Original Target Series

The previous studies on this issue have all commenced with the records of the target series as held by the Society for Psychical Research. These are not known to be digitally available. In any case, the most accurate record of the series was retained by Pratt (1974); Markwick’s (1978) findings were, indeed, checked with this record before publication. However, these records, too, have not been published, and Pratt’s archives are uncatalogued (Matlock, 1987). This situation necessitated an economical reliance on target series that have been published. Complete 25-digit series for 18 runs have been published in the papers of Scott and Haskell (1974) and Markwick (1978),⁴ while Medhurst (1971) reproduced two 12-digit series, and 10 9-digit series. In total, a sample of 30 target series, from perhaps as many runs, from eight sittings, comprising a total of 564 digits, could be obtained from published sources. While limited in representativeness of the 372 runs for which random numbers were reportedly prepared, this sample represented more than twice as many series as tested by Medhurst. Sampling limitations are addressed in the Discussion, but it should be noted at the outset that the sample was largely constituted of series previously implicated in claims of fraud.

With respect to the Scott and Haskell (1973, 1974) critique that suggested a non-uniform distribution of the target digits, it should be preliminarily noted that the distribution of digits within this sample of target series was quite uniform, well reproducing the distribution within the table of 7-figure logarithms themselves (outside of the digits 0 and 6–9); see Table 1. With respect to Markwick’s (1978) finding that some series were copied but with reversal of order, the order of final use on the score sheets was here of interest, such that both the reversed and nonreversed series were represented in the sample.

Results

Discussion

In comparison to empirical expectation, the number of retrievals of the Soal–Goldney target series from a source such as Chambers’ Tables was observed, in Experiment 1, to be exceedingly more likely than that obtainable on the basis of chance. Not only were significantly more retrievals of the published target series possible from the source and by the method originally reported by Soal and Goldney (1943), but the particular series that were retrieved bore the character of having been constructed by a process identical to that described in their report, and as suggested by reviewing the procedure and conditions. That is, we can quite graphically and statistically appreciate that a published table of 7-figure logarithms was sourced by skipping its first four pages, preferentially taking entry-points within its first 20 pages for antilogarithms greater than 10,000, then the next 20 pages, and thereon making a continuous sweep through the publication.

As the results were reliable over differently constructed chance-control sources, we can be confident that the results were independent of any question of the underlying randomness of the source. That is, it made very little difference whether the chance-control sources werecomposed of digits that were identical in their sequencing and frequency distribution with a table of the final digits of 7-figure logarithms, or merely shared their frequency distribution, or were, indeed, fully generated as independent (“truly random”) events.

One 9-digit match was obtained with a series sourced from Medhurst’s (1971) study, i.e. within Sitting 16. That this was not identified by Medhurst as a match exposes the limitation of his method: He did not permit matches from all possible segmentations of a series, but only those he himself delimited on some arbitrary basis.

An arguable limitation to conclusions is that the gross number of series retrieved (see the “N series” column in Table 2) did not significantly deviate from the numbers retrieved by chance. However, this level of analysis does not take into account the number of possible segmentations of each series. There is always the possibility—especially for the longer, 25-digit series—that some segments of the series should be more likely to be retrieved than others, given the assumptions reviewed in an earlier section (“Revisiting the Assumptions”). The number of times any segment of the series could be retrieved therefore represents the most appropriate level of analysis—not the number of series themselves that can be retrieved. On this basis, there was no question that the 7-figure logarithmic source yielded significantly more retrievals than expected by chance. Naturally, some retrievals were merely chance matchings, and we must include overlapping and repeated segments of the series in the counts. This was theoretically necessary as we have no a priori basis for segmenting the series. Yet the present method ensured that such overlaps and repeats were just as likely to occur from the chance-control sources as they were from the key 7-figure logarithmic source. We can therefore be confident that the results, based on the number of segments of each series that was retrieved, reliably lead us to support of the alternative hypotheses.

This reliable confirmation of the hypothesis of early entry into Chambers’ Tables suggested that an even more stringent test would involve requiring that the match of any segment of a series, at the largest possible length, that occurred at the earliest point into the Tables, should be taken exclusively of any other possible matches. Any remaining digits in the series should be treated in the same way, so that the searchstrategy is both exclusive and exhaustive. This would have the particular advantage of permitting the chance-control sources to return retrievals at greater lengths than found for the “true” source. This reiterative, exclusive, and exhaustive search strategy formed the basis of a second experiment.

ExPERIMENT 2. ExHAUSTIvE, ExCLUSIvE, AND EARLIEST RETRIEvALS

On the basis of the results of Experiment 1, it was feasible to attempt to retrieve the Soal–Goldney target series by an exclusive and exhaustive search. Also, it would be desirable to not have a fixed segment length for all series, but to vary the length, starting from the size of the series itself, and reiteratively searching for retrieval, by an ever-decreasing range, until a maximal series length. Effectively, we abandon Hypothesis 1, concerning the gross number of matches, while Hypothesis 2—the hypothesis of early entry—remains relevant. Replication of the post hoc confirmation of the hypothesis of early entry for the first 5,000—rather than 10,000—of entry-points into (or 10 rather than 20 pages of) Chambers’ Tables was also hypothesized.

Moreover, it was considered that this procedure should force the chance-control sources to represent Benford’s Law, while the corresponding antilogarithms with a leading digit of 1 by way of the “true” source should surmount Benford’s Law. This would naturally arise when the search strategy preferentially sampled entry-points into the table with the leading digit 1, with the leading digits 2, 3, and up to 9 being successively ever less likely. Perhaps a limitation was that the range of antilogarithms covered only one order of magnitude: After leaving the first range with leading digits of 1 (from 10,000 to 19,999), we do not return to them by sampling for logarithms beyond 99,999. Still, it could be reasonably hypothesized that Benford’s Law would predict the distribution of figures so contrived, and that the search strategy reasonably fulfilled what is known of the Law’s conditions (Fewster 2009, Hill, 1995, Raimi, 1976). In this way, hypothesis testing could involve theoretical as well as empirical expectation.

Method

The sample of target series, and the chance-control files, were identical to those used in Experiment 1. The search logic, however, was modified so that, initially, only the one maximally sized retrieval was accepted for each target series, and then only by the earliest of entry-points retrieving this length. Whatever portion was not thus matched was then searched for independent retrieval at another entry-point. Segments of less than five digits in length were not searched; this criterion being imposed prior to any searches being conducted, in the interest ofeconomizing on search time, and considering that the approach would be sufficiently informative without these additional searches.

An exceptionally long amount of time was required to search for retrievals on these bases, and, accordingly, the chance-control sources were identical to those used in Experiment 1, excepting that the first 240 of the 1,000 files of the shuffled source (and so as many as constituted the random source) were tested. Testing these samples itself required 12 days of continuous computerized searching to complete.

Results

The retrieval counts by the method of exhaustive search for maximallength segments at the earliest antilogarithms within each source are presented in Table 3. Naturally, as expected, the gross number of matches was identical over each source of digits. However, there was a divergence in derivation frequency via the “true” 7-figure logarithmic source versus chance expectation, per each chance-control source, for the number of retrievals within the first 5,000 digits. Specifically, 18 retrievals were obtainable by way of the “true” source, compared to an average of about 11.67 from each chance-control source (with SDs ranging from 2.93 to 3.38). Values of p_e are listed in Table 3, and these can be compared with those for the neighboring region of the next 5,000 digits. There was a total conformance within this neighboring range to empirical expectation, while the small values of p_e for the primary range were strongly consistent with the hypothesis that the target series were derived by early entry into a table of 7-figure logarithms.

Figure 2 permits assaying this result in the context of counts from all entry-point intervals. The observably smooth decline in counts per chancecontrol sources well reflects the bias toward early intervals incorporated in the search criteria. In fact, the number expected within each range on the basis of a logarithmic number line—given in Figure 2’s dotted line—was tightly matched by the chance control sources. These represent a near-perfect representation of Benford’s Law. This can be better appreciated by considering the 10,000-digit intervals. For example, for the shuffled source, there was an average of 19.98

459

TABLE 3 Retrieval Counts for Longest Non-Overlapping Segments per Series

Source N Matches % Matches p_e % Matches p_e 10,000–14,999 15,000–19,999

7-figure logarithms 64.00 28.13 12.50 Shuffled source (N = 240) 65.39 17.83 .063 12.72 .596 Permuted source (N = 119) 65.51 17.88 .042 13.13 .613 Random source (N = 240) 65.27 17.81 .017 12.24 .521

retrievals from the first 10,000-digit interval up to 20,000; comprising exactly 30.56% of the average total (over the 240 files) of 65.39 retrievals, which is precisely the proportion of antilogarithms with the leading digit of 1 predicted byBenford’s Law. With the average count in the next 10,000-digit interval being 11.10, we have 16.98%, where Benford’s Law predicts 17.61%; a match-tolaw with a trivial sampling error. The corresponding proportions were 31.02% and 17.44% for the permuted source, and 30.05% and 17.24% for the random source. Quite unambiguously, chance matching of the logarithmic final digits conformed to law for the antilogarithmic leading digits.

Figure2. Retrievalcounts(mean+SE)perentry-pointintervalanddigitsourcesbetween 10,000 and 100,000, with exhaustive search of earliest and longest segments, and per Benford’s Law for leading digit i (Equation 1).

Still with reference to Figure 2, search of the Soal–Goldney target series from the “true” source, as reported by Soal, yielded a very sharp deviation at the outset from chance-wise and lawful expectation; but an always chance-wise and lawful decline thereafter. The Soal-wise count of antilogarithms with 1 as their leading digit comprised 41% of all retrievals—clearly surmounting the 30.10% predicted by Benford’s Law for a chance operation, and reproduced by each of the chance-control series. With 18.75% in the next 10,000-digit interval, we havea sudden return to Law. The minor peaks observable in Figure 2 at intervals starting at 25,999 and 65,999 were not significant; e.g., p_e = .163 and .0750, respectively, in comparison to the number of random retrievals at these intervals.

Discussion

The results of Experiment 2 immediately recall the quote from Soal’s (1940) Fresh light report detailing, with examples, the procedure he followed in deriving the target series. Soal gave examples of his entry-points as antilogarithms commencing with the digit pair 10—and this is precisely what we find substantiated by attempting to retrieve the allegedly fraudulent target series by retracing the method he reported for their construction. The deviations we find all bear the mark of a human hand drawing digits from Chambers’ Tables by preferential entry into its earliest pages—clearly beyond the natural bias for entries in this range.

More generally, the results showed that the effect observed in Experiment 1 indicative of the Soal–Goldney target series having been sourced by early entry within Chambers’ Tables—as originally reported—was robust even under the conditions that (i) only the earliest of entries from all sources should serve as the basis of chance estimation and observed count; and (ii) the longest possible retrievals from each target series were exhaustively obtained. Under these conditions, a positive disproportion of matches was expected for the retrieval counts true to Soal’s method, versus those expected on the bases of chance and Benford’s Law. With already stringent search criteria and sampling that might well have favored the chance-control sources, this hypothesis was confirmed at ever more concise resolutions of the relevant antilogarithms.

Summarily, there was no reason to doubt that the target series were sourced other than how they were originally reported to have been, while positive evidence was acquired that they were indeed thus sourced.

ExPERIMENT 3: ALTERNATIvE LOGARITHMIC SOURCES

Upon concluding his study, Medhurst (1971) confronted Soal with the conclusion that the target series could not have been compiled as Soal had originally reported. Trusting the validity of Medhurst’s finding, Soal—the octogenarian—was distressed by this information, and he offered in explanation that, perhaps, he had formed the erroneous recollection that the publication he used in 1941 to prepare the target series was the same that he had used in his earlier studies; perhaps he had even sampled the digits at intervals other than 100 (Soal, 1971).

If a table of logarithms other than Chambers’ Tables had, then, been used, it might offer a larger number of retrievals than that obtainable from a source of 7-figure logarithms. Alternatively, if the 7-figure logarithms produce the highest retrieval counts, we could be only more confident in the accuracy of Soal’s reporting, and all the more assure ourselves that the number retrieved from the 7-figure logarithms has not been a mere fluke based on some fortuitous distributional properties of the digits 1–5 in the 7-figure logarithms. Markwick

(1978), indeed, offered a test based on a table of 6-figure logarithms, and one allowing for discrepancies from intervals of 100. Readers were informed that these efforts “met with no success.”

What other forms of logarithmic tables might be tested? From surveys and catalogues of mathematical tables (especially Comrie, 1948, Fletcher, Miller, Rosenhead, & Comrie, 1962, Henderson, 1926), it can be learned that, by 1941, a wide array of tables was available, differing by logarithmic precision and antilogarithmic range. The 4-figure tables appear to have been the most popular, followed by 6-figure tables; but also prominent were 10-, 12- and 20-place logarithms for integers in the range of 1 or 10,000 to 100,000, plus several series offering 5- to 8-place logarithms in a similar range. Additionally, a search of library reserves for such materials revealed (most conspicuously) a 7-place source for the numbers 20,000 to 200,000 (Sang, 1915), and a 16-place source of natural logarithms for the numbers 1to 100,000, in two volumes (and the decimal numbers from 0.0001 to 10.0000 in a further two) (Lowan, 1941).

There is no positive indication that Soal resourced any of these publications, and his suspicion that he was in error in describing Chambers’ Tables as his source was only made—as we may thus far be obliged to conclude—under dubious compulsions to do so. However, as a test of alternative logarithmic sources could provide a comparative assay of the robustness of the idiosyncrasy thus far observed for the 7-figure source, searches for the target series were conducted on these sources, with thesame hypotheses as applied to Experiment

1. Given the arbitrariness of the possible interval alternative to 100 through the tables, this factor cannot here be informatively pursued.

Method

The method was identical to that used in Experiment 1, including the same sample of target series and the fixed-length search logic, but only attempting retrievals of eight digits in length. Perl functions were again used to produce the files of final digits for logarithms taken at precisions of 4, 5, 6, 8, 10, 12, 16, and 20 decimal places, always assuming the same method of construction as used for Chambers’ Tables.¹⁰ These files were searched in the range of 10,000 to 99,999. Additionally, 7-figure logarithms in the range of 20,000 to 199,999, were searched.

For the lower range, no new chance-control sources were necessary; expectation and variance could be reliably obtained from the random source, which offered a normal distribution of retrieval counts. In order to provide for empirical expectations up to 200,000 digits, the economical approach was taken of combining and shuffling the digits from each one of the 240 random source files with a file randomly selected without replacement from the shuffled source, thus producing 240 files of 200,000 digits 0–9.

Results

Retrieval counts are presented in Table 5. For ease of comparison, the first row re-presents the counts obtained via the “true” source of 7-figure logarithms from Experiment 1. It will be readily noted that the alternative precisions produced retrieval counts that were generally within the range of chance expectation. The proportional frequencies of counts within the random chance-control source that were at least as great as that observed per source are represented in the p_e column of the table. Assuming normality and using the mean and variance of the random source as the basis of assaying deviation essentially yielded the same indications of significance. For the chancecontrol files of the count from the table of 7-figure logarithms up to 200,000, we obtain Z = −0.859, 1p = .805. However, there were some deviations among the alternative sources with respect to the gross number of retrievals that will merit discussion.

As for the proportion of counts that fell in the critical range of entry-points (<20,000), the eight alternative sources generally produced what could be expected of a uniform distribution of counts over the nine possible intervals (about 11%). As listed in Table 5, the p_e values for deviation of the counts from expectation indicated that, according to the criterion of early entry, there was absolutely no practical use of logarithmic tables apart from the table of 7-figure logarithms. For the two-volume set of 16-figure natural logarithms, an indication of artifactuality would have been obtained if, at the start of the second volume (from 50,000), some disproportion in matching were obtained; however, quite unlike the result for the first volume, there was not a single retrieval among the first 5,000 antilogarithms of this second volume. For the source of 200,000 entries, we can only start with the second interval of 20,000 to 29,999; and here, as already indicated in the results for Experiment 1, there was some modest deviation—secondary in significance to the first interval—of the “true” count of 13 from that predicted by the chance control, in this case

4.833. All the counts at further intervals into the tables, up to 199,999, were quite in conformity with chance; the highest remaining count being only 7, for the very last interval.

Discussion

A search for the sample of the Soal–Goldney target series from logarithmic sources other than those reported by Soal and Goldney (1943) failed to reproduce the results of Experiment 1 obtained with the “true” reported source. This was with respect to the number of matches obtainable by any segmentation of the target series, and the particular number of matches that fell in the critical range that amounted to the first 20 pages of Chambers’ Tables.

General Discussion

After examining the question as to the source of the Soal–Goldney target series, it appeared inadvisable, on historically evidential and logical grounds, to assume capacity to retrieve complete runs of 25 digits by retracing the manner in which they were reportedly generated (Soal & Goldney, 1943). Not even any long series of a particular length could be assumed to be retrievable, contrary to the unstated but apparent assumptions of Medhurst (1971) and those who replicated his efforts (Markwick, 1978, Scott & Haskell, 1974). This critique accorded with comments by Pratt (1971) on Medhurst’s research; and, when so informed, searches were conducted that indicated that the null hypothesis of non-derivation from a published method of using 7-figure logarithms was most unlikely, relative to chance-control sources based on shuffling or permuting the final digits of the logarithmic source, and randomly sampling its range of digits. Also, the proportion of entry-points in the range 10,000–19,999, among these matches, was consistently greater for the 7-figure logarithmic than the chance-control sources. These entry-points from the logarithmic source represented those within the first 20 pages of Chambers’ Tables (starting from 10,000) that accorded with the finer details of Soal’s description of his method. By effectively reducing the size of the bins within which the antilogarithms were tested—from 10,000 to 5,000 and then 1,000—this pattern was reliably observed, without any increase in sample size, to be restricted to what amounted to the first 10, and then the first two, pages of Chambers’ Tables. This sign of early entry into the Tables was clearly indicated as non-artifactual by its lack of reproduction by alternative logarithmic sources as well as each chance-control source. It was also robust against the allowance for chance-control retrievals to be entirely based on the earliest of entry-points, and in relation to Benford’s Law concerning the distribution of leading digits. Given that it was reasoned that Soal would have started his searches, in the main, within these first 20 pages, and the extant evidence indicated as much, these results suggest that the target series in the sample were generally obtained in the manner originally reported by Soal and Goldney (1943).

Relation to Earlier Efforts at Target Retrieval

The conclusion suggested by these results is at odds with those offered by Medhurst (1971), Scott and Haskell (1974), and Markwick (1978) for their searches of the target series. This discrepancy is simply effected given that the earlier reports lacked explicit statement of their search assumptions, and the criteria by which the likelihood of derivation from Chambers’ Tables could be adduced. Earlier results were compared with neither theoretical nor empirical expectation; readers were only offered something of a standalone statistic: “enough of them,” “no identifiable match,” and “drew a blank.” In contrast, the present findings have been based on judging retrieval against reliable and replicable empirical and—where appropriate—theoretical values of what should be expected; and the approach has been consistent with all the documented facts, and has relied on no novel assumptions regarding how the digits were sourced, compiled, and eventually used. Given these differences between the studies, we can expect no comparability of their results.

Generalizability

The present results have been based on 30 target series, from perhaps as many runs, and eight sittings, from a possible372 runs over 40 sittings. Arewe yet permitted to draw conclusions about the population of target series from these results?

Simply in terms of sample size, and the publication status of previous assays of the target series, this appears, by precedent, to be permissible. Table 6 presents the counts and proportions relevant to this comparison. In comparison to Medhurst’s (1971) study, which involved only 12 segments from 6 to perhaps 12 runs, the sample is extremely ample; and, indeed, encompasses and goes beyond his sample. Also, the scale of the present sample is similar to that on which claims of data manipulation have been based. Specifically, Hansel’s (1959) statistics were based on 51 runs within 6 sittings, restricted to those involving a “rapid-rate” of target assignment, but then not even all runs under this condition. While Scott and Haskell (1973, 1974) tested all of the relevant target series, their eventual claim pertained to the 14 runs of Sitting 16, selected on the basis of Albert’s allegation, and then the 12 runs of Sitting 8, and the first 6 of the 20 runs of Sitting 17, selected on the basis of a search for a data pattern identical to that observed in Sitting 16; see Pratt (1974:104) concerning this post hocbasis of their results, and Scott and Haskell (1975:222) for a rebuttal. Markwick’s (1978) final statistical result—as best as can be figured from her tables—was based on only 13 runs administered within 7 sittings, the 41 “extra digits” encompassed within these runs selected from an original yield of 93. Several subjective criteria were employed in this cull, e.g., of what were described as “weak,” “ambiguous,” and “apparently discrepant” “extra digits,” as well as “unmatched” digits. Then, the statistical result offered on this remainder amounted to a conservative confirmation of the null. In contrast, the present results are based on an objectively limited sample of 30 target series from 8 sittings that often equates with or exceeds these former studies in terms of sample size, while relying on neither subjective nor post hoc processes in its constitution, and being based on conventional statistical arguments. However, the entire target series were not available for testing the hypotheses; the sample was drawn from what of the population has been published, rather than randomly selected, such that it might be reasonably argued that conclusionsshould be constrained to the sample itself but could be well predicted for the population.

TABLE 6 Sample Sizes in This and Previous Studies of the Soal-Goldney Target Series

Note: For Medhurst (1971), and those series of the present study derived from his study, run and trial numbers are the maximum possible, including all tested digits, given that runs were not identified in the report. For Scott and Haskell (1974), N trials is given by the number of +1 trials in their 32 runs. For Markwick (1978), N trials consists of the length of the “interrupted duplicated sequences”listed in her Table 7, less those involving the Stewart study. Percentages of trials are taken with respect to the number of possible +1 hits (or +2 for rapid-rate) conducted under telepathy, clairvoyance, and other conditions, less missed trials (Soal & Goldney, 1943:95–97); or for on-trial hits if there was a suggestion to score on-trial (Soal & Goldney, 1943:49,55).

Implications for the Fraud Scenarios

What implications of these results are there for the fraud scenarios? Most clearly, Medhurst’s (1971) conclusion that the target series could not have been derived as originally reported must be queried: When respecting the conditions of their production, particularly as advised by Pratt (1971), the target series bear the marks of having been produced from a table of 7-figure logarithms, as originally reported. Scott and Haskell repeatedly referred to Medhurst’s null conclusion as corroborative but necessary evidence in support of their investigation of Albert’s allegation; they defined its evidentiality as equal to the loss of the original records, stating (somewhat ambiguously) that “These had to happen on our hypothesis and the fact that they did happen provides a little further confirmation” (Scott & Haskell, 1975:222). This necessary “evidence” can, in the light of the present results, be judged to be of quite arguable merit, if not nonexistent—even if we only consider the results for the sub-sample tested by Medhurst and upon which Scott and Haskell relied.

However, while the present results confirm Medhurst’s original hypothesis, they do not serve his objective to vindicate Soal; and it would be wrong to interpret these results as somehow implying Soal’s innocence. Our methods only involved, and the results confirmed, that “retrieval” should be possible for any small segments of target series. This was a necessary assumption given such procedures as haphazardly compiling the series into, and drawing them from, a pool of digits; copying errors;reusing prior series with various transformations, including omission of prior hits and reversals—and/or such manipulations as stacking the series with 1s, and altering the 1s into other digits. In this way, it involves no paradox to hold that the target series were sourced as originally reported, but also manipulated.

Furthermore, we need not compel the fraud scenario to predict that the points at which the “retrieved” series were segmented should, by and large, be those points at which Markwick found “extra digits” to occur.¹¹ Consider, for instance, the manner in which Experiment 2 offered “retrieval” of run 25-1a. This involved, firstly, a 10-digit match, surrounded by two 6-digit matches, accounting for 22 of its 25 digits, as follows (the different segments separated by dashes).

(5)143125(32)-2543251314-232154

Those digits that did not match (shown above in parentheses) were two from the start of the series, and two before the 10-digit match commenced. Those digits that Markwick identified as manipulated (underlined in the above) were included in the “retrieved” 10-digit segment. This cannot be used to suggest, however, that the indication of these digits as manipulated is somehow “wrong.” We have, after all, only obtained this “retrieval” by a statistical process, and can make no claim that any particular case represents the “real” source of the digits from Chambers’ Tables. Secondly, we cannot be sure that in any case of reuse the first-used series is the original and the later-used series is its copy—such that any “extra digits” may, in fact, be the result of omissions from one or the other series, rather than insertions into the later-used series. Also, the scenario of manipulation suggested by Markwick’s study does not limit manipulation to those digits that appear to be altered in the process of duplicating an already-used series; that limitation is only in the nature of her evidence, not in its implications. The above example could, after all, have been drawn from the original pool of digits, and manipulated at that point. The unmatched digits, then, could point to manipulated digits that were not apparent by Markwick’s method.

The original objective, as stated in the beginning, was, indeed, to identify the source of the target series so that Markwick’s findings of data manipulation could be extended beyond cases of reused target series. Some ideas for future research, consistent with the interest in extending if not confirming Markwick’s findings, can be offered. The present research suggests that we can pursue this objective by simulating the pool on the basis of its reported source, i.e. Chambers’ Tables. However, we need to suppose that Soal made use of “extra digits” and other alterations not only when reusing target series from one run to another, but also when transcribing the digits from the pool onto the target sheets—and perhaps even when copying digits from the Tables into the pool. Then, on the basis of Markwick’s and the present findings, we could hypothesize that those digits within a series that fall outside the starts and ends of any particular “retrievals” are more likely than not to occasion hits; and that they could mostly have been additional 1s that were altered into 4s and 5s. But this cannot be predicted for all series; the manipulations would not, for instance, offer any advantage when Soal had no sure access to the target sheets. So we could restrict our hypothesis to those runs when Soal acted as the agent’s experimenter, and/or was involved in the final scoring. When, as in Sittings 23, 24, and 25, sheets with “extra digits” appear to have been used in runs when Soal was responsible for the guess sheets, we could suppose that this simply occurred because Soal had prepared many sheets for manipulation, and he only needed to reduce the proportion of 1s before the sitting to that predicted by chance, and then manipulated (only) the guesses during the sitting. In this case, the hits should not tend to fall at the junction of any “retrievals.” Clearly, to support the necessary assumptions and various factors implied in these predictions, a larger sample of target series than has been presently available is required.

There are a couple confirmations of previously raised points in the context of the fraud scenario that can be mentioned. Firstly, the present results inform against the particular contra-fraud model of fortuitous reuse. This is because it is far less likely that duplications should fortuitously arise if Soal commenced his searches within only some few early pages of the Tables rather than freely entered them all over the volume. This interpretation is not clear-cut; as previously noted, for Soal’s 10,078 example of an entry-point, we can find several long duplications commencing with antilogarithms less than 11,000. Still, it might be tentatively concluded that support for the hypothesis of early entry is incompatible with a model of fortuitous reuse. Additionally, the footnoted statement by Soal and Bateman (1954) that specified the dates of the sittings for which Tippett’s tables were used must be inaccurate. The statistical details suggest that Tippett’s tables were used for all the runs within these sittings. As the deviations from chance in the present study were somewhat dependent on including runs from these sittings, accepting the present results implies that the targets for these runs were more likely to have been generated from a table of 7-figure logarithms than a source such as Tippett’s.

In summary, the present results suggest that (1) the fraud scenarios cannot

be clearly rationalized on the basis that the target series cannot be retrieved

from the reported source, but that (2) extending the search for evidence of data

manipulation by identifying the source of the target series might well return to

the originally reported source.

Notes

¹ It must be noted that Medhurst was chronically ill by the time he composed this report (Barrington, 1971, Goldney, 1974); it was published two months after his decease. What was published appears to be an unrevised manuscript, given, as others subsequently noted (Scott, 1971; Editor’s note on p. 203 of the same volume), that it contains several statistical and linguistic errors and ambiguities, including a crucial statistical test that Medhurst offered regarding the target digits, whereas it is clear that he confused these with the response digits. The presently described limitations—suggesting at least a hurriedness in Medhurst’s conclusion—must be put to the same account.

² Soal and Goldney (1943) did not specify the runs for which Tippett’s tables were used. Yet Soal and Bateman (1954:137n) later specified these as Sittings 24, 25, and 26. Still, the method of entering Tippett’s tables was not described, nor was it stated that they were used for all 42 runs of these sittings. This might, however, be assumed to have been the case, as, following this statement, the result for “480 (+1) trials” on these sittings was given. This number corresponds to the 20 telepathy runs, at “normalrate,” with the usual agent (Elliott), among the 42 runs that were administered within these 3 sittings.

³ An alternative and somewhat more economical approach would be to generate random target series, or reorder and permute the original targets, and test them for retrieval from the 7-figure logarithms in comparison to the original targets. This approach should yield the same results as using the true target series against randomly constructed search lists. Initial indications were, indeed, that the approaches yielded identical results. The present approach was, however, adopted, as it was considered to offer greater face validity to apply randomization to the source digits rather than to “tamper” with the original target series.

⁴ A report by Pratt (1951) reproduced targets and responses for 2 25-digit runs from Sitting 32. However, these were represented by the letters A to E, which, the text explained, substituted for the target initials (E, G, L, P, Z). As the digits 1 to 5 were randomly assigned to the targets upon every second run, it is not clear how they correspond to letters A to E. Being reliant on the record of digits, the present study could not include these series in the sample.

⁵ This copy of the Tables in fact extends to 100,009; but in deference to other editions that were available at the time, the present study sought for matches only up to the logarithm for 100,000.

⁶ This was as implemented by the Perl module Math::Random::MT, available from http://www.cpan.org

⁷ Specifications of this software are available from http://comscire.com/Home

⁸ Some readers might prefer to calculate p_e with the addition of 1 to the numerator, and perhaps also with the addition of 1 to the denominator. Multiplying the stated probabilities by the sample size, as given in Table 2, permits such calculations. The results obtainable thereby represent no meaningful restriction on the results here reported. Additionally, standard normal deviates and associated probabilities were calculated. The chance-control counts were not always normally distributed, as indicated by the (extremely conservative) Kolmogorov–Smirnov test, although the values for skewness and kurtosis were always quite small, and there was typically little difference between the mean and median counts. These values are available from the author by request.

⁹ For example, the random source yielded 654 among 15,664 entry-points that commenced with the digits “10”. This compares with 648.375 entry-points according to Benford’s Law (i.e. when entering “10” into Equation 2, we obtain .04576, which we then multiply by 15,664). The binomial distribution is referred to in relation to 15,664 trials, 654 “hits,” and a theoretical probability of .04576.

¹⁰ With respect to conventional limits in computational resources, a different set of functions—from the Perl module Math::BigFloat—was required to generate and store the decimal strings for the 16- and 20-figure logarithms than those with shorter decimal strings. Testing this module by using it to also produce the final digits of base-10 7-figure logarithms, and comparing them with those produced by the method otherwise used (which simply relied on Perl’s log and sprintf functions) revealed 110 discrepancies, presumably attributable to different rounding conventions. A sample of about 20 of these discrepancies was checked for identity with Chambers’Tables (Pryde, 1930), and the standard (log-sprintf) method was found to give final digits always agreeing with the Tables. Accordingly, results for the 16- and 20-figure logarithmic sources should be treated with particular reserve.

¹¹ This point is raised in consideration of an interpretation canvassed by an anonymous reviewer of this article.

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