Recently, there has been renewed interest in so-called evidence-based policy making. Enticed by the grand promises of Big Data, public officials seem increasingly inclined to experiment with more data-driven forms of governance. But while the rise of Big Data and related consequences has been a major issue of concern across different disciplines, attempts to develop a better understanding of the phenomenon's historical foundations have been rare. This short commentary addresses this gap by situating the current push for numerical (...) evidence within a broader socio-political context, demonstrating how the epistemological claims of Big Data science intersect with specific forms of trust, truth, and objectivity. We conclude by arguing that regulators' faith in numbers can be attributed to a distinct political culture, a representative democracy undermined by pervasive public distrust and uncertainty. (shrink)

Núñez and Fias raised concerns on whether our results demonstrate a linear number-space mapping. Patro and Nuerk urge caution on the use of animal models to understand the origin of the orientation of spatial–numerical association. Here, we discuss why both objections are unfounded.

Knowledge reduction of information systems is one of the most important parts of rough set theory in real-world applications. Based on the connections between the rough set theory and the theory of topology, a kind of topological reduction of incomplete information systems is discussed. In this study, the topological reduction of incomplete information systems is characterized by belief and plausibility functions from evidence theory. First, we present that a topological space induced by a pair of approximation operators in an (...) incomplete information system is pseudo-discrete, which deduces a partition. Then, the topological reduction is characterized by the belief and plausibility function values of the sets in the partition. A topological reduction algorithm for computing the topological reducts in incomplete information systems is also proposed based on evidence theory, and its efficiency is examined by an example. Moreover, relationships among the concepts of topological reduct, classical reduct, belief reduct, and plausibility reduct of an incomplete information system are presented. (shrink)

Socio-economic status and mathematical performance seem to be risk factors of mathematics anxiety in both children and adults. However, there is little evidence about how exactly these three constructs are related, especially during early stages of mathematical learning. In the present study, we assessed longitudinal performance in symbolic and non-symbolic basic numerical skills in pre-school and second grade students, as well as MA in second grade students. Participants were 451 children from 12 schools in Chile, which differed in (...) school vulnerability index, an indicator of SES. We tested an explanatory model of MA that included SES and longitudinal performance in basic numerical skills as predictors. The results showed a direct effect of SES on MA and a mediating effect of performance in symbolic and non-symbolic comparison tasks in pre-school. However, in second grade, only performance in symbolic comparison significantly mediated the SES-MA relationship. These findings suggest that performance in non-symbolic comparison plays an important role in explaining MA at initial stages, but that its influence is no longer significant by the time children reach formal instruction in second grade. By contrast, as children’s formal educational experience in mathematics increases, MA becomes linked primarily to symbolic numerical tasks. In sum, SES affects MA and this is due in part to the effect of SES on the development of numerical learning in pre-school, which in turn has an impact on subsequent, more complex learning, ultimately leading to differences in MA. We discuss the implications of these findings for preventing and acting upon the emergence of MA. (shrink)

The idea that there is a “Number Sense” (Dehaene, 1997) or “Core Knowledge” of number ensconced in a modular processing system (Carey, 2009) has gained popularity as the study of numerical cognition has matured. However, these claims are generally made with little, if any, detailed examination of which modular properties are instantiated in numerical processing. In this article, I aim to rectify this situation by detailing the modular properties on display in numerical cognitive processing. In the process, (...) I review literature from across the cognitive sciences and describe how the evidence reported in these works supports the hypothesis that numerical cognitive processing is modular. I outline the properties that would suffice for deeming a certain processing system a modular processing system. Subsequently, I use behavioral, neuropsychological, philosophical, and anthropological evidence to show that the number module is domain specific, informationally encapsulated, neurally localizable, subject to specific pathological breakdowns, mandatory, fast, and inaccessible at the person level; in other words, I use the evidence to demonstrate that some of our numerical capacity is housed in modular casing. (shrink)

In this article, we describe and attempt to solve a puzzle arising from the interpretation of modified numerals like less than five and between two and five. The puzzle is this: such modified numerals seem to mean different things depending on whether they combine with distributive or non-distributive predicates. When they combine with distributive predicates, they intuitively impose a kind of upper bound, whereas when they combine with non-distributive predicates, they do not. We propose and explore in detail four solutions (...) to this puzzle, each involving some notion of maximality, but differing in the type of maximality involved and in the source of maximality. While the full range of data we consider do not conclusively favor one theory over the other three, we do argue that overall the evidence goes against the view that modified numerals lexically encode a ‘standard’ maximality operator, and suggests the need for a pragmatic blocking mechanism that filters out readings of sentences that are generated by the grammar but intuitively unavailable. (shrink)

The concept of evidence is central to both epistemology and the philosophy of science. Of course, ‘evidence’ is hardly a philosopher's term of art: it is not only, or even primarily, philosophers who routinely speak of evidence, but also lawyers and judges, historians and scientists, investigative journalists and reporters, as well as the members of numerous other professions and ordinary folk in the course of everyday life. The concept of evidence would thus seem to be on (...) firmer pre-theoretical ground than various other concepts which enjoy similarly central standing within philosophy. (Contrast, for example, the epistemologist's quasi-technical term ‘epistemic justification’.). (shrink)

In this book, Reiss argues in favor of a tight fit between evidence, concept and purpose in our causal investigations in the sciences. There is no doubt that the sciences employ a vast array of techniques to address causal questions such as controlled experiments, randomized trials, statistical and econometric tools, causal modeling and thought experiments. But how do these different methods relate to each other and to the causal inquiry at hand? Reiss argues that there is no "gold standard" (...) in settling causal issues against which other methods can be measured. Rather, the various methods of inference tend to be good only relative to certain interpretations of the word "cause", and each interpretation, in turn, helps to address some salient purpose but not others. The main objective of this book is to explore the metaphysical and methodological consequences of this view in the context of numerous cases studies from the natural and social sciences. (shrink)

The use of numerical arguments has become part and parcel of evidence-based policymaking, serving increasingly as scientific evidence which is used to back up policy decisions and to convince citizens of the acceptability of those decisions. But numerical arguments and their quality and potential persuasive role in the specific institutional context of policymaking have received little treatment within argumentation theory. This paper endeavours to explain the forms, functions, and quality of numerical arguments in policymaking.

The use of numerical arguments has become part and parcel of evidence-based policymaking, serving increasingly as scientific evidence which is used to back up policy decisions and to convince citizens of the acceptability of those decisions. But numerical arguments and their quality and potential persuasive role in the specific institutional context of policymaking have received little treatment within argumentation theory. This paper endeavours to explain the forms, functions, and quality of numerical arguments in policymaking.

In this study, we test whether children whose culture lacks CWs and counting practices use a spatial strategy to support enumeration tasks. Children from two indigenous communities in Australia whose native and only language (Warlpiri or Anindilyakwa) lacked CWs and were tested on classical number development tasks, and the results were compared with those of children reared in an English-speaking environment. We found that Warlpiri- and Anindilyakwa-speaking children performed equivalently to their English-speaking counterparts. However, in tasks in which they were (...) required to match the number of objects in a display, they were more likely to reconstruct part or all of the spatial arrangement of the target than were their English-speaking counterparts. Following John Locke's interpretation of users of similar American languages and in contrast with later Whorfian interpretations, we suggest that CWs may be strategically useful, but that in their absence, other task-specific strategies will be deployed. (shrink)

The consideration of deliberate versus automatic processing of numeric representations is important to math education, memory for numbers, and decision-making. In this commentary, we address the possible roles for numeric representations in such higher-level cognitive processes. Current evidence is consistent with important roles for both automatic and deliberative processing of the representations.

Memory for numbers improves with age. One source of this improvement may be learning linear spatial-numeric associations, but previous evidence for this hypothesis likely confounded memory span with quality of numerical magnitude representations and failed to distinguish spatial-numeric mappings from other numeric abilities, such as counting or number word-cardinality mapping. To obviate the influence of memory span on numerical memory, we examined 39 3- to 5-year-olds’ ability to recall one spontaneously produced number (1-20) after a delay, and (...) the relation between numeric recall (controlling for non-numeric recall) and quality of mapping between symbolic and non-symbolic quantities using number-line estimation, give-a-number estimation, and counting tasks. Consistent with previous reports, mapping of numerals to space, to discrete quantities, and to numbers in memory displayed a logarithmic-to-linear shift. Also, linearity of spatial-numeric mapping correlated strongly with multiple measures of numeric recall (percent correct and percent absolute error), even when controlling for age and non-numeric memory. Results suggest that linear spatial-numeric mappings may aid memory for number over and above children’s other numeric skills. (shrink)

Contentless experience involves an absence of mental content such as thought, perception, and mental imagery. In academic work it has been classically treated as including states like those aimed for in Shamatha, Transcendental, and Stillness Meditation. We have used evidence synthesis to select and review 135 expert texts from within the three traditions. In this paper we identify the features of contentless experience referred to in the expert texts and determine whether the experiences are the same or different across (...) the practices with respect to each feature. We identify 65 features reported or implied in one or more practices, with most being reported or implied in all three. While there are broad similarities in the experiences across the traditions, we find that there are differences with respect to four features and possibly many others. The main difference identified is that Shamatha involves substantially greater attentional stability and vividness. Another key finding is that numerous forms of content are present in the experiences, including wakefulness, naturalness, calm, bliss/joy, and freedom. The findings indicate that meditation experiences described as contentless in the academic literature can in fact involve considerable variation, and that in many and perhaps most cases these experiences are not truly contentless. This challenges classical understandings in academic research that in these so-called contentless experiences all content is absent, and that the experiences are therefore an identical state of _pure consciousness_ or consciousness itself. Our assessment is that it remains an open question whether the experiences aimed for in the three practices should be classed as pure consciousness. Implications of our analysis for neuroscientific and clinical studies and for basic understandings of the practices are discussed. (shrink)

Numerical studies of the Hubbard model and its strong-coupling form, the t-J model, show evidence for antiferromagnetic, $d_{x^{\text{2}} - y^2 } $ -pairing and stripe correlations which remind one of phenomena seen in the layered cuprate materials. Here, we ask what these numerical results imply about various scenarios for the pairing mechanism.

Medicine is moving away from reliance on the proclamations of authorities to the use of numerical methods to estimate the size of effects of its interventions. But a rumbling note of uneasiness underlines present-day medical progress: the more we know, The more questions we encounter about what to do with the hard-won information. The essays in Where's the Evidence examine the dilemmas that have arisen as the result of medicine's unprecedented increase in technical powers. How do doctors draw (...) the line between "knowing" (the acquisition of new medical information) and doing" (the application of that new knowledge)? What are the long-term consequences of responding to the demand that physicians always do everything that can be done? Is medicine's primary aim to increase the length of life? Or is it to reduce the amount of pain and suffering? And who is empowered to choose when these ends are mutually exclusive? This engaging collection of essays will be of interest to professionals interested in the evidence-based medicine debate, including epidemiologists, neonatologists, those involved in clinical trials and health policy, medical ethicists, medical students, and trainees. (shrink)

A Bayesian measure of evidence for precise hypotheses is presented. The intention is to give a Bayesian alternative to significance tests or, equivalently, to p-values. In fact, a set is defined in the parameter space and the posterior probability, its credibility, is evaluated. This set is the “Highest Posterior Density Region” that is “tangent” to the set that defines the null hypothesis. Our measure of evidence is the complement of the credibility of the “tangent” region.

This article discusses the nature of numerals and the plausibility of their special semantic and epistemological status as proper names of numbers. Evidence is presented that minimizes the difference between numerals and other devices of direct reference. The availability of intensional contexts within formalised metamathematics is exploited to shed light on the relation between formal numerals and numerals.

In order to perform certain actions – such as incarcerating a person or revoking parental rights – the state must establish certain facts to a particular standard of proof. These standards – such as preponderance of evidence and beyond reasonable doubt – are often interpreted as likelihoods or epistemic confidences. Many theorists construe them numerically; beyond reasonable doubt, for example, is often construed as 90 to 95% confidence in the guilt of the defendant. -/- A family of influential cases (...) suggests standards of proof should not be interpreted numerically. These ‘proof paradoxes’ illustrate that purely statistical evidence can warrant high credence in a disputed fact without satisfying the relevant legal standard. In this essay I evaluate three influential attempts to explain why merely statistical evidence cannot satisfy legal standards. (shrink)

Previous accounts of hedges assume that they cause language to become vague or fuzzy ; however, hedges can actually sharpen numerical concepts by giving explicit information about approximation, especially where bare numbers appear misleadingly round or precise. They can also tell hearers about the direction of approximation. This article provides a first empirical account of interactions between hedging and rounding in numerical expressions. We demonstrate that hedges occur more commonly with round numbers than with non-round ones. However, we (...) also provide evidence from user studies that in the absence of hedges, readers interpret round numbers as approximations and non-round ones as precise; and that placing a hedge before a round number has no effect on its interpretation, whereas placing it before a non-round number shifts people’s interpretations from precise towards approximate. We attempt to explain this conundrum. (shrink)

Cohen Kadosh & Walsh (CK&W) argue that recent findings challenge the hypothesis of abstract numerical representations. Here we show that because, like many other authors in the field, they rely on inaccurate definitions of abstract and non-abstract representations, CK&W fail to provide compelling evidence against the abstract view.

Previous accounts of hedges assume that they cause language to become vague or fuzzy (Lakoff 1973); however, hedges can actually sharpen numerical concepts by giving explicit information about approximation, especially where bare numbers appear misleadingly round or precise. They can also tell hearers about the direction of approximation (greater or less than). This article provides a first empirical account of interactions between hedging and rounding in numerical expressions. We demonstrate that hedges occur more commonly with round numbers than (...) with non-round ones. However, we also provide evidence from user studies that in the absence of hedges, readers interpret round numbers as approximations and non-round ones as precise; and that placing a hedge before a round number has no effect on its interpretation, whereas placing it before a non-round number shifts people’s interpretations from precise towards approximate. We attempt to explain this conundrum. (shrink)

Due to their frequent uneven cognitive profiles, genetic developmental disorders allow researchers to investigate which numerical sub-system of those present in typically developing infants best predicts subsequent numerical abilities. More importantly, they can provide evidence of which other cognitive abilities outside number are necessary for the successful development of these numerical sub-systems. We discuss evidence from cross-syndrome comparisons of adults, adolescents, children, and infants with Williams syndrome and those with Down syndrome to show that the (...) approximate magnitude sub-system is crucial for later number development. In addition, we show that specific problems outside the number domain, and within basic-level visual and attention systems contribute to an explanation of the difficulties and proficiencies observed within each genetic disorder. Finally, we argue that a truly developmental approach is critical when using the cross-syndrome design in order to reveal subtle differences that impact over time on the development of cognitive abilities. (shrink)

. With an overall objective of establishing association between air pollutants and incidence of respiratory diseases, the environmental professionals and medical practitioners have made significant contribution, using statistical mechanics in modelling epidemiological data, population characteristics, and pollution parameters. Broadly speaking, the studies have shown that the increase in vehicular traffic has been one of the causes of respiratory diseases. However, the WHO Centre for Environment and Health, Europe in its 2005 document states: “There is little evidence for a causal (...) relationship between asthma prevalence/incidence and air pollution in general, though the evidence is suggestive of a causal association between the prevalence/incidence of asthma symptoms and living in close proximity to traffic”. Decision making process in a real world is invariably based on perceptions which are expressed in words or may be in numeric terms and not in probability terms. In the paper, we made an attempt to model the perceptions of experienced pulmonologists in arriving at their combined degree of belief/plausibility/ignorance for all the possible combinations of identified respiratory diseases, using evidence theory and fuzzy relational calculus without collecting sizeable parametric data accumulated over a period of years. Tightening pollution norms by the regulatory authorities is an overall objective of the global efforts on greenhouse gases reduction in general, and air pollution mitigation in particular. The concept of solar battery operated electric vehicles for road transport is advocated, initially for two/three wheelers, and extending it to four wheelers, especially in the developing countries. (shrink)

Numerous recent discoveries in China of ancient tombs have greatly increased our knowledge of ritual and religious practices. These discoveries include excavated oracle bones, bronze, jade, stone and pottery objects, and bamboo manuscripts dating from the twelfth to fourth century BCE. Inscribed upon these artifacts are a large number of records of numerical sequences, for which no explanation has been found of how they were produced. Structural links to the Book of Changes, a divination manual that entered the Confucian (...) canon, are evident; yet, the algorithm described therein dates to the slightly later second to first century BCE. By combining archeological and statistical evidence, we propose a new methodology that enables us to reconstruct and test cleromantic techniques which can explain how these numerical sequences were generated. Dice and divination stalk use, either in combination or separately, appear in fact to have been underlying the rather stable numerical patterns in ancient China all the way back to the late Shang dynasty (1300–1046 BCE). Bringing to light such a long-standing technique, which awaits further confirmation from the ever-growing database of newly discovered numerical and textual records, can change drastically our understanding of early Chinese history and of the historical development of sophisticated arithmetical practices and the rationalization of chance. (shrink)

Two kinds of probability expressions, verbal and numerical, have been used to characterize the uncertainty that people face. However, the question of whether verbal and numerical probabilities are cognitively processed in a similar manner remains unresolved. From a levels-of-processing perspective, verbal and numerical probabilities may be processed differently during early sensory processing but similarly in later semantic-associated operations. This event-related potential (ERP) study investigated the neural processing of verbal and numerical probabilities in risky choices. The results (...) showed that verbal probability and numerical probability elicited different N1 amplitudes but that verbal and numerical probabilities elicited similar N2 and P3 waveforms in response to different levels of probability (high to low). These results were consistent with a level-of-processing framework and suggest some internal consistency between the cognitive processing of verbal and numerical probabilities in risky choices. Our findings shed light on possible mechanism underlying probability expression and may provide the neural evidence to support the translation of verbal to numerical probabilities (or vice versa). (shrink)

An astonishing volume and diversity of evidence is available for many hypotheses in the biomedical and social sciences. Some of this evidence—usually from randomized controlled trials (RCTs)—is amalgamated by meta-analysis. Despite the ongoing debate regarding whether or not RCTs are the ‘gold-standard’ of evidence, it is usually meta-analysis which is considered the best source of evidence: meta-analysis is thought by many to be the platinum standard of evidence. However, I argue that meta-analysis falls far short (...) of that standard. Different meta-analyses of the same evidence can reach contradictory conclusions. Meta-analysis fails to provide objective grounds for intersubjective assessments of hypotheses because numerous decisions must be made when performing a meta-analysis which allow wide latitude for subjective idiosyncrasies to influence its outcome. I end by suggesting that an older tradition of evidence in medicine—the plurality of reasoning strategies appealed to by the epidemiologist Sir Bradford Hill—is a superior strategy for assessing a large volume and diversity of evidence. (shrink)

The question of whether non-human animals are conscious is of fundamental importance. There are already good reasons to think that many are, based on evolutionary continuity and other considerations. However, the hypothesis is notoriously resistant to direct empirical test. Numerous studies have shown behaviour in animals analogous to consciously-produced human behaviour. Fewer probe whether the same mechanisms are in use. One promising line of evidence about consciousness in other animals derives from experiments on metamemory. A study by Hampton (Proc (...) Natl Acad Sci USA 98(9):5359–5362, 2001 ) suggests that at least one rhesus macaque can use metamemory to predict whether it would itself succeed on a delayed matching-to-sample task. Since it is not plausible that mere meta-representation requires consciousness, Hampton’s study invites an important question: what kind of metamemory is good evidence for consciousness? This paper argues that if it were found that an animal had a memory trace which allowed it to use information about a past perceptual stimulus to inform a range of different behaviours, that would indeed be good evidence that the animal was conscious. That functional characterisation can be tested by investigating whether successful performance on one metamemory task transfers to a range of new tasks. The paper goes on to argue that thinking about animal consciousness in this way helps in formulating a more precise functional characterisation of the mechanisms of conscious awareness. (shrink)

This study investigated their views concerning evidence and expert opinion of 10th‐grade students, accessed by an open‐ended questionnaire in the context of a socio‐scientific issue: the cause of flood disasters, and personal epistemology identified by the Learning Environment Preference Questionnaire . Students' responses to the open‐ended questions showed that when thinking about the flood issue, most students rely heavily on direct and numerical data to draw their conclusions, while experts represented a source of conclusive information. The LEP scores (...) indicated that, in terms of epistemological development, students were mostly at the stage of the ‘multiplicists’ in Perry's model. The statistical analysis suggested that view towards evidence and expert were associated with personal epistemology. (shrink)

The use of neuroscientific evidence in criminal trials has been steadily increasing. Despite progress made in recent decades in understanding the mechanisms of psychological and behavioral functioning, neuroscience is still in an early stage of development and its potential for influencing legal decision-making is highly contentious. Scholars disagree about whether or how neuroscientific evidence might impact prescriptions of criminal culpability, particularly in instances in which evidence of an accused’s history of mental illness or brain abnormality is offered (...) to support a plea of not criminally responsible. In the context of these debates, philosophers and legal scholars have identified numerous problems with admitting neuroscientific evidence in legal contexts. To date, however, less has been said about the challenges of evaluating the evidence upon which integrative mechanistic explanations that bring together evidence from different areas of neuroscience are based. As we explain, current criteria for evaluating such evidence to determine its admissibility in legal contexts are inadequate. Appealing to literature in the philosophy of scientific experimentation and theoretical work in the social, cognitive and behavioral sciences, we lay the groundwork for reforming these criteria and identify some of the implications of modifying them. (shrink)

Many theories about human number representation stress the importance of a central semantic representation that includes the magnitude information of small integer numbers, and that is conceived as an abstract, compressed number line. However, thus far there has been little or no direct evidence that units and teens are represented on the same number line. In two masked priming experiments, we show that single-digit and two-digit Arabic numerals are equally well primed by an Arabic numeral with the same number (...) of digits as by an equally distant Arabic numeral with a different number of digits (e.g. the priming effect of 7 on the target 9 is the same as the priming effect of 11 on the target 9). The finding was obtained both with a number naming task and with a parity judgement task. This is in line with the hypothesis that units and teens are part of a continuous number line. (shrink)

Criminal defendants cannot be punished unless found guilty “beyond a reasonable doubt”. Under probabilistic accounts, this means that the probability of guilt given the evidence is above a certain numerical threshold, such as 0.9. Under psychological accounts, by contrast, what is essential is that a factfinder reaches a certain psychological attitude toward guilt, such as certainty or unwavering belief, when contemplating the evidence. An adequate account should provide a normative explanation for why guilt BARD warrants punishment. Psychological (...) accounts are more likely to meet this challenge than probabilistic, it is argued. The permissibility of punishment is likely to be relative to the cognitive attitudes of the punisher, but not to the conditional probability of guilt in light of the evidence. (shrink)

Objective: To examine the role of explainability in machine learning for healthcare (MLHC), and its necessity and significance with respect to effective and ethical MLHC application. Study Design and Setting: This commentary engages with the growing and dynamic corpus of literature on the use of MLHC and artificial intelligence (AI) in medicine, which provide the context for a focused narrative review of arguments presented in favour of and opposition to explainability in MLHC. Results: We find that concerns regarding explainability are (...) not limited to MLHC, but rather extend to numerous well-validated treatment interventions as well as to human clinical judgment itself. We examine the role of evidence-based medicine in evaluating unexplainable treatments and technologies, and highlight the analogy between the concept of explainability in MLHC and the related concept of mechanistic reasoning in evidence-based medicine. Conclusion: Ultimately, we conclude that the value of explainability in MLHC is not intrinsic, but is instead instrumental to achieving greater imperatives such as performance and trust. We caution against the uncompromising pursuit of explainability, and advocate instead for the development of robust empirical methods to successfully evaluate increasingly inexplicable algorithmic systems. (shrink)

Scholars in philosophy, law, economics and other fields have widely debated how science, environmental precaution, and economic interests should be balanced in urgent contemporary problems, such as climate change. One controversial focus of these discussions is the precautionary principle, according to which scientific uncertainty should not be a reason for delay in the face of serious threats to the environment or health. While the precautionary principle has been very influential, no generally accepted definition of it exists and critics charge that (...) it is incoherent or hopelessly vague. This book presents and defends an interpretation of the precautionary principle from the perspective of philosophy of science, looking particularly at how it connects to decisions, scientific procedures, and evidence. Through careful analysis of numerous case studies, it shows how this interpretation leads to important insights on scientific uncertainty, intergenerational justice, and the relationship between values and policy-relevant science. (shrink)

A new Evidence Test is applied to the problem of testing whether two Poisson random variables are dependent. The dependence structure is that of Holgate’s bivariate distribution. These bivariate distribution depends on three parameters, 0 < theta_1, theta_2 < infty, and 0 < theta_3 < min(theta_1, theta_2). The Evidence Test was originally developed as a Bayesian test, but in the present paper it is compared to the best known test of the hypothesis of independence in a frequentist framework. (...) It is shown that the Evidence Test is considerably more powerful when the correlation is not too close to zero, even for small samples. (shrink)