Tests of randomness for pseudorandom number generators

This study examined the “randomness” of the numbers generated by the VIC-20 computer. Using eight standard tests, it was determined that the numbers provided by the VIC-20 are adequately random for practical purposes. The tests are applicable to other computer systems and provide a general means for evaluating random-number generators.     

Making heads or tails of probability: An experiment with random generators

Background. The equiprobability bias is a tendency for individuals to think of probabilistic events as ‘equiprobable’ by nature, and to judge outcomes that occur with different probabilities as equally likely. The equiprobability bias has been repeatedly found to be related to formal education in statistics, and it is claimed to be based on a misunderstanding of the concept of randomness. Aims. The aim of the present study was to examine whether experimenting with random generators would decrease the equiprobability bias. Sample. The participants were 108 psychology students whose performance was measured either immediately after taking part in a training session (n= 55), or without doing any training exercises (n= 53). Method. The training session consisted of four activities. These included generating random sequences of events, and learning about the law of large numbers. Subsequently, the participants were tested on a series of equiprobability problems, and a number of other problems with similar structure and content. Results. The results indicated that the training successfully decreased the equiprobability bias. However, this effect was moderated by participants’ cognitive ability (i.e., higher ability participants benefitted from the training more than participants with lower cognitive ability). Finally, the training session had the unexpected side effect of increasing students’ susceptibility to the representativeness heuristic. Conclusions. Experimenting with random generators has a positive effect on students’ general understanding of probability, but the same time it might increase their susceptibility to certain biases (especially, to the representativeness heuristic). These findings have important implications for using training methods to improve probabilistic reasoning performance.     

Cautions regarding random number generation on the Apple II

Undocumented characteristics of the pseudorandom number generators in Applesoft BASIC and Apple Pascal are described that cause identical sequences to be generated on different executions of programs written in those languages. Although it is relatively easy for this problem to escape notice, in both cases the problem is easily corrected once its existence is known.     

Effects of regular switching between languages during random number generation

Random number generation is a task that engages working memory and executive processes within the domain of number representation. In the present study we address the role of language in number processing by switching languages during random number generation (numbers 1-9), using German (L1) and English (L2), and alternating L1/L2. Results indicate large correspondence between performance in L1 and L2. In contrast to nonswitching performance, randomization with alternating languages showed a significant increase of omitted responses, whereas the random sequences were less stereotyped, showing significantly less repetition avoidance and cycling behavior. During an intentional switch between languages, errors in language sequence appeared in 23% of responses on the average, independently of the quality of randomization but associated with a clear persistence of L2. These results indicate that random number generation is more closely linked to auditory-phonological representation of numerals than to visual arabic notation.     

Manipulating attentional load in sequence learning through random number generation

Implicit learning is often assumed to be an effortless process. However, some artificial grammar learning and sequence learning studies using dual tasks seem to suggest that attention is essential for implicit learning to occur. This discrepancy probably results from the specific type of secondary task that is used. Different secondary tasks may engage attentional resources differently and therefore may bias performance on the primary task in different ways. Here, we used a random number generation (RNG) task, which may allow for a closer monitoring of a participant's engagement in a secondary task than the popular secondary task in sequence learning studies: tone counting (TC). In the first two experiments, we investigated the interference associated with performing RNG concurrently with a serial reaction time (SRT) task. In a third experiment, we compared the effects of RNG and TC. In all three experiments, we directly evaluated participants' knowledge of the sequence with a subsequent sequence generation task. Sequence learning was consistently observed in all experiments, but was impaired under dual-task conditions. Most importantly, our data suggest that RNG is more demanding and impairs learning to a greater extent than TC. Nevertheless, we failed to observe effects of the secondary task in subsequent sequence generation. Our studies indicate that RNG is a promising task to explore the involvement of attention in the SRT task.     

A theoretical and empirical comparison of mainframe,microcomputer, and pocket calculator pseudorandom number generators

This article presents an extensive theoretical and empirical analysis of the pseudorandom number generators provided by subroutine libraries (NAG, CERN, IMSL, and ESSL), statistical and simulation software packages (GLIM, SAS, SPSS, DATASIM, ESSP, and LLRANDOMII), builtin functions of programming languages (APL, Turbo Pascal, Advanced BASIC, GW-BASIC, and QBASIC), and autoimplemented algorithms (Fishman & Moore, 1986; Wichmann & Hill, 1982; Van Es, Gill, & Van Putten, 1983). On the basis of these analyses, it is concluded that most of the built-in functions of the software packages that were tested can be used safely. In addition, it is concluded that the Wichmann and Hill algorithm is a good choice if only single-precision arithmetic is available, but that a prime-modulus multiplicative congruential generator with modulus 2³¹ ?1 and multiplier 16,807 is a better choice if double-precision arithmetic is available, and that the same generator with multiplier 62,089,911 or 742,938,285 is the best choice if extended-precision arithmetic is available. A Turbo Pascal and a VS FORTRAN program for the latter are given in the Appendixes.     

Effective cutoffs for detecting random, partially random, and nonrandom 370-item MMPI-2 short form protocols

The ability of the 370-item short form MMPI-2 (Butcher, Graham, Ben-Porath, Tellegen, Dahlstrom, & Kaemmer, 2001) validity scales to detect random protocols was investigated using samples of 500 nonrandom protocols, 250 half-random protocols, and 250 all-random protocols. The long-form cutoff of VRIN ≥ 80 was unable to detect protocols with either level of randomness. The long-form cutoffs of Fp ≥ 100 or F ≥ 100 were able to detect all-random but not half-random protocols. Alternative cutoffs for VRIN, Fp, and F were investigated and short-form subscales of those scales were developed to improve detection of partially random protocols. An algorithm using alternative cutoffs for the scales and the new subscales was highly effective, detecting almost all of the random protocols as well as the nonrandom protocols. A follow-up cross-validation study was conducted that confirmed the effectiveness of the algorithm.     

Examining psychokinesis: the interaction of human intention with random number generators--a meta-analysis

Séance-room and other large-scale psychokinetic phenomena have fascinated humankind for decades. Experimental research has reduced these phenomena to attempts to influence (a) the fall of dice and, later, (b) the output of random number generators (RNGs). The meta-analysis combined 380 studies that assessed whether RNG output correlated with human intention and found a significant but very small overall effect size. The study effect sizes were strongly and inversely related to sample size and were extremely heterogeneous. A Monte Carlo simulation revealed that the small effect size, the relation between sample size and effect size, and the extreme effect size heterogeneity found could in principle be a result of publication bias.     

A hardware random number generator for use with computer control of probabilistic contingencies

A solid-state random number generator is described, which uses the leakage across a reversed biased diode as the random source. Statistical tests carried out on sequences of numbers failed to show any biases from randomness at sampling rates up to 3 kHz.     

Unidirectional interference in use of nondominant hand during concurrent Grooved Pegboard and random number generation tasks

The interference effect between Grooved Pegboard task with either hand and the executive task of cued verbal random number generation was investigated. 24 normal right-handed subjects performed each task under separate (single-task) and concurrent (dual-task) conditions. Articulatory suppression was required as an additional secondary task during pegboard performance. Analysis indicated an unambiguous distinction between the two hands. Comparisons of single-task and dual-task conditions showed an asymmetrical pattern of unidirectional interference with no practice effects during pegboard performance. Concurrent performance with nondominant hand but not the dominant hand of random number generation performance became continuously slower. There was no effect of divided attention on pegboard performance. Findings support the idea that the nondominant hand on the pegboard and random number tasks draw from the same processing resources but that for the executive aspect random number generation is more sensitive to changes in allocation of attentional resources.     

Visual comparison of words and random letter strings: Effects of number and position of letters different

Memory & Cognition - Ss classified tetragrams as either “same” or “differen”. Stimuli were either words or consonant strings. In the case of different pairs, the number...     

Long-term benefits of seeding the knowledge base

Exposure to numerical examples (seed facts) produced a substantial long-term reduction in domain-specific innumeracy. In particular, learning the populations of 24 seed countries improved accuracy of estimates of the populations of 75 untrained countries, both at the time of learning and 4 months later. Consistent with abstraction-based theories of learning and memory, the benefits of having been exposed to the seed facts were as large 4 months after the exposure as immediately after it, despite the specific populations of the seed countries having been forgotten during the interval.     

Intracerebral source generators characterizing concentrative meditation

Previous researchers have studied meditation practices as a means to understand consciousness as well as altered states of consciousness. Various meditation techniques, such as Transcendental Meditation (TM) and Qigong, have been explored with source localization tools; however, the concentrative meditation technique has yet to be fully studied in this manner. The current study demonstrates findings, which outline differential activation in a self-referential default network during meditation in participants who espouse themselves as regular concentrative meditation practitioners, as well as comparisons with a control group practicing a modified version of the relaxation response. The results are compared with other putative experimental findings employing other meditation techniques, and the findings outlined in the current study are discussed with respect to changes in perceptual awareness often reported by meditators.     

Memory seeding: representations underlying quantitative estimations

N. R. Brown and R. S. Siegler (1996) found that training participants on a subset of country populations improved estimations for novel transfer country populations, an effect called seeding that remained intact over time. They attributed this effect to the abstraction by participants of a general metric framework for estimating populations not dependent on specific country anchors. In a series of 3 follow-up experiments, the authors found that training on seed populations produces both general metric information and durable specific country information. Moreover, minimal amounts of general (mean or range of populations) or specific (1 or 3 countries) information made available for inspection while estimating produced a significant seeding effect. Retention over long intervals was facilitated by both presenting 3 seed countries as opposed to 1 and providing names for the seed countries.     

Facilitation of attention to number and conservation of number

This study examined the relation between number conservation and attention to number, density, and length or area in kindergarteners (age 5 years, 8 months) and third graders (age 8 years, 11 months). Attention was assessed by the dimensional preference technique, using triads. Both the attention and conservation tasks included stimulus factors known to facilitate or hinder conservation: small or large numbers of objects, nonlinear or linear arrays, and trials ordered from easiest to hardest, hardest to easiest, or randomly. The number of objects and type of array affected both tasks similarly and there was an order effect in the attention task. There was an increase in attention to number from kindergarten nonconservers to kindergarten conservers to third-graders. Possible developmental interactions between attention and conservation were suggested.     

Some random observations

Conclusion Of course, the rationale of PME is so different from what has been taught in orthodox statistics courses for fifty years, that it causes conceptual hangups for many with conventional training. But beginning students have no difficulty with it, for it is just a mathematical model of the natural, common sense way in which anybody does conduct his inferences in problems of everyday life.The difficulties that seem so prominent in the literature today are, therefore, only transient phenomena that will disappear automatically in time. Indeed, this revolution in our attitude toward inference is already an accomplished fact among those concerned with a few specialized applications; with a little familarity in its use its advantages are apparent and it no longer seems strange. It is the idea that inference was once thought to be tied to frequencies in random experiments, that will seem strange to future generations.     

How Spencer made number: First uses of the number words

This article describes the development of number concepts between infancy and early childhood. It is based on a diary study that tracked number word use in a child from 12 to 38 months of age. Number words appeared early in the child’s vocabulary, but accurate reference to specific numerosities evolved gradually over the entire 27-month period. Reliable performance on standardized number and counting tasks lagged behind accurate number use in informal settings by many months. Although there was evidence that this child labeled mentally represented sets, these instances comprised a very small proportion of the total sample and were usually references to anticipated sets rather than remembered sets.     

The unimportance of being random

This note fleshes in and generalizes an argument suggested by W. Salmon to the effect that the addition of a requirement of mathematical randomness to his requirement of physical homogeneity is unimportant for his ontic account of objective homogeneity. I consider an argument from measure theory as a plausible justification of Salmon's skepticism concerning the possibility that a physically homogeneous sequence might nonetheless be recursive and show that this argument does not succeed. However, I state a principle (the Generalized Salmon Thesis) that is intuitively plausible and reflects this skepticism. The principle entails that one should be just as certain that the limit of such an infinite sequence is irrational as one is certain that the sequence is not computable. But I claim that this consequence is acceptable.