The Moon Illusion,
by Donald E. Simanek
|Moon rising over Seattle. Multiple exposure.|
Credit and copyright: Shay Stephens
The moon does vary in angular size due to the eccentricity of its orbit. When the moon is closest to earth its angular size is about 11% larger than when it is most distant. The moon illusion describes the variation of apparent size during a much shorter time period of a few hours between its maximum elevation in the sky and its rising and setting.
Another real physical effect causes the angular size of the moon's image on the retina to be about 2% smaller when the moon is near the horizon, compared to its size near the zenith. This is due to the fact that the moon is one earth radius farther away when observed on the horizon. This size change from zenith to horizon is much smaller, and in the opposite sense to the moon illusion. A change of such a small amount is not large enough to be noticed with our unaided visual system.
Some people suppose the moon illusion to be due to atmospheric refraction. Refraction effects can be measured with instruments or cameras, and we find that refraction actually makes the moon's disk subtend a smaller angle in the sky than it would have if the atmosphere were not present. These refraction effects make the moon's apparent horizontal angular diameter still smaller (by about 1.7 percent) when the moon is near the horizon! These physical effects can be confirmed with telescopes and cameras.
The moon's vertical angular diameter at the horizon is even smaller, causing the moon to appear "flattened." At the horizon the light must pass through a greater distance in our atmosphere than when the moon is higher in the sky. This size change is also opposite (and much smaller than) the psychological moon illusion!
The reason that the atmosphere makes the moon's disk appear smaller is interesting in itself. Consider looking at a star directly overhead. The ray from this star takes the path AO to the observer at O. It passes through the atmosphere without refraction, along the radial line AC. Another star at lower elevation sends light along the line DF. This light is continuously refracted along the curved line FO, its path bending toward the radial line BC, finally reaching the observer at O. But the observer judges the direction of this ray to be along the line OE, which makes a smaller angle with AO than does the line DF. Therefore the angular separation of the stars seems smaller than it would be if the atmosphere were not present. This diagram is exaggerated. In the case of the moon, the angular separation of the edges of the moon is much smaller (about half a degree), but the same principle applies.
Note: Some people report that they do not experience the moon illusion at all. The literature on the illusion is largely silent on the reasons for this fact.Explanations of illusions must be taken with appropriate skepticism. Many are of the nature of "plausible hypotheses." Most are not (as yet) such that they can be independently verified in terms of physical processes in the brain. Also, we know that our visual perceptions arise because our brain synthesizes multiple cues. Our brain weighs these cues; some dominate in certain conditions, while others are "weaker" and are ignored. But the weightings shift in strength according to the nature of the stimuli. Many classic visual illusions arise from conflicting sensory cues of nearly equal "strength". By controlled experiments, we may often rule out some of the inadequate hypotheses about particular illusions. But when one devises a test for one specific illusion hypothesis, it's difficult to avoid introducing some other type of illusion into the procedure.
Let's consider some of the proposed hypotheses about the moon illusion. Each will be presented, followed by evidence and arguments for and against it.
A Ponzo illusion?
Illusion of size. The two figures are exactly the same size, yet many people judge the upper one to be smaller. But not all persons do. This illusion illustrates how our judgment of shape and size of an object can be influenced by the shapes and sizes of other objects nearby in the field of view.
The Ponzo illusion. The two circles are the same angular diameter. Yet many people judge the right one to be smaller. The two converging straight lines nearby influence our judgment. If the lines are replaced with solid black areas, the illusion remains.
The figure shows convergent lines that we imagine as parallel lines converging to a common "vanishing point" on the horizon. The two black bars are the same angular size, but we think that the one labeled B is farther away, and therefore many people think it is "larger". Not all persons perceive this size difference, but very few think A is larger. Remove the convergent lines, and nearly everyone judges the bars to be equal length.
Contextual effects; reference cues in the field of vision.
When we judge the size of an object near the horizon our perception is influenced by familiar terrestrial objects in the field of view (trees, houses, roads). We know from everyday experience that many of the recognizable things we see in the distance are quite far away. But when our gaze is upwards, we have no reference cues for distance, and judge things near the zenith to be closer than those on the horizon. Ibn Alhazan proposed this explanation for the moon illusion around 1000 CE.
Some experiments seem to support this explanation.
A memory map.
Perceptions are influenced by our past experience. One model of visual perception postulates that when we perceive a new and unfamiliar phenomenon our brain interprets it by comparing it to a mental map or model of our memory of previous sensory experiences. Of course, this represents just one of the cues that the brain must sort out, weighing it against other cues. Conflict between sensory cues is the basis of many common visual illusions.
When we observe the horizon moon and think, "My, that moon seems unusually large," we are comparing it to past experience of moons seen higher in the sky. The moon illusion, seen "in the wild" always involves a comparison of the horizon moon being seen now with a memory of an overhead moon seen earlier, or vice versa. There's about six hours between the horizon moon and the overhead moon (more or less, depending on the tilt of the moon's orbital plane and the season). Too often discussions bury this important point in a lot of moonshine. Perhaps it's more accurate to say that we are comparing our immediate perception of the moon with all of our memories of moons seen in the past, under a variety of conditions. We have only one moon, and cannot observe it in two positions at the same time.
|The sky-dome model of the moon illusion,|
from the 1872 "Buch der Erfindungen"
(book of inventions) vol. 2, p.239.
This is an example of a misleading and nearly
incomprehensible illustration that explains nothing.
Exercise: list all the mistakes in the picture.
If there's a mental map or model, it need not be assumed to be located in a particular place in the brain. Very likely it is an interactive linking of specific visual functions distributed over various locations in the physical brain.The mental sky-dome model.
This commonly seen explanation postulates that we have in our minds a mental map or model of the shape of the sky. We use this model as a real-time reference surface for objects too distant for stereoscopic distance cues, even when there are no other cues in the visual field to serve as distance benchmarks. This mental model does not picture the sky in the shape of a hemisphere, but is more like a relatively shallow, inverted soup-bowl. When we view something on the horizon we perceive that it's located on a portion of the sky farther away than an object of the same real angular size at the zenith.
This figure shows the comparison between a spherical sky dome and the shallow apparent sky bowl. When an object of constant angular size is seen near the horizon at A, and then near the zenith at B, we judge that both are "on" the apparent "flattened" sky dome. But we judge A' to be farther away than B', and the moon at A'the is therefore judged to be a larger object on that dome than B'.
The important point of the argument is that our judgment starts with the stimulus of the retinal image of the moon, which is very nearly the same size for horizon moons or overhead moons. Stated in an equivalent way: The moon has nearly the same actual (physical) angular size wherever it is in the sky. So we mentally assume that an object of this retinal size lies on the perceptual sky dome. That dome is perceived as more distant at the horizon than overhead.
This is the same as saying, "What is our judgment of actual size of two things at different perceived distances, even though they have the same angular size?" The answer is that the one assumed nearer is judged to be smaller. This conclusion is consistent with the mental judgment that the horizon moon is farther from us.
This process supposedly operates even (especially) in the absence of any other visual cues. But the process is confused when we have our heads in an unusual position. This may be the result of our knowledge of the orientation of our head, from visual cues, and perhaps from information from the balance-sensing mechanisms of our inner ear. When there are competing sensory cues, our judgment of angular size can be altered by them, which may account for the confusing results of experiments designed to show that visual cues are the sole reason for the moon effect.
Any hypothesis that depends on a mental model of the sky requires that we have some way to know, at least approximately, which direction is "up". The illusion is "anchored" on our judgment of "up" and "down".
What's wrong with the sky-dome model as an explanation of the moon illusion?
One reader of an earlier version of this document mistakenly assumed I was promoting and defending the sky-dome model as an explanation of the moon illusion. That was not my intent at all, so I must increase the wordiness of this document by pointing out the obvious objection. Let me be very clear. The evidence for the illusion of the sky as an inverted soup-bowl is, in my view, abundant and undeniable, and must be dealt with if we are ever to understand the moon illusion. However, to use it as an explanation of the moon illusion is specious.
Consider an equally plausible argument using the sky-dome. We know that the full moon displays constancy of appearance (surface features) no matter where it is in the sky. It is the same moon wherever we see it. Few persons, when asked, would claim that the moon actually changes its physical size like a deflating balloon as it moves up in the sky. We adults agree that the moon has size constancy (children may not have formed such a conclusion). If we observe the moon in a starry sky, most persons say that the apparent distance of moon and surrounding stars is the same. There's no visual clue (such as parallax) to lead us to think otherwise. All of these "sky objects" are judged to be on this flattened "sky dome." If that is so, then the overhead moon should be judged larger in angular size, since it is on a portion of the sky that we judge is nearer to us.
Comparing these two hypotheses with their contradictory conclusions leads us to see the inadequacy of both of them, and the emptiness of this sort of argument. It forces us to examine just what we mean by "distance", "size", and "angular size". We will return to this point later. This confusion does not invalidate the perception of a flattened sky dome, but does reveal the dangers in trying to use one illusion (the sky dome) to explain another (the moon illusion).
I hate to suggest this, but many of the arguments about moon and sky illusions are similarly flawed, and seem only to be playing verbal games with the reader. They "sound good" but fall apart on close examination.
See A New Theory of the Moon Illusion for a satirical example of how a bogus explanation can seem quite plausible. Or is it bogus?
The flattened sky-dome hypothesis examined.
So why should we have this shallow-bowl model of the sky? Or, to put the question another way, why do we have a cognitive processing mechanism (however complex) that, in effect, gives us a perception of the sky distorted to conform to such a model? Two suggestions have been made. (1) The mechanism is hard-wired into our brains from birth. (2) The mechanism is built up through experience, by daylight sensory experiences, from a host of visual cues in everyday life.
Studies of children from age 4 to adult suggest that the moon illusion is present in children, and is stronger than it is in adults. It decreases in strength with age. [Liebowitz, H. and Hartman, T. "Magnitude of the Moon illusion as a Function of the Age of the Observer." Science, 130, 569-570.] This study was done indoors in a large darkened room (no other visual cues) with artificial moons at distances of 85 feet.
Studies with children can be confusing, for a child will often reach out to touch a distant object, like the moon on the horizon. Children haven't yet developed the same mental model of visual space that adults have. This study suggests that visual experience as one matures to adulthood modifies the illusion by improving our judgment of horizontal distances, thereby decreasing the illusion. This casts doubt on any explanations that assume that the moon illusion is a result of visual experience and suggests that the illusion itself may be innate and present at birth.
The hard-wired hypothesis supposes that natural selection has shaped those brain mechanisms that process and interpret sensory data, devoting more resources to those things that are important to survival. This results in brain resources being biased toward things seen in front of us, fewer resources to things overhead. Similar imbalance of perception details are present in animals.
Running great risk of over-simplifying a complex problem, we might say that things seen overhead appear smaller in angular size because our brain never evolved adequate resources to interpret data we judge to be "overhead". That data was not important to our species' survival. Our visual space allocates less detail and poorer judgment of relative distances for objects at large elevations up, or down.
Anisotropy of visual space.
Luneberg (1947) proposed a theory of vision that attempted to relate physical space with virtual (perceptual) space. He concluded that virtual space is non-Euclidean space of negative curvature: a hyperbolic space. Visual space does not have the same metric properties in all directions. (This idea is mentioned here for completeness, not to claim that Luneberg developed the idea fully and successfully, nor that he applied it to the moon illusion.)
This hypothesis is most helpful in understanding the fact that all lines of a set of physically parallel lines are also perceived as straight. Certainly they are not rendered straight on our retinas, and their curvature on the retina changes in complicated ways as we move our eyes right/left and up/down. Yet at any instant, they seem perceptually straight. Our brain is continually recalculating the data from the retinal image, to give a perception of straightness. Is our cognitive apparatus biased to render "straightness", or is this merely a by-product of cognitive processes that "correct" the "warped" retinal image to resolve visual contradictions when our eyes scan the real visual world? We live in a modern world with many straight and parallel lines: streets, walls, railroad tracks, etc. Yet the mechanism for dealing with the visual world evolved to its present form long before our ancestors ever experienced such geometric regularities. Clearly this process has more fundamental importance to our vision than rendering straight lines "correctly". I am not arguing that the curved retina is the reason for the anisotropy of visual space. I'm simply pointing out that the brain has mechanisms for dealing with the retinal shape, and the dynamically changing retinal image (as the eyes scan a scene), to produce a stable, and reasonably consistent geometry of visual space. This is what psychologicsts call "perceptual space".Once we recognize that visual space may not be anisotropic, and that it is a somewhat consistent distortion of real space, we have a new way of thinking about the moon illusion and related sky illusions.
The larger visual space.
Spherical perspective of parallel lines
perceived from wide-ranging sweeping of the eyes
over more than a whole hemisphere.
This is similar to a fish-eye lens photograph.
Z (zenith), N (nadir), V (left and right vanishing points).
The insets show how parallel lines are
rectified as straight in the smaller visual field.
Only when we scan our eyes around this geometric world, consciously trying to get the "bigger picture", do we become aware how our eye/brain mechanism handles this problem. Try this experiment. Look at a long straight wall. The wall seems to have straight and parallel lines when our line of sight is perpendicular to the wall. But when looking parallel to the wall, those same lines appear to be straight lines converging to a "vanishing point" on the horizon. Shifting our gaze from one end of the wall to the other we integrate all of these views and finally perceive that those parallel lines appear as curved lines diverging from a point at the horizon, becoming nearly parallel, then converging to a point at the other horizon. This is the geometric world of our wider visual field, one of curved lines, a "Remannian" space. But when we fix our gaze in one direction, our brain straightens out those curves, producing a result like the "Euclidean" space of an artist's strict perspective rendering.
This process does not, however, consistently "correct" angles between lines. Right angles in a perspective drawing are usually not right angles on the paper or canvas. Nor are angles correctly perceived by the eye when the plane of the angle is tilted with respect to our line of sight.
Anyone can appreciate the character of this anisotropy by looking at the daylight cloud-covered sky in relatively flat terrain. Physically the cloud canopy is nearly a flat plane, as is the earth under our feet, because the radius of their curvature is so large compared the distance to the visual horizon. Think of observing the sky from a ship in the middle of an ocean. The ocean does appear nearly flat, but the cloud cover appears as an inverted shallow bowl. This shape can be appreciated even better if the observer slowly moves his eyes to scan around the horizon and up from horizon to zenith, with conscious intent to get a feeling for the shape of things in this larger visual space. Some people even perceive that the ocean curves somewhat upward toward the distant horizon just as the cloud cover curves downward. In any case, most people judge the sky at the horizon to be the same distance as the horizon.
|A uniform cloud cover seems more like a flat roof than a spherical dome.|
We assume the cloud's average size is constant with distance.
We see a linear decrease in the cloud's apparent sizes, just as expected for a flat surface.
This visual cue dominates, for there are no others.
Such vusual experiences form our mental model of the shape of the sky.
How does this impression square with the situation in real (physical) space? The physical distance to the most distant object one can see on the horizon depends on the elevation of the observer's eye above the ground. One can derive the formula for it, in terms of the earth's radius. For an eye elevation of six feet, the things we see on the horizon are actually about 3 miles away. Alto-cumulous clouds are about 2 to 3.5 miles overhead. So, physically, the distances horizontally and vertically are nearly the same, yet the overhead clouds seem much closer to most people than those near the horizon. This calculation may not seem quite fair, for we can see clouds that are physically well beyond the surface horizon, perhaps 10 miles away, due to their height above the earth surface. But can any reader and observer honestly claim that the clouds at the horizon seem farther away than the horizon? I've never found anyone who would make that claim.
Since the clouds seen "at" the horizon are about 10 miles away, the physical ceiling of a cloud-covered sky is a very flattened bowl (or saucer), 10 miles in radius and only about 3 miles high.
We have a strong impression that the cloud cover "joins" the horizon. Can this simply be that there's absolutely no visual cue to suggest that they are at different distances? Our brain may be making the simplest reconciliation of the situation where there are no visual clues to relative distance.
Why does the cloud cover appear to tilt downward at all? This is seen even when there are individual clouds with cues of shape, size, and shadowing. Beyond a certain distance, a distance set by other visual cues, our brain refuses to place objects at a greater apparent distance, especially if there are no local cues indicating objects between the horizon and the cloud cover. By "local" I mean, visually contiguous objects near the horizon line. Our cognitive mechanism seems to prioritize decisions, favoring reconciliation of "adjacent" cues at the expense of those displaced from one another by larger visual angles.
These facts illustrate that when we are dealing with these great distances, some of our usual geometric logic about distances is essentially useless to describe what we perceive. Once this fact of perception is realized, one also realizes that many of the experiments and theories of the moon and sky illusions and many of the published papers on the subject of visual illusions are simply irrelevant to explain our judgments of distances of very distant objects.
Why is visual space anisotropic? Specifically how does the anisotropy work? Why does the "metric" of this anisotropy shift in response to visual cues, such as objects at various distances in the field of view, and even on past visual experience? And why does some anisotropy remain even if nearly all visual distance cues are absent? Those are questions that need to be addressed. Little progress has been made in that direction. But we can't solve the moon and sky illusions until we deal with those questions.
The size-distance paradox.
Most people have a cognitive model that relates angular size to distance. This is called the "size-distance invariance law". When one observes two persons, one subtending a small angle, one a larger angle, one judges the person subtending the small angle to be more distant. This works for objects that we know or have already judged (from memory and other cues) to be "really" about the same size.
As usually seen, the sky-dome model "explains" the moon illusion in this manner: The retina receives a certain size stimulus, the same angular size for horizon or overhead moons. The brain has a model of the sky that causes us to judge that the overhead moon is the nearer one. Therefore since the two moon images have the same retinal size (stimulus), we judge the overhead moon to have smaller actual size. As indicated earlier, this over-simplified "explanation" seems to me empty and misleading.
Those who object to the "sky-dome" model in any of its variants point out that when subjects are asked which moon is larger, they answer "The horizon moon." When asked which moon is nearer, most will say "The horizon moon." This, critics say, contradicts the simple sky-dome explanation, and therefore invalidates it. They call this the "size-distance paradox". This objection, in my opinion, carries no weight. [I state again that I'm not a defender of the sky-dome "explanation" of the moon illusion, but just pointing out that some arguments against it are as weak as the model itself.]
In these discussions we must distinguish three levels of sensory processing that play a role in the final judgment.
What is the meaning of 'size'? When dealing with physical measuring instruments such as cameras and telescopes, the relations between angular size, linear size and distance is a straightforward application of trigonometry.
tan(angular size) = (linear size)/(distance)
Or tan(α) = S/D
But visual judgments of linear size and distance are clearly confounded by visual illusion effects of various kinds. So "apparent linear size" and "apparent distance" were introduced, and the size-distance-invariance hypothesis was written:
tan(angular size) = (apparent linear size)/(apparent distance)
Or tan(α) = S'/D'
On this view, if distance cues predominate, altering D to D', then the apparent size S' will be "computed" by the brain as necessary to interpret the angular size A of the image on the retina. But if size cues predominate (as when you are looking at a person whose size is familiar from long experience), the apparent distance D' will be computed by the brain.
This hypothesis has an underlying assumption: that the angular size of importance is simply the physical size of the retinal imagethe visual stimulus. But is it possible that perceived angular size is also independently subject to illusory effects? If so we should write:
tan(α') = S'/D'
This view, and this equation, were first proposed by Don McCready . He developed this more fully in his 1985 paper, and applied it to the moon illusion . His website provides a very thorough presentation of these ideas.
This raises the question: What are we doing when we judge the "size" or "distance" of something (like the moon) for which we've had no experience close-up, and whose physical linear size and distance are too large to perceptually comprehend anyway? When we say a moon is "larger", are we talking about angular size or linear size?
Some argue that the basic moon illusion is one of angular size. Certainly all the experiments that ask subjects to compare the moon's image (or a simulated moon) with a physical object (or a simulated physical object) are asking for a comparison of perceived angular size. Also, the interpretation of experimental studies may be confounded by ambiguity of the meaning of the word "size", as understood by the subjects and by the experimenters. Even if the experimenter carefully explains to naive subjects the difference between angular size and linear size, and between "real" size and "perceived size", a subject may be incapable of making those distinctions when describing an observed object.
I will not belabor the three size equations given above, for they may not even be relevant to the question of the moon illusion. The moon illusion operates at real distances essentially infinite, and real sizes beyond direct apprehension. The only relevant variables are angular size, real and apparent. However we must admit that experimental studies of illusions seen in nearby space may help us understand the cognitive mechanisms that may also apply to the moon illusion.
Some studies have asked subjects "which moon seems nearer", which seems an unfair question, but still one cannot deny that we have mental processes that produce at least an approximate subconscious judgment of "near and far" for things in the sky, and this is certainly relevant to the sky illusions. The cognitive judgment of apparent angular size is made at a very early level of cognitive processing, without "thinking about it" consciously (deliberately). When asked which moon is larger, we then must consciously process this question (without being consciously aware of our previous unconscious determination of its apparent angular size).
Suppose an observer has subconsciously determined that the horizon moon has larger angular size than the zenith moon. This is the "perceived" angular size, α'. When asked which moon is nearer, the observer may consciously reason that the horizon moon must therefore be nearer. Never in this process is this observer aware of the unconscious process by which his brain already concluded that the horizon moon had larger angular size. No contradiction or paradox is present here, for these two levels of mental processing are not simultaneous. The contradiction comes only when we think about what we said.
Most visual illusions contain similar judgment contradictions. When the Penrose impossible tribar is seen, a subject may judge that the right side is more distant than the left, based on the way they connect at the top vertex. But then, looking at the constant angular width of the lower horizontal bar, the subject is forced to conclude that the two bottom corners are the same distance away. Contradiction. Yet each individual judgment seems "right" and is based on habitual visual associations of everyday experience. This is what makes illusions tantalizing.Observers schooled in physics and math, physicists, astronomers, and amateur sky-gazers, have learned somewhat different conscious models. When asked "Which moon is nearer?" they may respond "That's an unfair question; it's still the same moon, and in both cases the distance and size are too great to make a distance judgment." If a follow-up question is asked, "Which one seems nearer?" the answer is often "Well, you can't trust your eyes in such matters." Many scientists have learned (by their mistakes) not to trust their eyes in certain kinds of informal observations. As Helmholtz said, "I would never believe anything solely on the evidence of my eyes." Unlike Helmholtz, most physical scientists do not inquire into the reasons behind visual deceptions, they simply try to stick to observations and instruments that can be shown to produce reliable and observer-independent results.
Have you noticed that the moon, projected onto a planetarium dome with exactly the correct angular size, seems the same apparent size wherever it is located on the dome, and therefore appears "too small" when on the horizon, compared to our memory of the setting or rising moon in the real sky?
I've been informed (Jan 2000) that some researchers claim that the threshold for detecting depth due to retinal disparity is 1 arc second, so persons with normal vision can detect distance differences between the moon and objects up to at least 100 meters away. Such experiments are done under "ideal conditions" and usually involve judging whether two points of light are equally far away. My comments about the planetarium sky do not depend on the correctness of this figure, since the planetarium dome is clearly well within the range of stereoscopic vision, whichever figure you accept for the maximum distance for stereo vision.The hypothesis of the importance of eye-convergence (stereoscopic vision) in the moon illusion can be tested. Suppose we suppress stereoscopic cues in the planetarium by covering one eye. Experiments related to the moon illusion, especially those where nearby objects (nearer than about 50 feet) are in the field of view, should also be done with one-eyed viewing to suppress stereoscopic cues. This has not always been recognized as an important visual cue in these experiments. However, I think they play little role in the moon illusion itself, or any related illusion related to objects beyond the range of stereoscopic vision, except to establish the distance of nearby reference objects in the field of vision.
This illustrates why experimental psychology can be every bit as difficult as physics. As McCready says (August 1999): "Clearly, no theory has been fully accepted by the experts yet."
 This raises a question. What if we inverted the visual surroundings of the moon when the moon is near the horizon? It wouldn't be hard to design a wide view binocular telescope with magnification equal to one, or a mirror system, that would do this. Has it been done? I haven't seen it reported in the literature. Looking at this inverted view, would observers perceive the moon illusion?
A simple inverting (Keplerian) telescope would not suffice for this experiment, for it magnifies the image and sees only a narrow field of view, suffering the same problems as looking at the moon through a cardboard tube.
In this document I have not attempted to give a complete history of ideas about the moon illusion, nor have I tried to indicate in every case who first proposed a hypothesis or who did the experimental studies, or when. I had no intention to write a definitive history of the subject. Besides, it's often impossible to know who first had a particular idea, as distinct from who was first to publish it. Bart Borghuis' web document may be consulted for such details, and the anthology by Hershenson  contains detailed review and bibliography of the literature of the subject.
Listing these web sites here does not constitute endorsement of the validity of all of their content. Many websites present only one or a few "explanations", without mentioning others. Some are a bit "crankish" in promoting a pet theory. From my perspective as a physicist, much of this seems like overkill in framing grand theories based on studies where there are too many counfounding variables, few data points, small sample size of subjects and often poor or ambiguous data.
Input and suggestions are welcome at the address shown to the right. When commenting on a specific document, please reference it by name or content.
Most recent edit September 2016.