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A psychological study of leg-before-wicket judgments in cricket.

Cricket belongs to the family of games that includes baseball and rounders. One player (the bowler) attempts to strike a target of three wooden upright sticks (the wicket) by projecting a ball towards it from a distance of about 20 metres (22 yards). Another player (the batsman) stands close to the wicket, and attempts to defend it by hitting the ball with a bat. Where it is consistent with successful defence, the batsman also aims to hit the ball as far as possible, or to direct it out of the reach of the opposing players, thereby gaining an opportunity to score for the batting side. If the bowler succeeds in striking the wicket with the ball, the batsman is 'out' and takes no further part in that phase of the game (and obviously cannot further increase his team's score).

The laws of cricket (Marylebone Cricket Club, 1992) state that the batsman may use only the bat (or a hand holding it) to defend the wicket. If the ball collides with any other part of the batsman or the batsman's clothes, the batsman can also be adjudged 'out' by the umpire, if the umpire believes that the ball would otherwise have hit the wicket. As it is almost invariably the batsman's padded leg that obstructs the flight of the ball, a batsman given out in this way is said to be out 'leg-before-wicket', or LBW.

The umpire's task is not an enviable one. Firstly, there are few other situations in sport where the official must make a judgment not merely about what events did take place, but about what events would have taken place had certain other events not happened. Secondly, in addition to extrapolating the ball's trajectory, the laws require the umpire to observe also where the ball bounces on the ground on its journey towards the wicket (it usually does bounce once), where the batsman's leg is when struck by the ball, and whether the batsman attempts to hit the ball with the bat. These judgments are all made with a ball that may be travelling at 90 miles per hour; the whole sequence of events lasts less than a second. Finally, LBW decisions are important: the course of a whole game (possibly lasting five days) may be altered depending upon the result of a single LBW decision, and with it the course of players' careers. Not surprisingly, LBW decisions are a source of much controversy and acrimony.

To the author's knowledge there are no published studies of the ability of umpires to make LBW decisions. Such studies may provide information that would be useful in the training of umpires, and we might also learn something general about people's ability to perceive the trajectories of moving objects. These experiments seek to fill this gap, by using simulated stimuli and the procedures of psychophysics to determine the limits of participants' ability to make LBW decisions. In doing so the possibility of applying the same methods to similar decisions in other sports is demonstrated.

Good bowlers will seek to make the ball's trajectory vary in order to deceive the batsman. Is it the case that they unwittingly deceive the umpire, too? Are there some parameters of the ball's trajectory that can be estimated and allowed for when making the LBW decision, and others that cannot be allowed for? The experiments reported here were done to find out whether people can detect and make allowance for variations in 'swing' (lateral curvature of the ball's flight) and variations of the point of release of the ball by the bowler.

Measures of performance

To quantify umpires' ability to make LBW decisions we can treat them as measuring devices. The measurement in question is where the trajectory of the ball would have intersected the plane of the wicket had the ball not been obstructed, from which measurement the umpire can decide whether the ball would have struck the wicket or not. We cannot directly observe the results of any such measurement, but we can infer the statistical properties of the measuring device (i.e. a person) by analysis of their performance in a 2-AFC task (the details of this inference are dealt with in the Methods section). It is informative to consider measurement errors as arising from two sources. The first is the participant's random error: if the same measurement is made several times, the results of the measurement will show variation, conveniently quantified by the standard deviation of the sample of measurements. The second source of error is the participant's bias (or constant error, or systematic error): if the same measurement is made several times, the results may consistently err in the same direction from physical reality. The obvious measure of bias is the mean of the differences between the participant's measurements and the corresponding physically correct measurements. Both the random error and bias are expressed as distances measured in the plane of the wicket.

It is usual to assume that, for motivated participants, the magnitude of the random error is beyond voluntary control. Participants may also exhibit biases over which they have no control, and it is such biases that are a major concern of the experiments reported here. However, people are also able to voluntarily adjust their biases to some extent, the usual reason for adjustment being that some classes of errors carry heavier costs than others. For example, in cricket, it is considered a worse error for an umpire to incorrectly give a batsman 'out' than it is for the umpire to incorrectly reprieve them, and the umpire is in principle able to adjust the bias in their perceptual measurements to achieve the appropriate compromise between the two classes of mistaken decision. This matter is taken further in the Discussion.

These well-established ideas of random error and bias provide a succinct and informative summary of people's ability to make LBW decisions. They could equally well be used for characterizing decisions in other sports, such as line judgments in tennis and its relatives.

Strategy

Although, as we shall see, the LBW decision is suitable for study by psychophysical methods, it poses one particular problem of its own, namely, the large dimensionality of the stimulus. The position of the ball when released by the bowler can vary in three dimensions, as can its velocity, and a good bowler will try to vary his deliveries in this way. A fast or medium-pace bowler can orient the ball's prominent seam in such a way as to impart a sideways curvature, or' swing', to the ball's path. A slow bowler can spin the ball to cause it to curve in the air, to cause it to deviate when it bounces on the pitch, or to cause it to bounce more or less steeply than previous deliveries. A ball delivered by a fast bowler can also deviate on bouncing, either because the bowler has spun ('cut') the ball, or because the ball happens to have struck an irregularity in the surface. Finally, the trajectory of the pad as it moves to meet the ball can vary in many, less easily specified, ways. In practice, a bowler can only manipulate the point of release of the ball significantly in one direction, and some of the parameters are not wholly independent of each other, but the combinatorial problem posed by even a few degrees of freedom is still severe.

The strategy adopted in these experiments was to ask questions of the form: 'can an umpire take account of variations in parameter P when making an LBW decision?' For example, variation in the position of the release point of the ball may or may not cause corresponding biases in the umpire's decisions. If a hypothetical umpire were to make the decision purely on the basis of the position of the ball when it strikes the pad, then variation in the direction of the ball as it strikes the pad (caused by variation in release point) will cause a characteristic pattern of bias. If, on the other hand, the umpire is truly able to extrapolate the ball's trajectory, then the bias in the participant's decisions should not vary systematically with release point.

In the experiments to be reported, some questions of the above form are asked by measuring the participant's bias as a function of the value of the chosen parameter. All other parameters are held at constant, typical values. There are two closely related reasons why one might feel uncomfortable with an approach of such draconian reductionism. One is the restricted volume of the stimulus parameter space that one is investigating (along the axes), and the other is the neglect of possible interactions between the effects of different stimulus parameters (again caused by dedicating attention to the axes). These concerns could be dealt with by doing factorial experiments, but even with only two factors, such experiments would take six or seven times as long as the experiments reported here. At this exploratory stage it seems wise to restrict one's ambitions until it is clearer what are the important variables to investigate.

A further strategic decision was to provide feedback to the participant after every trial. The experiments are concerned with the limits of human ability to make LBW decisions, and it makes sense to take all steps to allow participants to reach those limits. They had plenty of practice at the task: in the course of the experiment, each one made as many LBW decisions as a professional umpire would in several years' work. Within the terms of this experiment, their lack of previous practice is not a disadvantage: they had ample opportunity to become good at the task. Despite their experience, professional umpires will have had little in the way of direct, objective feedback. Once the techniques of studying LBW decisions have been established on other participants, a future study could see what effect this extensive unmonitored practice has had on the way that professional umpires make their decisions.

Terminology

A right-handed batsman will stand to the right of the wicket as viewed by the umpire at the bowler's end of the pitch ([ILLUSTRATION FOR FIGURE 1 OMITTED] shows an umpire's view of the pitch and wicket, but no batsman is included). The three wooden sticks that stand in line to form the wicket are conventionally referred to, from left to right, as the off stump, the middle stump and the leg stump. The notation is reversed for a left-handed batsman, but for this study, a right-handed batsman is assumed. There is another identical wicket at the bowler's end (out of sight in the illustration). The leg stump, and the stump on the same side at the bowler's end, between them define an imaginary vertical plane referred to as the 'line of leg stump'. A ball which bounces to the right of this plane (seen from the umpire's end) is said to have 'pitched outside the line of leg stump'. Another, parallel, plane is defined by the off stump and its counterpart at the umpire's end, and is called the 'line of off stump'. Both of these planes are referred to in the LBW law (though, curiously, the law does not specify which parts of the stumps define the lines of leg stump and off stump, nor which is the critical part of the ball).

Unless otherwise stated, the positions of things are given in a system of Cartesian coordinates, the origin of which lies at the base of the middle stump at the umpire's end. The x axis runs horizontally at right angles to the line joining the middle stumps (coordinates increasing rightwards as seen by the umpire), they axis runs vertically (coordinates increasing upwards), and the z axis runs along the line joining the middle stumps (coordinates increasing away from the umpire). The position of the ball is defined to be that of its centre.

The passage of the ball from the bowler's hand through the air towards the batsman's end of the pitch is usually referred to as a 'delivery'.

In the following text, any phrases referring to the position of the ball 'where it passes the wicket' (or similar wording) should be taken as meaning the place at which the ball would have passed the plane of the wicket, had it not been obstructed by the batsman's leg. This usage is adopted for brevity.

Methods

Participants

Two participants were used in the experiment. One, the author, had some informal experience of umpiring in low-grade cricket, and had a lot of experience of the experiment obtained in pilot experiments. The second was a paid volunteer with no previous experience. They were both male, aged 32 and 23 respectively, with normal visual acuity as assessed by a Snellen chart.

Psychophysics

The LBW decision is ideally suited to study by standard psychophysical methods. Taking as an example the judgment of whether the ball would have struck the wicket, there is a continuously variable independent variable (the position of the ball as it passes the plane of the wicket had it not been obstructed) and a discrete response variable (a judgment that the ball would have missed on the side of the off stump, struck the wicket, or missed on the side of the leg stump).

This three-alternative task can be treated as a two-alternative task if we think about the two sides of the wicket separately. Consider the decision as to whether the ball would have missed the off stump or struck the wicket. The x coordinate of the position of the ball as it passes the plane of the wicket (had it not been obstructed) will be denoted by x, and the x coordinate of a ball just grazing the off stump will be denoted by p. The experimenter provides stimuli with a range of values of x, and on each trial the participant's task is to decide whether x exceeds p or not. For this task, the responses 'hit' and ' miss to the leg side' both correspond to x [greater than] p and can be pooled. A similar argument can be applied, mutatis mutandis, when the decision regards the leg stump. Thus what is a three-alternative task to the participant can be analysed by the experimenter as two two-alternative tasks, one about whether the ball would have passed outside the off stump and one to do with whether the ball would have passed the leg stump (henceforth referred to as off- and leg-stump decisions).

The unit of the experiment was a block of 80 trials of a single class of decision. For the following description we will assume the decision was an off-stump decision (left stump as the umpire looks at it). The cue for the participant's decision is the difference between the x coordinate of the ball as it passed the wicket and the x coordinate of a ball that was just grazing the outside of the off stump. Several trials were presented for each value of the cue, and the fraction of times that the participant reported the ball as either hitting the wicket or passing to its right was calculated. At the end of the run, a psychometric function was plotted giving these empirical probabilities as a function of the value of the cue. The response rate rises from zero, where the cue is strongly negative, to 1.0, where the cue is strongly positive. Where the cue is zero, and the ball is just grazing the outside of the stump, we would expect an unbiased participant to respond at chance levels. If we assume that his random error distribution is Gaussian, the psychometric function will, in the limit of a large number of trials, take the form of a cumulative Gaussian. In practice, we do a finite experiment and use Probit analysis (Finney, 1971) to fit a cumulative Gaussian to the data. The standard deviation of the Gaussian distribution underlying the fitted curve is our estimate of the standard deviation of the participant's random error distribution. The mean of the underlying Gaussian is our estimate of his bias: it corresponds to the 50 per cent point on the psychometric function, and tells us the value of the cue for which he was guessing. By way of example, if the bias were estimated at -0.01 m, the conclusion is that, if presented with a series of trials on which the ball passed 0.01 m to the left of the off stump, the participant would produce 'hit' and 'miss' responses equally often.

Although we assume for the sake of the analysis that his error distribution is Gaussian, it appears to be of little practical consequence if the distribution is of a different shape. Statistical studies were undertaken of the complete psychophysical procedure, including stimulus selection (see next paragraph), using a simulated participant whose internal error distribution could be chosen at will. Rectangular and triangular error distributions were used, and it was found that the psychophysical procedure estimated the standard deviations of these distributions almost as accurately as it estimated the standard deviation of a Gaussian error distribution. In other words, we can get a good estimate of the standard deviation of the internal error distribution even if it only approximates to the Gaussian that the analysis supposes. In practice, all psychometric functions were inspected, along with the fitted curves, and there was no indication that a Gaussian error distribution was not a satisfactory approximation to the actual error distribution.

For efficiency, an adaptive psychophysical procedure, Adaptive Probit Estimation (Watt & Andrews, 1981), was used to select stimulus levels on the basis of the participant's previous responses. The distribution of stimuli presented to the participant thus concentrates about the regions of uncertainty; an example of an actual distribution is given with the results of Expt 1. In all of the experiments, several independent blocks of 80 trials were randomly interleaved and run in parallel. This enabled leg- and off-stump decisions to be combined in one run of trials. The experiments were particularly concerned with the effects of certain physical parameters of the ball's path (e.g. lateral acceleration of the ball) on the participant's ability to make LBW decisions. By interleaving blocks of trials with different settings of a certain parameter, the effect of different values of the parameter on the participant's performance could be measured, in a realistic situation where he is exposed to a different and unpredictable value of the parameter from trial to trial.

Some of the experiments required runs of nearly 1500 trials. Every 80 trials, the participant was given the chance to continue or to take a rest as he wished.

Stimuli

The studies described in this paper were done using a computer simulation of the umpire's view of a cricket pitch. There are advantages to using a computer simulation compared to the alternative of observing real LBW decisions. (a) All aspects of the flight of the ball are under the experimenter's control. New combinations of parameters can be easily tested. (b) Because the trajectory of the ball is calculated, one can determine exactly where the ball would have passed the plane of the wicket, and give accurate feedback. (c) Many trials can be undertaken in a short period of time. Of course, there are also certain disadvantages to using the simulation. The validity of the results depends upon the fidelity of the stimulus to the real situation. The real stimulus is very complex and variable, and compromises have to be made in reproducing it on a computer monitor.

1. The most significant of these compromises was that no attempt was made to depict a batsman; the pad was simply represented by a white rectangle of the appropriate size. As the experiments were concerned not with the effect of the batsman's behaviour on the umpire's decision, but with the effects of variations in the ball's flight, this simplification was considered tolerable.

2. In a real situation, the umpire's field of view is larger than that provided by the stimuli. In particular, the umpire will have extra knowledge (from direct observation of the bowler) about the bowler's position at the moment that the ball is released.

Apart from these simplifications, much attention was given to constructing stimuli that were physically realistic. The stimuli were complex to construct and will only be outlined here; full details are given in the Appendix.

The stimuli reproduced, in a greyscale image, the essential features of an umpire's view of a cricket pitch in a perspective projection on the monitor face [ILLUSTRATION FOR FIGURE 1 OMITTED]. The scene included a set of stumps and the standard pitch markings. The pitch itself was represented by an area of visual texture; as the pitch receded into the distance, the texture altered in scale appropriately to the distance, and also reduced in contrast as distance increased. The remainder of the playing area was drawn with broad, low-contrast stripes corresponding to the stripes present on a mown area of grass. Finally, the sky was represented by a light grey area. The ball, when present, was a black circle, the size of which varied according to its distance from the umpire. The viewing arrangements ensured that the participant viewed the stimulus from the viewing position assumed in the perspective calculations. Different views were presented to each eye, providing appropriate binocular cues to depth.

The trajectory of the ball was calculated according to the physical model given in the Appendix (from Daish, 1972). The x component of the initial velocity of the ball (as it leaves the bowler's hand) was adjusted so as to direct the ball at the desired place relative to the wicket. The position of the ball in three dimensions was calculated at intervals of 1/72 s (corresponding to the successive frames of the 72 Hz non-interlaced monitor). For each position an appropriately sized and positioned ball was drawn and stored in computer memory. When the stimulus was presented, the stored representations of the ball were drawn one after the other on successive cycles of the monitor. A small Gaussian blur on the image of the ball (and also the stumps and pitch markings) ensured that the ball appeared circular despite its representation on a pixelated display, and that it could be positioned to sub-pixel precision.

The batsman's pad was represented by a simple white rectangle which appeared a short time before the ball struck it and which disappeared immediately after the impact. The position of the pad in the direction was estimated from video-recordings of international cricket matches, and is roughly the position of the pad for a batsman playing neither forwards nor backwards. (A batsman will usually make a definite step either forwards or backwards to play the ball, and failing to do either is poor technique that often leads to the loss of one's wicket. Hence, this would be a typical position for a batsman involved in an LBW decision. In addition, because of the uncertainties involved, umpires rarely give batsmen out LBW if they have played well forward.) The time of the appearance of the pad corresponded roughly with the time at which the batsman would place his foot down in order to play a shot. The time of disappearance was a more arbitrary choice. For the particular questions being asked in these experiments, it is unlikely that the precise timing of the pad's appearance and disappearance would affect the pattern of the results.

A sequence of three tones told the participant that a trial was about to begin. Then the ball appeared from high on the left as it would to an umpire, travelled down the pitch, bounced once, struck the pad, and bounced and rolled away finally coming to rest. After the participant had responded and received feedback, the next trial began almost immediately.

Participant's task

The task was to decide whether the ball would have missed the wicket on the left (off) side, struck the wicket, or missed it on the right (leg) side. The participant used the '4', '5' and '6' keys of the computer's numeric keypad to enter responses. Following his response on each trial, feedback was provided. On every trial, the ball was drawn where it would have passed the plane of the wicket had it not been intercepted. On trials where the participant's response was wrong, the correct response was indicated by the use of symbols.

Apparatus

The stimuli were generated on a Viglen Genie 4DX66 computer and displayed on a Viglen Envy 15 in. colour monitor. The monitor screen measured 25.5 cm by 19.0 cm. There were 640 pixels horizontally and 480 pixels vertically. Attached to the front of the monitor was a viewing tunnel, coloured black inside. The tunnel was divided vertically to restrict each eye's view to the corresponding half of the display. The eye holes in the viewing tunnel were positioned so that the display occupied the same position with respect to the participant's eyes as assumed in the perspective calculations. The eye holes were each equipped with a suitable converging lens and prism so that he was both accommodated and converged at infinity when viewing the stimulus.

When viewed through the tunnel, the animation gave a compelling impression of a ball travelling in three dimensions and behaving as a real ball should when moving under gravity. (Even if the display was presented monocularly, and with the pictorial depth cues provided by the background removed, participants spontaneously reported seeing a ball travelling rapidly away from them in depth.)

EXPERIMENT 1

A medium-pace or fast bowler can often make the ball's path curve sideways. The following experiment was done to discover whether participants' LBW decisions are biased by such 'swing'. Fourteen parallel blocks of runs were undertaken in each experimental session. These covered off- and leg-stump judgments for seven different degrees of swing. It was expected that, if the participant failed to take the swing into account, then his decisions would suffer a bias that was regularly related to the amount of swing. On the other hand, if he could take account of the swing, any bias would be independent of the amount of swing.

Cricket balls are encased in leather and possess a prominent equatorial seam. To swing a cricket ball, the bowler releases the ball with the plane of the seam at an angle to the direction of the ball's motion. Different conditions of airflow on the two sides of the ball cause a lateral acceleration. The mechanics of the process are very complex, and no attempt was made to derive the force on the ball from physical principles. Instead, a simple constant acceleration parallel to the x axis was applied to the ball throughout its flight. Further experiments would be needed to test for sensitivity to higher-order derivatives of the x component of position.

All parameters of the ball's flight, except the swing and the initial x velocity, were held constant. They and z components of its initial velocity were -4.4 m [s.sup.-1] and 35 m [s.sup.-1], corresponding to a fast bowler at international level. The x, y and z coordinates of the release point were -0.05 m, 2.5 m and 1.5 m. Seven values of swing were used, being lateral accelerations of -1.5, -1.0, -0.5, 0.0, 0.5, 1.0 and 1.5 m [s.sup.-2]. These give maximum lateral deviations of the flight of 24 cm, a range which includes the range of swings likely to be obtained in practice. The 1120 trials in each run were done in a single session, with breaks every 80 trials at the participant's discretion. Participant 1 (BJC) did five runs, and participant 2 (EMS) did six runs. The entire experiment was spread over several days.

Results

The data were subjected to a three-way analysis of variance, the factors being amount of swing, stump (side of wicket) and run. Because only a single measure was taken for each combination of factors, no within-condition error term was available, and so the three-way interaction was used as the error term. Looking first at participants' biases, we find significant effects of swing on bias for both participant B.J.C. (F(6,24) = 2.94, p [less than] .05) and participant E.M.S. (F(6,30) = 26.27, p [less than] .001). Figure 2 shows the participants' biases measured as a function of the lateral acceleration of the ball, averaged over all other factors. The biases are expressed as deviations from veridical perception (see Methods section). There was also a significant effect of stump for B.J.C. (F(1,24) = 133.6, p [less than] .005) and E.M.S. (F(1,30) = 52.9, p [less than] .001). Figure 3 shows participants' biases as a function of stump, averaged over all other factors.

Both participants also showed a significant effect of run upon bias (B.J.C.: F(4,24) = 6.88, p [less than] .005; E.M.S.: F(5,30) = 5.86, p [less than] .005), but this result is hard to interpret: a reduction in bias may be manifest as either a reduction or an increase in the numerical value of the bias depending upon what the sign of the initial bias was. In order to see whether bias was being reduced by practice, an analysis of variance was performed using unsigned biases as the performance measure. If participants are successfully using the feedback provided to alter their behaviour in such a way as to bring all their biases closer to zero, then the unsigned bias should show a decline with practice. No significant (p [less than] .05) effects of run upon unsigned bias were found. In support of this finding, there was no significant interaction between swing and run in the original analysis of signed bias.

The data for the participants' random errors are less consistent. Only E.M.S. showed significant main effects of the experimental manipulations. He showed main effects of stump (F(1,30) = 19.5, p [less than] .001) and run (F(5,30) = 33.38, p [less than] .001), and interactions between swing and run (F(30,30) = 2.42, p [less than] .01). Figure 4 shows the participants' random errors (standard deviation of their internal error distributions) averaged over value of swing, as a function of the run number, plotted separately for left and right stump judgments. E.M.S. shows a clear effect of learning, with the random error declining by a factor of 2 approximately, and a suggestion that there is further improvement to come. The graphs suggest that the improvement is greatest for the left stump judgments (B.J.C. actually appears to get worse for the right stump judgments) and the interaction between stump and run is indeed significant for both (E.M.S.: F(5,30) = 4.7, p [less than] .005; B.J.C.: F(4,24) = 3.29, p [less than] .05). The effect of stump found for E.M.S. is rather small.

The distribution of stimuli presented to B.J.C. in this experiment is shown in Fig. 5. The value of the cue is expressed as the x coordinate of the ball as it passes the plane of the wicket. Balls grazing the off stump and leg stump have x coordinates of -0.15 and 0.15 m, respectively. It can be seen from the distribution that the adaptive psychophysical method caused the stimuli to be clustered about these two regions of doubt. Nearly three-quarters of the stimuli were within 1 standard deviation (of the participant's error distribution) of the stump-grazing cases, and almost all were within 3 standard deviations. In other words, the participant was rarely given easy decisions to make. (The peaks at [+ or -]0.31 m and [+ or -]0.01 m correspond to the large cue values with which each run began.) The distribution of stimuli presented to E.M.S. is rather broader (because he showed larger random errors) but is otherwise very similar and is not plotted.

Discussion

At the time of completion of the experiment, E.M.S. had performed approximately 480 trials for each condition of the experiment and a total of over 6500 trials altogether, all with feedback. Nevertheless, he still showed biases depending upon the swing on the ball and the stump at which the ball was aimed. B.J.C., with perhaps three times as much experience of the experiment from pilot runs, showed a much weaker effect of swing on bias, but an equally strong relative bias between leg- and off-stump judgments (in effect, both were behaving as if the wicket was about 1 cm wider than it really was). We cannot conclude that the difference in their biases was due to different amounts of practice, because the data provide no evidence that they were using the feedback to reduce their biases: there were no main effects of run upon unsigned bias and none of the interactions of swing or stump with run even approached significance (p [greater than] .25). The exact pattern of bias clearly changes from participant to participant, but it seems that there are biases in participants' LBW judgments which are resistant to elimination by feedback. Although they are statistically significant, these biases are small in physical terms and in relation to the participants' random error. The practical consequences of the size of the biases will be considered in the general discussion.

A second effect of feedback could be to reduce participants' random errors. The data for E.M.S. show a very strong effect of practice upon random error. In view of this apparent practice effect, the absence of a learning effect for B.J.C. may simply be due to large amounts of previous practice.

Both participants showed significant effects of swing on bias, and for both, the bias became more positive as the swing on the ball became more positive, although the relationship was weak for B.J.C. The meaning of this relationship is most easily seen in the data of E.M.S. Where there is no swing, his judgments are very nearly unbiased. When the swing is positive, and the ball curves to the right, his point of indecision is a little to the right of where it should be. It is as if he fails to take complete account of the curvature in the hall's path. A similar pattern is observable for B.J.C., though the compensation for swing is much more complete, and the relation of swing to bias is superposed on an overall tendency for his points of indecision to be to the left of where they should be. One can speculate about what sort of decision rule E.M.S. would have to be using to explain the pattern of bias observed. For example, he might have used a rule-of-thumb based only upon the position of the ball at the instant of interception by the pad, or a rule based upon the position and the direction of travel of the ball at interception. Simple calculations show that he was using neither of these rules: the first predicts a bias twice that observed, and the second a bias much smaller than that observed.

EXPERIMENT 2

Skilled bowlers will vary the point in space at which they release the ball. The height at which the ball is released is more or less fixed by the bowler's personal dimensions, and although slow bowlers do occasionally bowl the ball from further than the regulation distance in order to upset the timing of the batsman's stroke, the main variation used is in the lateral (x) position of the point of release. This experiment had the same design as Expt 1, except that the swing on the ball was zero throughout, and the x coordinate of the release point was varied instead, taking values of -1.8, -1.5, -1.2, -0.9, -0.6, -0.3 and 0.0 m, covering the entire range of positions available to a bowler bowling 'over the wicket' (i.e. to the left of the wicket at the umpire's end).

Results

As in Expt 1, a three-way analysis of variance was done, using release point, stump and run as factors. The general pattern of results is very similar to that of Expt 1. Considering signed biases first, B.J.C. and E.M.S. both showed significant main effects of release point (F(6,24) = 18.0 and F(6,30) = 9.4, respectively, p [less than] .001 for both) and stump (F(1,24) = 44.5 and F(1,30) = 106.9, respectively, p [less than] .001 for both). B.J.C. showed a significant interaction between release point and stump (F(6,24) = 5.09, p [less than] .005), but this interaction appears to be due to a single aberrant data point. Figure 6 shows the biases for each participant, plotted as a function of release point and averaged over all other factors. Figure 7 shows the biases as a function of stump, collapsed over all other factors. The biases are calculated as deviations from veridical perception. E.M.S. showed a significant effect of run (F(5,30) = 4.54, p [less than] .005), and B.J.C. a significant interaction between stump and run (F(4,24) = 4.99, p [less than] .01). As in Expt 1, these changes in bias are hard to interpret and consequently an analysis of variance was performed upon the unsigned biases, to see whether the effect of practice upon bias constituted an improvement. This analysis showed no significant effects of run.

As with Expt 1, the data for the participants' random errors are less consistent. The most notable finding is the effect of run upon random error, which is significant for both participants (B.J.C.: F(4,24) = 3.6, p [less than] .05; E.M.S.: F(5,30) = 3.12, p [less than] .05). Fig. 8 plots random error as a function of run, for both participants. B.J.C. also showed a significant effect of release point (F(6,24) = 6.17, p [less than] .005); his random errors for left-stump judgments declined steadily by a factor of 2 as the release point changed from the most extreme to the most central position (nearest the line joining the wickets).

The distributions of stimuli presented to the participants were very similar to that shown in Fig. 5 and are not plotted.

Discussion

The results of Expt 2 are similar in form to those of Expt 1. Participants show biases related to the release point of the ball and to the side of the wicket on which they are making the decision. The absolute magnitude of these biases does not appear to be reduced by practice in the course of the experiment. Both participants showed a reduction in their random errors with practice (compared to one participant in Expt 1).

The relationship between bias and release point shown by the participants is consistent but puzzling. If they were making their decisions simply on the basis of, for example, where in relation to the wicket the ball struck the pad, without regard to the direction in which the ball was travelling at the time, then we would expect their biases to follow a monotonic trend as a function of release point. Instead, they both show a U-shaped dependence, with the maximum leftward bias being observed near the middle of the range of release points. A possible explanation is that the participants suffer from two biases that oppose one another. For example, they may systematically misperceive the release point of the ball, and quite independently apply a systematically incorrect compensation for the perceived release point. If these two biases operate in opposite directions as release point changes, the U-shaped dependence of bias upon release point could result. The much greater similarity between the participants in this experiment compared to the last one may be because participant 1 was not more practised at this particular task.

As in Expt 1, participants behaved as if the wicket were about 1 cm wider than it really is.

GENERAL DISCUSSION

Two main conclusions can be drawn from the results of these experiments. (1) Participants appear to suffer from biases in their judgments, caused by variations in the swing on the ball, the point of release and the side of the wicket towards which it travels. These biases appear to be resistant to elimination by feedback. (2) Participants are able to use feedback to reduce the random errors in their judgments.

Biases

The biases exhibited by the participants are rather small: typically the systematic error is 1 cm or less, and is never more than 1.5 cm. From the umpire's viewpoint, 1 cm is equivalent to a visual angle of about 1.5 arc minutes; it is a quarter the thickness of a stump and one-seventh the diameter of the ball. It is also (as shall be discussed at more length below) only one-third of the size of the participant's random errors. This performance is achieved in a situation where the participant never actually gets to see the physical state of affairs about which he is to make a judgment.

Small as the biases are, they are certainly present, and it is not clear why they should be so resistant to elimination. One possibility is that participants are unable to extract the relevant parameters of the flight of the ball and therefore cannot take account of them when extrapolating the ball's trajectory. Some progress towards testing this explanation could be made by explicitly measuring their sensitivity to the various parameters of a ball's flight, for example, asking them to decide whether a given delivery was swinging further left or further right than the previous one. A suitably low sensitivity to swing in a participant would explain why swing causes bias in that participant. The relative bias observed between off- and leg-stump judgments might be similarly explained because the ball necessarily approaches the two sides of the wicket at different angles. Insensitivity to changes in the angle of approach would account for the bias. This class of explanation does not easily account for the U-shaped dependence of bias upon point of release of the ball. If the participant were merely taking no account of the variation in release point, we would expect a monotonic relationship between release point and bias. That we do not get such a relationship suggests that perhaps the participant's estimate of release point does not vary monotonically with actual release point, or that, as suggested earlier, the compensation applied by him for the perceived release point may be inappropriate. The changing balance of the two biases could cause the U-shaped function observed.

Whether the biases observed have significant effects on the overall performance of umpires is discussed in the next section.

Random errors

The absolute values of the errors made by the participants are of practical interest, but should be treated with a little caution. The stimuli were created in such a way as to include all the major cues available to a real umpire, but were certainly not identical to the real thing. Therefore we might expect that, although we can detect the effects of changing stimulus parameters upon certain performance variables with some confidence, our measures of the absolute values of those performance variables are less certain. Participants' random errors (standard deviations of their internal error distributions) are of the order of 3 cm in the plane of the wicket, which is almost half the width of the ball, and, for an umpire standing 2 m behind the wicket at the other end, is about 5 arc minutes of visual angle.

In a real situation, the umpire's task is more demanding, for the decision about whether the ball would have hit the wicket also includes a decision (not required of the participants in this study) about whether the ball would have passed over the top of the wicket. In addition, the complete LBW decision involves other aspects of the ball's journey down the pitch: the umpire must make judgments about where (with respect to the lines of leg stump and off stump) the ball struck the ground and where it struck the batsman's leg. These multiple judgments could interact in the sense that the requirements to make all of them could increase the amount of error in each individual judgment. There is evidence that such interaction occurs: experiments conducted but not reported in detail here show an increase in random error of about 25 per cent in a task where the participant had to make additional judgments on every trial about where the ball pitched and where it struck the batsman's leg. Finally, there are attentional factors to consider. In these experiments, the participants knew that they had to make an LBW decision on every trial, whereas in a real situation an umpire is required to make an LBW decision only rarely and unexpectedly, which may make the task harder.

A further complication arises when we consider the effects of bias and random error. Data for different values of the dependent variables of swing and release point were analysed separately. This approach allowed us to isolate the effects of each of these variables on each participant's bias, and it was found that they do indeed cause bias in the participants' judgments. Now in a real situation, the umpire has no control over the values of these variables: they are effectively random. The umpire therefore suffers an extra source of random variation in his or her judgments caused by his or her inability to detect and/or accurately compensate for random variations in the swing and release point. If we consider all the umpire's judgments together, the biases caused by these variables will now contribute to the umpire's overall random error. It follows that the umpire's random error as measured in a realistic situation is likely to be larger than random errors measured for subsets of the data where the characteristics of the delivery are held constant within each subset.

The importance of this effect depends upon the relative sizes of the participant's intrinsic random error and the range of biases experienced by him. To estimate the effective increase in random error caused by uncontrolled bias, the data for Expt 2 (where both participants suffered strong biases) were reanalysed. New psychometric functions were assembled, combining data from all trials in a given run regardless of the value of release point, and random errors were extracted from these functions. After collapsing over run by taking the root mean square (RMS) of the data for each value of run, there were four such overall random errors, being the left and right stump judgments for each of the two participants. These overall random errors were compared to the RMS of the corresponding random errors obtained from the original analysis, where separate random errors were calculated for each value of release point.

The ratio of the overall random errors to the RMS original random errors ranged from 1.06 to 1.18, which does not represent a very dramatic increase in random error. (This result can be appreciated intuitively by comparing the range of biases observed with the sizes of the participants' random errors (compare [ILLUSTRATION FOR FIGURES 2 AND 4 OMITTED], and [ILLUSTRATION FOR FIGURES 6 AND 8 OMITTED]): the range of biases is generally small compared to the participants' random errors, and therefore random errors account for most of the variation in their performance.) In addition, the range of release points used in the experiment, while representing the range legally available to the bowler, is unlikely to reflect the range actually used by a bowler, or even a group of bowlers, during a match. Consequently, the analysis just referred to is very likely to have overestimated the effect of bias upon a participant's overall random error, and we can conclude that uncontrolled bias does not make a large contribution to the random error of umpires.

Practical consequences

So far, we have observed behaviour by participants in an experiment, and summarized that behaviour in terms of the mean and standard deviation of a hypothetical internal error distribution. Cricket enthusiasts will now want the process to be reversed: given what we know about our participants' error distributions, what can we conclude about the accuracy of umpires' decisions on the field? It is impossible to give a general answer to the question 'what percentage of LBW decisions will be correct ?', because the answer depends upon the circumstances. For example, if an umpire was given an unvarying diet of 'plumb' deliveries, i.e. balls heading directly for the middle stump, the data show that they would almost never fail to give the batsman out, but if all of the decisions concerned balls directed towards the edges of the wicket, they would of course be wrong much more often. However (with the earlier caveat about the absolute values of the random error in mind), we can examine some specific cases. The requirement that an umpire give the benefit of any doubt to the batsman adds an extra layer of complication, and we ignore it for the time being.

Assuming that umpires are perceptually unbiased, how does the correctness of their judgments vary with the trajectory of the ball? The line marked with solid circles in Fig. 9 shows the percentage of correct judgments made for an umpire for whom the SD of the random error is 3 cm, plotted as a function of the point at which the ball passes the plane of the wicket in the region of the off stump. The same diagram, reflected, would apply to judgments concerning the leg stump. Marked on the graph are the positions of the ball for which it would be clear of the stump by its (the ball's) own diameter, just grazing the stump, just occluding the stump, and just entirely within the bounds of the wicket. It can be seen that any delivery which causes the ball to pass more than about half its diameter from just-grazing will give rise to a correct judgment 90 per cent of the time.

The line marked with open circles in Fig. 9 shows the percentage of correct judgments made by an umpire with the same random error but with a bias of -0.01 m. This umpire behaves as if the edge of the wicket is 1 cm further left than it really is. Thus, the percentage of judgments that the ball would have hit the wicket starts to rise above 50 per cent while the ball is still 1 cm from the off stump, and reaches 63 per cent when the ball is almost but not quite grazing the stump. However, as the ball is not actually going to hit the wicket in these cases, the percentage of correct responses declines below 50 per cent correct to 37 per cent correct in the not-quite-grazing case. As the ball is positioned further right, the participant's percentage of 'hit' judgments continues to rise smoothly, but these judgments abruptly become correct, because the ball really is now hitting the wicket. Thus we see below-chance performance in some cases, and a discontinuity in the function corresponding to deliveries grazing the stump.

Taking a broader view, the effect of a bias of -0.01 m is to change the percentage of correct responses by at most a quarter, a figure that declines as the judgments become easier. (The percentage of incorrect responses shows much larger fractional changes, but only where the error rate is small anyway.) In this case, with a leftward bias and considering off-stump judgments only, the changes in error rate caused by the bias are all in favour of the bowler. However, if the bowler is contemplating trying to swing the ball leftwards to deceive the umpire in this way (as [ILLUSTRATION FOR FIGURE 2 OMITTED] suggests is possible), he or she should remember that what is gained on the off stump will be lost on the leg stump, for there (if the umpire's bias remains the same) the judgments will change to favour the batsman. Nevertheless, as more deliveries are, on the whole, directed in the region of the off stump than are directed at the leg stump, it appears that a bowler could gain a modest advantage by bowling outswingers (deliveries that curve to the left). The data concerning the effects of release point on bias in Fig. 6 suggest that similar deceptive possibilities are available to the bowler by varying the release point. However, we should not take this idea of bowling to deceive the umpire very seriously: LBW decisions are fairly uncommon, and, as there are other ways to dismiss a batsman, there is much more to be gained by trying to deceive the batsman than by trying to take occasional advantage of the umpire's perceptual limitations.

Decision strategy

Does the particular value of the random error have any consequences for the way an umpire goes about his job? Obviously, the larger an umpire's random error, the less confidence we (and they) will have in the correctness of their decisions. The usual practice in cricket is that the umpire should rule in favour of the batsman where doubt exists. Obeying this constraint means that the umpire will inevitably reprieve the batsman on some occasions when the ball was really going to hit the wicket. The next few paragraphs explore the nature of the compromise that has to be reached by the umpire.

The laws of cricket offer no guidance to the umpire about the principles he or she should apply in order to give the batsman the benefit of any doubt. Indeed, the general principle of erring in favour of the batsman, although universally accepted, is nowhere explicitly stated in the laws. Umpires are therefore free to interpret the principle as they wish. Let us suppose that umpires act like psychologists, and are prepared to tolerate no more than a 5 per cent probability of giving the batsman out when in fact the conditions for an LBW dismissal have not been physically met. In a situation where all the other conditions for an LBW dismissal have been met beyond doubt, the 5 per cent criterion applies to the judgment about whether the ball was going to hit the wicket. The case in which the umpire is most likely to make such a 'false alarm' judgment is where the ball just fails to graze one of the outer stumps, so we assume that the umpire must act to keep the false-alarm rate for this case below 5 per cent. (How the umpire might monitor the false-alarm rate is another matter.)

Suppose also that the umpire makes an LBW decision as follows. On each delivery the umpire generates an estimate of where the ball would have passed the stumps. There is a critical rectangular region of the wicket, extending the height of the wicket, and placed centrally with respect to the wicket. If the umpire's estimate of where the ball would have passed the plane of the wicket falls within this region, the batsman will be given out LBW; if the umpire's estimate is that the ball would have struck the wicket outside this region, the batsman is reprieved. If the critical region is physically appropriate, i.e. the same width as the wicket plus the ball's radius, then the false-alarm rate for the just-not-grazing case will be an unacceptable 50 per cent, because half the time the umpire will estimate that the ball would have struck the wicket. To keep the false-alarm rate acceptable, the umpire must operate a critical region that is narrower than physically appropriate. Specifically, the umpire must use a critical region such that, for the just-not-grazing case, there is a probability less than .05 of the estimated position of the ball falling within the critical region. If the umpire's (Gaussian) error distribution is of standard deviation 3 cm, it turns out that the critical region must be narrowed by 5 cm on both sides of the wicket. This value means that the umpire will give the batsman out (all other conditions being fulfilled) if his estimate of the ball's position as it passes the plane of the wicket is such that more than two-thirds of the ball overlaps the wicket. As the umpire's random error rises, the critical region must get narrower to keep the false-alarm rate acceptably small.

Of course, making the critical region narrower will reduce the probability of the umpire correctly giving the batsman out on occasions when the ball really was heading for the wicket. Knowing the umpire's random error, we can work out the probability with which the umpire will estimate the ball to be going to pass within the critical region (the 'hit rate'), for any physical position of the ball as it passes the plane of the wicket. The easiest case as far as the umpire is concerned is where the ball is, in reality, heading for the exact centre of the wicket; the hit rate will decline for balls heading towards the edges of the wicket. If we assume that the ball is equally likely to pass any part of the wicket, we can average these hit rates to produce an overall hit rate. Figure 10 plots overall hit rate calculated as a function of the umpire's random error, where the size of the critical region is adjusted in each case to make the false-alarm rate for the just-not-grazing case equal to 5 per cent. For realistic values of the random error, the relationship between hit rate and random error is very nearly linear, the hit rate falling by roughly 10 per cent per centimetre of the umpire's standard deviation. For the participants in this experiment, the hit rate is about 70 per cent. Note that an umpire whose random error was greater than about 10 cm would hardly ever be able to give a batsman out, given the constraint on the false-alarm rate.

In conclusion, these experiments are a demonstration of how standard psychophysical techniques can be applied without alteration to a real situation. Officials in other sports (e.g. tennis) make decisions which are similar in type to those made by cricket umpires, and the methods and measures used here could in principle be transferred to these other situations. Although a simulated stimulus was used in these experiments, the use of psychophysical methods does not preclude the use of real stimuli where a sufficient degree of control and/or physical measurement can be attained.

Participants exhibited biases and (inevitably) random errors in their judgments. In these experiments it was found that the random errors are more important than the biases in determining overall error rates: the biases are small compared to the random errors. It was found that random error can be reduced by practice with feedback.

It is in the nature of experimental psychology to draw attention to and to measure the limitations of human performance, and this study is no exception. In some quarters, such knowledge of the limits of human LBW judgments may be taken as an argument to augment or replace umpires with technology. The logic of this argument is not uncontentious, at least as far as umpires' random errors are concerned. Cricket is a game of many random variables: a shower of rain or an irregularity in the pitch can determine the outcome of a match. Knowledge of the umpire's limitations is no reason why we should stop accepting the stochastic nature of LBW decisions in the same way that we happily accept the other stochastic events that affect the course of a game.

Acknowledgements

I am grateful to V. Bruce, R. Campbell, P. Hancock and H. Hill for their comments on drafts of this manuscript, to R. Macdonald for statistical advice, and to E. McSorley for acting as a participant.

References

Barton, N. B. (1982). On the swing of cricket balls in flight. Proceedings of the Royal Society of London, Series B, 379, 109-131.

Daish, C. B. (1972). The Physics of Ball Games. London: English Universities Press.

Finney, D. J. (1971). Probit Analysis. Cambridge: Cambridge University Press.

Marylebone Cricket Club (1992). The Laws of Cricket (1980 code), 2nd ed. London: MCC.

Watt, R. J. & Andrews, D. P. (1981). APE: Adaptive Probit Estimation of psychometric functions. Current Psychological Reviews, 1, 205-214.

Appendix: Details of stimuli

Dimensions of a cricket pitch

The following dimensions are dictated by the laws of cricket. Each wicket consists of three vertical wooden sticks ('stumps') placed in a straight line in the ground, forming a rectangular target 0.71 m high and 0.22 m wide. The wickets are placed with their planes parallel and 20.12 m apart; the line joining the middle stump of each wicket is at right angles to the planes of the wickets. In front of each wicket (i.e. in the direction of the other wicket) a line (the popping crease) is drawn parallel to the wicket at a distance of 1.22 m. The bowler must have at least part of his front foot behind this line at the moment that the ball is released. On each side of the wicket, 1.33 m from the centre of the wicket, a line is drawn at right angles to and starting from the popping crease, to an undefined distance beyond the plane of the nearest wicket. The bowler must have at least part of his front foot within these two lines at the moment that the ball is released. A fourth line is drawn parallel to the popping crease and through the base of the wicket. These lines are drawn at both ends of the pitch (because bowlers change ends every six deliveries) but have no significance in the leg-before-wicket (LBW) law. However, they form part of the natural stimulus for LBW decisions and were therefore included in the experimental stimulus.

The following dimensions were used for the remaining parts of the stimulus. They are not specified by the laws of cricket but are typical. The wickets are placed upon an area of rolled turf 32 m by 6 m (represented by the texture described in the main text). The remainder of the playing area extends to a distance of 80 m from the nearest wicket. The grass is typically mown in a striped pattern which provides a powerful cue to depth. This pattern was represented in the stimulus by alternate light and dark grey stripes 4 m apart and parallel to the axis of the pitch. The sky was represented by a lighter shade of grey.

Physical model

The following physical model of the flight and bounce of the ball was obtained from Daish (1972) as follows. The mass of the ball was 0.16 kg and its radius r 0.036 m. At the velocities used in the experiment, the drag coefficient [C.sub.p] would be 0.45. The density p of the air was taken to be 1.22 kg [m.sup.-3]. The drag force F on a ball travelling at velocity v is then given by

F = 0.5 [C.sub.D][Rho][v.sup.2] A

where A is the cross-sectional area of the ball.

As mentioned in the main text, the' swing' of the cricket ball was modelled simply as a constant lateral acceleration. Much is often made of the phenomenon of 'late swing', where the ball appears to deviate sharply towards the end of its flight. To explain late swing, we have to understand why cricket balls swing in the first place. To make a ball swing, the bowler bowls the ball with the plane of the equatorial seam vertical, but rotated about a vertical axis so it makes an angle of 10-30 degrees with the long axis of the pitch. It is generally accepted that this practice causes the airflow to be laminar on one side of the ball, and turbulent on the other, with a consequent pressure difference that causes a lateral acceleration of the ball. Now if the ball is bowled above a certain critical velocity (30-40 m [s.sup.-1]), turbulent flow occurs on both sides of the ball, and the ball does not swing. Therefore, if a bowler can deliver the ball just above the critical velocity, the ball will not swing until, some distance down the pitch, its velocity has decreased enough for laminar flow to be re-established on one side. Thus a very fast bowler achieves late swing. However, the bowler has a difficult task, for above the critical velocity, the aerodynamic drag on the ball is greatly reduced, the ball losing only about 3 per cent of its speed on the way down the pitch. The bowler must therefore achieve great precision in the release velocity of the ball in order to make it start above the critical velocity but fall below that velocity before it reaches the batsman (Barton, 1982). Given the demanding nature of the conditions required to produce late swing, and the observation that late swing is also attributed to bowlers of lesser pace, we must consider a possible perceptual explanation, which is that even if the sideways acceleration of the ball is constant, the parabolic sideways deviation that it induces means that almost half of the lateral deviation will occur in the last quarter of the flight (Barton, 1982; Daish, 1972). In recent years the phenomenon of 'reverse swing', where the ball deviates in a direction opposite to that expected on the basis of the bowler's action, has been employed to devastating effect particularly by some Pakistani bowlers. Neither late swing nor reverse swing was included in the physical model.

When the ball strikes the pitch, the surface of the pitch deforms elastically (and therefore temporarily) to conform to the shape of the part of the ball that is in contact with it. The ball (which also deforms) therefore makes a significant saucer-shaped deformation of the pitch. Because the ball is rebounding not off a level surface but off the inclined face of the depression, it bounces higher and loses more speed than might be expected. Thus the effective coefficient of restitution is around .55, compared to the value of about .3 that is obtained for a ball dropped vertically, and the effective coefficient of friction is around .53, compared to the value of about .4 if the deformation of the pitch is not taken into account.

The coefficient of restitution between the ball and the pad was taken as .1, and the coefficient of friction as .5. Neither of these figures had any empirical basis other than that they were chosen by trial and error to make the collision between ball and pad look approximately realistic.

Calculation of the trajectory of the ball

The path of the ball was calculated numerically by considering sections of the flight 1/72 seconds long. For each section, the forces acting on the ball were calculated, and used to calculate the acceleration on the ball in three dimensions. These accelerations were used to adjust the three velocity components of the ball, and hence to adjust the three position components of the ball. When they component of position fell below zero for the first time (i.e. when the latest section to be calculated included the bounce of the ball), linear interpolation was used to calculate the x, y, z components of velocity and the x and z components of position at the instant of bouncing, that is, when they value of position was equal to the radius of the ball.

It is important that one sample correspond exactly to the bounce of the ball, because otherwise the bounce in the final animation does not appear sharp. To achieve this situation, a sequence of samples, all separated by 1/72 second and including the sample representing the bounce, was then calculated by linear interpolation between the existing calculated samples. The path of a cricket ball was very nearly straight, and simulations at higher sampling rates suggested that the change in final calculated position caused by linearly interpolating between samples at 1/72 second is of the order of a few millimetres. Given the physical approximations involved in calculating the trajectory, and the variations in the flight of real cricket balls caused by changes in the conditions of the ball (which wears and roughens a great deal during a day's play), we can conclude that the use of the interpolation procedure does not make the calculated trajectories less realistic or representative.

The calculation of the trajectory of the ball then proceeded to the collision with the pad, calculating samples at 1/72 intervals starting with the bounce. The position of the pad was adjusted in the x and directions away from its nominal position so that the collision of ball and pad coincided with one of the samples of the ball's flight. The effect of the collision with the pad on the bali's motion was calculated, and the calculation of trajectory continued as the ball flew off the pad, bounced a few times on the ground, and came to rest. This final phase of the ball's flight was included simply to make the simulation appear more realistic, and no special attention was paid to the sampling phase of the bounces.

On each trial the experimenter needs to direct the ball at a specified part of the wicket. The direction of the ball was controlled by varying the x component of the initial velocity. Because the ball is travelling almost perpendicular to the x axis, adjustments of the initial x velocity of the ball have a negligible effect on its overall speed. In the most simple case, the ball travels in a vertical plane, and the required x component of velocity can be calculated precisely.

When the ball suffers a sideways acceleration during flight (i.e. when the ball 'swings ') it no longer moves in a vertical plane, and an approximate calculation was made. If air resistance and the effects of the bounce on speed are neglected, the time taken for the ball to reach the far wicket can be calculated from the initial z component of velocity. The lateral deflection caused by the swing in this time can then be easily calculated, and used to adjust the calculation of initial x velocity. The subsequent detailed calculation of the trajectory reveals exactly where the ball would have passed the wicket, and this true value of the bali's position is used in the data analysis. In other words, although we cannot send the ball to exactly the place we want, we can determine exactly where it did end up going. Thus, any effects of the approximate calculation are restricted to a possible very slight reduction in the efficiency of the experiment, and in practice the intended and exact destinations of the ball were consistent within a small number of millimetres.

To provide accurate information about what the x and y coordinates of the ball on passing the wicket would have been, had the ball not been intercepted by the pad, the trajectory calculation was repeated with no pad present. The calculations were continued up to the point where the two most recent sample positions straddled the plane of the wicket. Linear interpolation was then used to calculate the x and y coordinates of the ball as it passed the plane of the wicket.

Drawing the stimuli. The stimuli were drawn as perspective projections onto the vertical monitor face. The umpire was assumed to have eyes 1.6 m above the ground, and to stand 2.0 m behind the nearest wicket. Different umpires choose to stand in different positions, but this position is fairly typical. The monitor face was positioned 50 cm from the viewing position with its centre 6 cm below eye level. The x, y and z coordinates of the ball at each sampling position were transformed into x andy coordinates on the monitor face, and appropriately sized images of balls were created in memory, and stored along with the x and y display coordinates where they were to appear. In order that features of the stimulus could be drawn at positions arbitrarily related to the pixel array, everything that was drawn was anti-aliased by applying a Gaussian blurring function with standard deviation 1.0 pixels.

The display was binocular. The scene before the umpire was depicted twice, the left eye's view being on the left half of the display and the right eye's view on the right half of the display. The two views were drawn on the assumption that the umpire's eyes were 6 cm apart.

The participant's view was that of an umpire looking along a cricket pitch [ILLUSTRATION FOR FIGURE 1 OMITTED]. The stumps and usual ground markings were drawn in white. The outfield was drawn with alternating light and dark grey stripes running parallel to the pitch, and above the far edge of the outfield a very light grey uniform region representing the sky extended to the top of the display.

To complete the impression of a scene in three dimensions, the surface of the pitch was drawn with a visual texture on it. In order for the Gaussian blurring of this texture to be achieved, it was necessary that the brightness value of the texture be rapidly calculable at any arbitrary position on the pitch surface. This constraint required that the texture be a simple function of two position variables. After some trials, a pattern of the product of two orthogonally oriented sine wave gratings was used. The texture was calculated in the coordinate system of the simulated pitch, and then blurred in the coordinate system of the display. The result was a texture gradient which decreased in contrast as the texture elements decreased in size, losing contrast altogether towards the far end of the pitch. A number of colleagues were asked to comment on this texture gradient. All agreed that the texture gradient appeared like a plane surface receding in depth, and none interpreted the variations in brightness on the pitch as being due to undulations in the surface.

The batsman's pad was represented by a simple white rectangle of width 0.15 cm and height 0.6 cm. It appeared at a nominal distance of 1.72 m from the wicket at the batsman's end; this distance was adjusted slightly on each trial as above to maintain suitable sampling phase at the collision. The pad appeared on the display as soon as the ball was less than 5.0 m from it, and disappeared on the frame after the ball had struck it. The reasoning behind the choice of these values is given in the main text.

The animation. The ball's trajectory was calculated in advance of each trial, and the corresponding image fragments stored in memory ready for use. The animation was achieved by displaying these fragments in sequence on the display. The rate at which the fragments were drawn was controlled by drawing one pair of left- and right-eye fragments (after deleting their predecessors) during the vertical retrace of the display. In this way, the samples of the ball were displayed at precisely the 1/72 separation at which they had been calculated, and because image alterations took place during the vertical retrace, no flickering or other degradation of the image occurred. In preliminary trials, a fast photometer and oscilloscope were used to make sure that the computer was capable of updating the display at the required rate.
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Author:Craven, B.J.
Publication:British Journal of Psychology
Date:Nov 1, 1998
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