	260	283	.057	2.811
.419	255.415	243	250	.603	.115
.426	254.654	284	269	3.382	.808
.433	217.661	230	208	.699	.429
.439	253.215	265	259	.548	.132
.446	252.425	251	250	.008	.023
.453	251.626	257	235	.115	1.099
.460	250.817	264	249	.693	.013
.467	249.999	256	231	.144	1.444
.474	249.170	249	271	.000	1.913
.481	248.333	244	268	.076	1.558
.488	247.486	233	274	.848	2.841
.495	246.630	239	246	.236	.002
.502	280.803	276	264	.082	1.005
.510	244.765	275	262	3.735	1.214
.517	243.881	249	254	.107	.420
.524	242.988	231	250	.591	.202
.531	242.087	223	242	1.505	.000
.538	275.555	288	256	.562	1.388
.546	240.126	229	228	.516	.612
.553	239.199	246	244	.193	.096
.560	272.224	282	281	.351	.283
.568	237.183	235	252	.020	.926
.575	236.231	255	229	1.491	.221
.582	268.802	255	273	.709	.066
.590	234.163	239	211	.100	2.291
.597	266.419	281	276	.798	.345
.605	232.062	218	220	.852	.627
.612	263.999	270	249	.136	.852
.620	262.692	261	285	.011	1.894
.628	228.776	253	207	2.565	2.073
.635	260.216	272	260	.534	.000
.643	258.881	256	265	.032	.145
.651	225.417	233	230	.255	.093
.658	256.352	263	270	.172	.727
.666	254.989	272	248	1.135	.192
.674	253.618	266	263	.605	.347
.682	252.239	245	235	.208	1.178
.690	250.849	255	274	.069	2.137
.698	249.453	239	240	.438	.358
.706	248.048	228	252	1.620	.063
.714	246.635	248	243	.008	.054
.722	245.215	230	254	.944	.315
.730	243.787	235	246	.317	.020
.738	272.544	296	240	2.019	3.886
.747	240.729	231	230	.393	.478
.755	239.279	240	250	.002	.480
.763	267.449	269	266	.009	.008
.772	236.177	229	256	.218	1.664
.780	263.944	292	283	2.982	1.376
.789	233.048	241	215	.271	1.398
.797	260.410	246	276	.797	.933
.806	258.526	239	227	1.475	3.844
.815	228.216	228	203	.000	2.786
.823	254.953	235	236	1.562	1.409
.832	253.050	262	250	.317	.037
.841	251.143	262	258	.469	.187
.850	249.228	241	261	.272	.556
.859	247.309	245	225	.022	2.012
.868	245.385	237	247	.287	.011
.877	270.387	253	270	1.118	.001
.887	241.307	220	245	1.881	.057
.896	239.369	237	231	.023	.293
.905	263.687	267	281	.042	1.137
.915	235.266	241	211	.140	2.503
.924	259.121	226	236	4.234	2.063
.934	256.713	266	252	.336	.087
.944	254.300	235	250	1.465	.073
.954	251.886	268	235	1.031	1.132
.964	249.469	272	244	2.035	.120
.974	247.051	238	249	.332	.015
.984	244.633	243	239	.011	.130
.994	242.212	230	231	.616	.519
1.004	263.640	274	260	.407	.050
1.015	237.132	250	250	.698	.698
1.025	258.052	267	270	.310	.553
1.036	255.128	256	241	.003	.782
1.047	252.207	219	254	4.372	.013
1.058	249.289	238	245	.511	.074
1.069	246.374	258	244	.549	.023
1.080	243.465	253	235	.373	.294
1.091	262.286	271	256	.290	.151
1.103	237.399	229	215	.297	2.113
1.114	255.683	267	265	.501	.340
1.126	252.252	280	275	3.052	2.051
1.138	248.830	243	227	.137	1.915
1.150	245.421	231	271	.847	2.666
1.162	242.022	256	234	.807	.266
1.174	258.370	270	263	.524	.083
1.187	254.414	235	256	1.481	.010
1.200	250.476	260	247	.362	.048
1.213	246.557	271	260	2.423	.733
1.226	242.659	255	227	.628	1.010
1.239	256.990	256	242	.004	.874
1.253	252.521	251	262	.009	.356
1.267	248.081	257	225	.321	2.147
1.281	243.672	244	220	.000	2.300
1.295	256.219	257	232	.002	2.289
1.310	251.234	273	215	1.886	5.226
1.325	246.291	244	241	.021	.114
1.340	257.309	249	285	.268	2.980
1.356	251.786	268	266	1.044	.802
1.372	246.320	240	229	.162	1.218
1.388	255.788	265	229	.332	2.805
1.405	249.751	252	278	.020	3.195
1.422	243.787	280	237	5.379	.189
1.439	251.707	268	267	1.055	.929
1.457	245.192	240	237	.110	.274
1.475	251.846	247	257	.093	.105
1.494	257.489	259	268	.009	.429
1.514	249.809	238	256	.558	.153
1.534	242.262	233	238	.354	.075
1.554	257.931	247	237	.463	1.699
1.576	249.141	242	263	.205	.771
1.598	251.267	239	265	.599	.751
1.621	242.072	236	249	.152	.198
1.644	252.941	252	222	.004	3.785
1.669	252.098	226	281	2.702	3.313
1.695	250.295	254	224	.055	2.762
1.722	247.560	263	276	.963	3.267
1.750	252.118	247	217	.104	4.892
1.780	246.755	265	265	1.349	1.349
1.811	255.166	255	257	.000	.013
1.845	246.473	246	226	.001	1.701
1.880	249.853	249	259	.003	.335
1.918	249.912	266	237	1.036	.667
1.959	252.136	248	238	.068	.793
2.004	249.874	249	251	.003	.005
2.053	252.080	239	273	.679	1.736
2.108	251.207	248	217	.041	4.658
2.170	249.038	238	251	.489	.015
2.241	250.376	242	251	.280	.002
2.326	250.143	226	240	2.330	.411
2.432	249.585	248	241	.010	.295
2.575	251.174	262	256	.467	.093
2.807	249.526	254	264	.080	.840
Total		50124	49876	158.967	187.450

A more refined test of the fit in the tails was made on the deviates exceeding 2.326 in absolute value. Eighty intervals (Mann and Wald, 1942) were used, each with an expectation of approximately 25. The χ² value was 76.26, with 80 d.f.; the probability of a larger value is about 0.61. The details of this test are given in Table 8.

Class Limits	Expected Number	Observed Number		(Obs.- Exp.)2/Exp.
		Left Tail	Right Tail	Left + Right Tails
2.326	26.366	29	22	.986
2.336	25.757	21	20	2.165
2.346	25.159	26	30	.960
2.356	24.574	17	24	2.348
2.366	23.999	19	21	1.416
2.376	25.748	24	22	.664
2.387	25.082	20	21	1.694
2.398	24.428	20	21	1.284
2.409	23.790	20	24	.606
2.420	25.240	30	35	4.672
2.432	26.524	32	36	4.516
2.445	23.748	23	26	.237
2.457	24.948	30	23	1.175
2.470	25.986	25	30	.657
2.484	25.098	23	14	5.083
2.498	24.236	20	23	.803
2.512	25.037	29	24	.670
2.527	25.682	28	25	.227
2.543	24.657	22	24	.304
2.559	23.669	16	16	4.970
2.575	26.868	28	25	.178
2.594	24.261	21	26	.563
2.612	25.652	24	18	2.389
2.632	24.336	24	31	1.829
2.652	25.321	21	23	.950
2.674	24.926	26	26	.093
2.697	24.412	29	32	3.221
2.721	25.624	31	26	1.133
2.748	24.639	31	24	1.659
2.776	25.135	27	25	.139
2.807	25.164	26	39	7.635
2.841	24.759	24	21	.594
2.878	25.087	25	28	.339
2.920	25.144	26	35	3.893
2.968	24.731	19	24	1.350
3.023	25.063	29	18	2.609
3.090	25.160	35	22	4.245
3.175	25.002	20	32	2.959
3.291	24.939	24	15	3.996
3.481	24.977	26	30	1.052
Total	1000.928	990	1001	76.263

The only tests made on the squares of the deviates consisted in computing sums of k squares and comparing the distribution of the sums with the χ² distribution with k d.f., employing again the standard goodness-of-fit test. This was done for k = 25, 50, 100, 300, with the following results:

k	Number of Sums	Number of Intervals (i)	χ2 with i – 1 Degrees of Freedom	Probability of a Larger χ2
25	4000	100	92.92	0.66
50	2000	100	92.45	0.67
100	1000	50	57.75	0.19
300	333	34	38.70	0.23

The fourth column gives the goodness-of-fit χ² value for the fit to the χ² distribution with k degrees of freedom. Intervals of approximately equal probability were used in all cases.

Use of the Tables

The lines of the digit table are numbered from 00000 to 19999. In any use of the table, one should first find a random starting position. A common procedure for doing this is to open the book to an unselected page of the digit table and blindly choose a five-digit number; this number with the first digit reduced modulo 2 determines the starting line; the two digits to the right of the initially selected five- digit number are reduced modulo 50 to determine the starting column in the starting line. To guard against the tendency of books to open repeatedly at the same page and the natural tendency of a person to choose a number toward the center of the page: every five-digit number used to determine a starting position should be marked and not used a second time for this purpose.

The digit table is also used to find a random starting position in the deviate table: Select a five-digit number as before; the first four digits give the starting line (the lines being numbered from 0000 to 9999) and the fifth digit gives the starting position in the line.

Ordinarily, the table is read in the same direction as a book is read; however, the size of the table may be effectively increased by varying the direction in which it is read. Thus, one may read columns instead of lines, may read the table backward, may read lines forward but pages from bottom to top, etc. Of course, care must be taken in using these devices to avoid introducing correlations when the table is used more than once on the same problem.

To obtain a random permutation of the integers 1, 2, . . . , n, select a random starting position; use the five-digit number containing the starting position and the following n – 1 five-digit numbers; put the integers in the same order as these n five-digit numbers. In case of ties among the five-digit numbers, use additional columns to the right to make six or more digit numbers. The same procedure is used to obtain a random permutation of n objects, some of which are indistinguishable, by merely numbering the objects arbitrarily from 1 to n.

To obtain random observations from any distribution G(x), use Eq. (1), substitute G(x) for F(x), and employ as many digits in D as required for the desired accuracy of the observations. Of course the negative exponent of 10 in Eq. (1) must be equal to the number of digits in D. If G(x) has a discontinuity at x₀, define it to be continuous on the right and take the solution of Eq. (1) to be x₀ when the left side of Eq. (1) falls between G(x₀-) and G(x₀). For example, if

and one is content with three-figure accuracy, then the three-digit number 082 determines an observation from a population distributed by G(x) as follows:

A technique suggested by von Neumann, called the "rejection method," enables one to substitute for the solution of Eq. (1) a stochastic process involving a much simpler computation; this technique is discussed in Kahn, 1956.

To obtain pairs of normal deviates with given correlation ρ, use pairs (x, y) of independent deviates from the table and transform them to

Thus for ρ = –.6, for example, if (.732, –1.205) are two deviates from the table, then

is a pair of deviates from a normal population with the desired correlation.^[3]

In general, to obtain a random observation from a bivariate population with distribution G(x,y), one uses a marginal distribution on one variate, say, G₁(x), and the conditional distribution, say, G₂(y/x), on the other. Two random numbers determine the observation: one determines x by employing G₁(x) in Eq. (1), and the other determines y by employing G₂(y/x) in Eq. (1). Thus, if a probability density is uniform (and equal to two) over the triangle bounded by x = 0, y = 0, x + y = 1 and is zero elsewhere, then

and two four-digit random numbers, 5402 and 1770, determine the observation (.3220, .1200). The direct generalization of this procedure will determine observations from multivariate populations.

Complete tables available for download at the top of the page ⤴

View addendum on data quality issues

References

1. Kendall, M. G., and B. B. Smith, Random Sampling Numbers, Cambridge University Press, 1939.

2. Juncosa, M. L., Random Number Generation on the BRL High-Speed Computing Machines, Ballistic Research Laboratories Report No. 855, Aberdeen Proving Ground, Maryland, 1953.

3. Meyer, H. A., L. S. Gephart, and N. L. Rasmussen, On the Generation and Testing of Random Digits, WADC Technical Report 54-55, Wright-Patterson Air Force Base, Ohio, 1954.

4. Moshman, Jack, "Generation of Pseudo-random Numbers on a Decimal Calculator," J. Assoc. Computing Machinery, Vol. 1, 1954, p. 88.

5. The Monte Carlo Method (Proceedings of a Symposium held in 1949), National Bureau of Standards Report AMS 12, Government Printing Office, Washington 25, D.C., 1951.

6. Curtiss, J. H., "Sampling Methods Applied to Differential and Difference Equations," Seminar on Scientific Computation, International Business Machines Corp., New York, 1949.

7. Kahn, H., and A. W. Marshall, "Methods of Reducing Sample Size in Monte Carlo Computations," J. Operations Res. Soc. of Amer., Vol. 1, 1953, pp. 263–278.

8. Kahn, H., Applications of Monte Carlo, The RAND Corporation, RM-1237, 1956.

9. Horton, H. B., and R. T. Smith, "A Direct Method for Producing Random Digits in Any Number System," Ann. Math. Statistics, Vol. 20, 1949, pp. 82–90.

10. Mann, H. B., and A. Wald, "On the Choice of the Number of Class Intervals in the Application of the Chi-Square Test," Ann. Math. Statistics, Vol. 13, 1942, pp. 306–317.

11. Tables of Probability Functions, 2d ed., U.S. Government Printing Office, Washington, D.C., 1948.

12. Good, I.J., "The Serial Test for Sampling Numbers and Other Tests for Randomness," Proc. Camb. Phil. Soc., Vol. 49, 1953, pp. 276–284.

Notes

• [1] For readers who are not statisticians: The χ² (chi-square) test is a standard statistical criterion used to measure discrepancy from expectation. The probability value associated with the criterion ranges between zero and one. A small probability value, e.g., less than .05, indicates the possibility of a discrepancy or bias. For very large samples, such as is the case here, with 125,000 digits, the test is extremely sensitive and will result in a small probability value even though the bias may be quite trivial from a practical standpoint.
• [2] Some issues concerning discrepancies in the numbers presented here are discussed in an addendum prepared in April 1997.
• [3] RAND has a table of deviations for Gaussian-Markov chains with a discrete time parameter. There is one chain of 10,000 observations for each of the correlations: .600, .800, .900, .950, .970, .980, .990, .995.