Tactics and strategy abound in the unfolding of a game of baseball. However, very little of the tactical side of a game is preserved in the customary summaries of completed games such as boxscores. Even the detailed play-by-play data from Retrosheet doesn't capture tactical decisions such as fielder positioning, sacrifice bunts and "hit and run" plays. Since 1955, one particular tactic, purposely issuing a base on balls, the IBB (Intentional Base on Balls or Intentional Walk), to a hitter has been part of the official record. The Retrosheet play-by-play record is complete from 1956 to 2007 encompassing all but the first year of the IBB as an official game statistic. While there are arguable reasons for giving an IBB there is also a cost in the form of an additional base runner. This paper describes a method of analysis and its results for determining the costs and benefits for the IBB and shows just how marginal the IBB is as a defensive tactic, that is its value in preventing a run. Briefly, I give a description of the neural net (statistical) technique used to characterize each plate appearance as being an IBB situation or not, describe long term IBB usage and examine run production following the first IBB situation in an inning focusing on the Slugging Average, SA, of the batter receiving the IBB, examine run production in situations suggesting the IBB, and document run production in innings based on the number of IBBs given and number of IBB situations.

Two categories of bases on balls are referred to throughout this presentation. The intentional kind are referred to by the standard acronym IBB while the Unintentional variety are referred to by the slightly non standard term, UBB. BB by itself is used to refer to all of them. With the exception of a new quantity, Intentional Walks Fraction, IWFR which is introduced and defined in this paper, I believe the other statistical terms and acronyms used are well known.

IBB situations are defined by the way are used by Major League managers and are considered a beneficial tactical option for the defense. The problem with this definition is identifying all IBB situations in the play-by-play record since IBBs are not offered in every IBB like situation. The play-by-play record for MLB documents what happened, not why it or what might have happened. Fortunately, there are a number of statistical techniques that can be used to construct a decision procedure based on actual data. I have chosen one of these, a particular kind of neural network, the back propagation neural net, as the method for making the decision whether a game situation warrants an IBB or not. The critical task deciding if a plate appearance (PA) warrants an IBB, thus is an IBB situation, is accomplished by the neural net evaluating ten parameters expressed numerically that describe the context of the PA in the game. The neural net is trained using the approximately 609000 UBBs and 62000 IBBs presently available in the Retrosheet play-by-play files. Prior to 1974 play-by-play data does not exist for all games and Retrosheet has included an extended boxscore for these missing games. Only BB occuring in the actual play-by-play record can be used in this analysis. The number of games expressed by the extended box score are tabulated in the BSEF column of the "Team Scoring Summary" tables linked from my season stats pages.

A neural net is a well defined numerical computation for converting the ten game parameters (see Table 1 for the parameters employed) describing the context of a PA expressed numerically into a single number that is compared to a decision value. If the neural net calculated value is greater than the decision value the game situation is called an IBB situation. A training procedure using game data establishes the behavior of the neural net. The neural net used to identify IBB situations will be referred to by the acronym NN. Since the NN has been determined by the actual IBB and UBB events it reflects the average practice of managers during the period studied.

For those technically inclined, the neural net used in this project is a standard back propagation net with one layer of hidden units. Ten input units are used corresponding to each parameter in the game situation with each base given a separate input unit (parameter) and eleven hidden units are used. A hidden unit is a weighted sum over the input unit values plus a constant. Thus its computation has the same form and complexity as a linear regression formula evaluation. The result of this sum is passed through an activation function, sigma(x), to generate the hidden unit output. The activation function in this case is sigmoidal, having asymptotic values of +1 and -1 and a slope of 1.0 at x=0. The output unit is the sum of a constant term and the weighted outputs of the hidden units. Subjecting the output unit sum to the activation function completes the neural net calculation. This neural net requires about twelve times the computation of evaluating a linear regression equation for the same input parameters in addition to the twelve activation function evaluations. It contains 133 coefficients which, unlike a linear regression formula, do not have an interpretation in terms of the input parameters. There are no requirements for the linear independence of the parameters or the range of values used. A training procedure using BB data from the play-by-play record determines the values of all the coefficients.

This particular neural net works best when the training set contains approximately equal numbers of IBBs and UBBs. Since the record contains about 10 times as many UBBs as IBBs, a training set uses all the IBBs and an equal number of UBBs selected randomly. Typically, the training procedure is run 100 times and the neural net yielding the highest value of the sum of UBB correct fraction and IBB correct fraction evaluated over the entire record is taken as the neural net, NN. Additional details of the neural net including a simple and easy to visualize example can be seen on this page.

When presented with a particular PA context NN generates a number between +1, most IBB like and -1 which is least IBB like. The decision point (-0.03 for the NN) in declaring a particular PA context UBB or IBB is set midway between the average of NN output values for the training set IBB event and the average of the UBB event outputs, and is essentially zero.

Table 1 lists the ten parameters used to characterize a PA in a particular game situation and indicates their relative importance using the results of individual linear regressions for each parameter (Column R). The SA and BA values are approximately 14 day running averages, not season values. The IBB or UBB value is the dependent variable. In these calculations an IBB is represented as a +1 value while the UBB is given a -1. R is the correlation coefficient. A negative R indicates anticorrelation, that is increasing the value of the variable in question reduces the chance for an IBB. As expected, a runner on first decreases the chance for an IBB as does a higher batting average for the on deck hitter. The square of R indicates the amount of variance "explained" by the variable. The most important indicator, largest R squared, for the IBB is a runner on second base, confirming conventional wisdom. The batter evaluator used is the season slugging average while the on deck batter evaluator is the season batting average. This choice of hitting evaluators leads to the most accurate neural net. Creating a set of 10 neural nets, each with one of the parameters removed, then dividing the error without the parameter by the error using all ten parameters and subtracting 1 leads to a similar estimate for the importance of the individual parameters (column "Drop") as is shown by the linear regressions. Perhaps the relatively low importance of the hitting measures to the NN classification accuracy is surprising.

Table 2 displays the results of NN classification for all BBs and HPs in the 52 season play-by-play record. The results for the two leagues are essentially identical. IBBs are 9.3% of all BBs. The fraction of UBBs in IBB situations, 8.2%, is slightly higher than overall incorrect fraction, 7.8%, perhaps evidence for the "pitching around" phenomena. The fraction of HPs in each NN defined category, IBB and UBB, is very close to the total for all BBs. Repeating the training using other randomly chosen groups of BBs produces essentially the same results. Splitting the data by league or chronologically, training with half the data and evaluating the remainder, produces equivalent results. Adding league and season as inputs to the NN makes a negligible improvement in the classification accuracy thus they are not used in its final version.

Figure 1 displays and Table 3 tabulates several event rates as a function of the NN value. The rates for UBBs, HP (hit by pitch) are relatively constant, independent of the calculated NN value. Slugging Average shows a noticeable rise for the higher NN values. IBB, UBB and HP values have been divided by PA to establish a rate. The actual decision value (-0.03) used in determining an IBB situation is sufficiently close to 0.0 that is is not indicated. The use of a logarithmic scale for the event rates tends to obscure the actual variation in these quantities. The line, PA FRAC, indicates the fraction of the plate appearances, PA, that occur for each NN value range. IBB situations, NN values that are greater than the decision value, total 7% of all PA. The line for IBBs given shows a extremely high correlation with the NN value. The rapid increase in number of IBBs as the neural value increases indicates the NN is capturing the essential decision making employed by managers. The relative constancy of the batter SA indicates that it is not an important factor in the decision to offer the IBB thus is consistent with relative unimportance shown in Table 1. Both UBBs and HPs actually decrease (perhaps not significantly because of the small numbers of these events) at the most IBB like values (positive) NN values suggesting that pitching around and intimidation are not an important substitute for the IBB.

IBB usage is characterized by a new measure, IWFR (Intentional Walks FRaction), defined as the number of IBBs given expressed as a fraction of the number of IBB situations. The meaning of IWFR is made precise by defining it as the number of IBBs received when the NN values for the PAs are computed as IBB like by the NN then expressed as a fraction. Figure 2 indicates the trends in IWFR for the 52 seasons that are the basis for this study and shows a long term trend towards a lower use of the IBB. Following the introduction of the designated hitter rule in 1973, the AL shows a definitely lower use of the IBB than the NL. (The only good thing I've ever been able to say about the DH.)

Figure 3 tabulates the IWFR as a function of batting order position. Again, the abrupt change caused by the AL use of the designated hitter since 1973 is evident. Without the DH the number 8 hitter draws a disproportionate number of IBBs due to the generally weak hitting of pitchers in both leagues.

Now the most crucial question: Does the IBB save runs? I have developed a convincing answer to this question by tabulating runs scored in an inning following the first IBB situation in an inning into two categories. The first is for the IBB not being given, the batter being challenged by the pitcher. The second is for the batter being given the IBB. About 45% of IBBs are given in the first IBB situation of an inning. Two or more IBBs in the same inning are not uncommon. Such cases are not distinguished by the first IBB situation criteria.

To make a comparison as meaningful as possible it is necessary to select situations that only differ by the presence or absence of the particular event being studied, the IBB. The NN identifies all PA where an IBB is an option. I have additionally limited the analysis to the first IBB situation in an inning as a way of minimizing the problems that would occur if the analysis had to deal separately with the many things including additional IBB situations that can follow in such innings. With these restrictions the differences in scoring can be attributed to the the IBB.

The tyranny of the "Law of Small Numbers", the need for an adequate amount of data, must be respected in any statistical study. The accuracy of averages or any other quantity computed from a series of measurements or events is highly dependent on the number of samples. In general, the accuracy of averages increases only as the square root of the number of samples: to achieve twice the accuracy requires four times the number of samples. Comparing the runs/inning in the two categories stated, IBB given or not given, requires comparing two such averages. A standard technique for assessing the significance of differences in these averages is Student's t-test (Numerical Recipes in C, 2nd Ed, pp 616-617). In the arcane language of the statistician, the t-test returns the probability that the null hypotheses is confirmed. That is, the probability that the two averages being compared are drawn from the same distribution of events. Probabilities less than 0.01 to 0.05 indicate lack of confirmation for the null hypothesis: the differences are significant.

Tables 4-9 summarize first IBB situation in an inning comparisons. The columns containing counts, Number of IBB Situations IBB Given (NISG) and Runs in IBB Situations, IBB Given (RISG) are followed by the average (RG/N). The IBB Not given values are indicated by replacing G with N in the column headings. The result of the t-test is given for the comparison of the two R/N columns in the T-TEST column. For the number of IBB situations tabulated for league seasons a difference in the run averages of 0.1 to 0.2 is significant. For the smaller number of events when comparing team season totals differences of 0.5 to 0.6 are needed to insure significance. Overall, the differences between the IBB given and not given categories is significant although several seasons of data must be used to show this clearly.

During the 52 seasons used in this study there have been 11101 player seasons having at least 300 plate appearances (PA). Table 4 tabulates the 11101 player seasons by SA ranges. Column NUMB is the number of player seasons in the SA range. This data is plotted in Figure 4 and provides the strongest evidence that there is a value of batter SA that seems to justify giving the IBB. The IBB not given curve is roughly linear in SA displaying the obvious expectation that a better batter will produce more runs than a weaker one will. The extra base runner due to the IBB clearly provides a greater chance for scoring for all but the very best hitters. There is a drop off in runs scored following an IBB for batters with SA > 0.60 . Even with the quantity of data available in this study, the t-test significance of the drop in run production for the two highest SA ranges after an IBB is only marginally significant.

Table 5 lists the 15 instances of player single season SA >= 0.700, thus its total line is the last line in Table 4. For comparison, Table 6 listing the group of hitters having the 15 highest season values for IWFR of the 11101 shows a higher rate of scoring runs after an IBB than when they are allowed to hit. Only Barry Bonds (2001, 2002, 2003, 2004) appears on both of these lists. Two of the top IWFR list were to hitters batting in the number 8 spot: Tony Pena and Adolfo Philips. Overall, in Table 5 the IBB reduces runs scored although there are exceptions. The t-test for the TOTALS line in Table 6 is 0.84 suggesting that the decrease in runs for this group of player seasons is not significant. The somewhat larger decrease for the TOTALS line in Table 5 has a t-test value of 0.064 suggesting some significance may be attached to this difference. Bonds' season values in Table 7 can be expected to show high variability because of small numbers of IBB related events in a single season. The lack of consistency and judgment in using the IBB is seen in the high IWFR group, Table 6, which shows essentially no difference in runs scored following the IBB compared to allowing the batter to hit. In both Tables 4 and 5 the "Law of Small Numbers" is evident. Individual players do not always follow the average results. Notable, but not statistically significant, deviations in the opposite direction from the averages are evident in both tables.

Table 7 displays the IBB given and not given results for Barry Bonds' career. The average number of runs scored following the first IBB situation is essentially the same for the two categories which is reflected by a t-test of 0.54 suggesting that even for a hitter of his caliber there is insufficient data to demonstrate the efficacy of the IBB applied to him. Mark McGwire's career hitting is similarly inconclusive regarding the IBB.

Table 8 summarizes runs per inning from the first IBB situation with and without the IBB given in ranges of the NN output value for the first PA having an IBB value. Even for the positive NN valued situations the IBB does not, on average, save runs. The two highest ranges, only one showing statistical significance, indicates a slight decrease in runs when the IBB is given compared to similar NN values where the IBB is not given. Overall, there is a statistically significant increase in runs score when an IBB is given.

Table 9 breaks the first IBB situation in an inning data into the 24 base runner-outs combinations. Sub tables include displaying the same data by only outs or base runner configurations. When broken down this way the results are ambiguous. Focusing on cases with a runner on second only representing a little more than half of IBB situations yields essentially no difference between giving and not giving the IBB. While there are significant differences in runs scored for each of the three values of outs, they essentially cancel. Aggregating by outs shows a significant increase, about 0.2 runs for giving the IBB with 1 out but no statistically significant differences for either 0 or 2 outs.

Another way to view the effect of the IBB is to tabulate average runs scored in an inning indexed by the number of IBBs given and the number of IBB situations that occur in it. More than 1 IBB in an inning is not uncommon. There are 2483 cases of 2 IBBs, 35 of 3 and a single case where 4 IBBs were given in an inning. Table 10 displays the average runs per inning as a function of the two quantities. IBBs per inning are given as rows and total IBB situations per inning in columns. As the number of IBB situations increases, the average number of runs goes up. For any number of IBB situations in an inning where there is sufficient data to compute reliable averages the number of runs scored increases with an increase in the number of IBBs. The totals column indicates the average number of runs scored for each value of IBBs in an inning and is independent of the number of IBB situations. If the IBB was a good tactic for reducing scoring the 1 IBB per inning line should show a decrease compared to the no IBB line. Similarly, the column totals represent the average number of runs score for the given number of IBB situations.

In a study of various evaluators for run production a linear regression was performed using 16 different parameters and is documented in Table 2 of the study. That regression, which was performed using team season data for the same 52 seasons as the present one, gives a weight of 0.156 runs per IBB. In the totals ine of Table 4 the difference between the IBB given (RG/N) and not given (RN/N) values is 0.159 runs per IBB. The good agreement of these two different ways of determining the effect of the IBB testifies to the reality of the net increase of runs by the IBB.

The extremes of IBB "respect" are Barry Bond's 2002, 2003 and 2004 seasons. These three are the highest season IBB totals on record. In 2004 he received a total of 232 BB including 120 IBB which are both single season records. His career BB total of 2558 (through the 2007 season) has eclipsed Ruth for first in this category. The other extreme among power hitters is Roger Maris (SA 0.620) who didn't get a single IBB during his home run record setting 1961 season. Of course, he was "protected" by Mickey Mantle who had an even higher SA, 0.687, that year. Maris' 42 career IBB are less than five single season totals for Bonds.

Extreme cases, while seldom proof of anything, are often very interesting. The 4 IBB in a single inning case occurred in the home half of the 8th inning, Kansas City Athletics at the Chicago White Sox October 1, 1965. Going into the bottom half of the inning the score was tied 1-1. Chicago scored 5 runs on 3 hits and the four IBBs winning 6-1. Interestingly, from the perspective of this article, there were only 2 IBB situations computed for this half inning and no UBBs. A play-by-play description is available on the Retrosheet web site (starting with the home page follow the links Regular Season -> 1965 -> Chicago White Sox -> Game Logs -> 10-1-1965 -> Box+PBP).

With the IBB the most extreme way to give one is to intentionally walk a batter with the bases loaded which happened Bonds in the bottom of the 9th in a game against Arizona May 28, 1998. Brent Mayne lined out to right to preserve a one run win for Arizona. The NN value for Bonds' IBB is -0.711, definitely not an IBB situation.

While it is not possible to know what a manager was thinking in a particular situation, using suitable statistical techniques the decision making about offering an IBB can be approximated. IBB situations can be clearly recognized as indicated by the 96% accuracy for the offering of IBBs in neural net defined IBB situations. However, managers don't call for the IBB in every possible situation it could be offered. This component of their decision making has not been modeled.

I have shown, the most incriminating data is in Table 4 and Figure 4, that the IBB creates runs when batter receiving it has season SA less than 0.600 . This statement is true as an average over a large number of events, many leading to unwanted runs by the fielding team. However, the converse is also true. There are many instances of fewer runs following an IBB. Generally, additional walks lead, again on the average, to additional runs. Only about 4% of the IBBs tabulated in the period covered in the study were given to batters during seasons where they has a SA greater than 0.600 . Giving the benefit of statistical doubt to the managers, at least 96% of the IBBs were offered in situations that increase the chances for a run being scored. There is some evidence that the use of the IBB is diminishing (Figure 2) but it is still greatly overused.

In the October 13, 1923 issue of Colliers, Walter Camp wrote: "Ruth, saving baseball with his terrific hitting and by inspiring the Hornsbys, the Walkers and the Williamses to go and do likewise, also made acute a baseball evil, an evil that must be destroyed if the rulers of the game mean to play fair with the fans. I mean, of course, the evil of the intentional pass. The daily patrons of the game have demonstrated that they like long hitting, and it is manifestly unfair to them when a pitcher deliberately passes a man like Ruth when the fans have come to see him hit". Substitute Bonds, A Rod, Howard and Fielder and this quote is equally appropriate for today's game. Giving fans what they want, especially when the IBB is shown to be a poor tactic the majority of times it is employed, is perhaps the best reason to minimize the use of the IBB.

Many different kinds of neural nets have been invented. There is no assurance that the back propagation net is the optimum method for this task. The theory of this particular form of the neural net is covered in “Neural Networks”, Laurene Fausett, Prentice Hall, 1994, and other textbooks.

A preliminary version of this paper won the USA Today Sports Weekly prize for Best Research Poster Presentation at the 2003 Society for American Baseball Research convention in Denver, CO (SABR33).

The author acknowledges his extreme gratitude to the Retrosheet organization. Without the detailed and complete play-by-play record provided by Retrosheet a study of this kind could not be made. He also thanks several sabermetrics colleagues who share his interest in baseball statistics and analysis for reading and commenting on this paper.

Table 1. Game Information Used in NN parameter Drop R R^2 Runner on First 0.1467 -0.3194 0.1020 Runner on Second 0.3597 0.6280 0.3944 Runner on Third 0.2655 0.4394 0.1931 Inning 0.1066 0.2892 0.0836 Outs 0.0300 0.2935 0.0862 Score Difference 0.0778 0.2789 0.0778 AT Bat SA 0.0129 0.0943 0.0089 On Deck BA 0.0067 -0.1306 0.0170 Left-Right Matchup 0.0216 0.1615 0.0261 Batting Order Position 0.0231 0.1266 0.0160

Table 2. IBB/UBB Classification Accuracy by NN UBB IBB UBB+IBB HP as cor incor frac cor incor frac cor incor frac crt BB IBB frac AL 288522 30144 0.9054 25399 809 0.9691 313921 30953 0.9102 0.9373 23727 2242 0.9137 NL 261269 28660 0.9011 35057 1169 0.9677 296326 29829 0.9085 0.9344 22045 2131 0.9119 ALL 549791 58804 0.9034 60456 1978 0.9683 610247 60782 0.9094 0.9358 45772 4373 0.9128

Table 3. 1956-2007 ML Batting Event Rates by IBB Neural Net Value NN RANGE PA /TOT AB SING DOUB TRIP HRUN UBB IBB HBYP BA SA UBB/PA IBB/PA HP/BA -1.00:-0.80 4414173 0.625 3980118 728369 178199 24951 101194 332218 175 29023 0.259 0.393 0.0753 0.0000 0.0066 -0.80:-0.60 1615895 0.229 1466698 267313 70857 9567 44668 124587 292 9194 0.268 0.420 0.0771 0.0002 0.0057 -0.60:-0.40 250749 0.035 219500 37313 9124 1300 4584 21913 290 1737 0.238 0.354 0.0874 0.0012 0.0069 -0.40:-0.20 122709 0.017 104335 18921 4851 823 2159 11870 332 930 0.256 0.381 0.0967 0.0027 0.0076 -0.20: 0.00 129609 0.018 109961 19253 5078 861 2701 13830 841 1014 0.254 0.389 0.1067 0.0065 0.0078 0.00: 0.20 108297 0.015 91008 18187 4910 793 3255 10952 1518 870 0.298 0.477 0.1011 0.0140 0.0080 0.20: 0.40 126876 0.018 104676 19803 5308 843 3178 13246 4044 931 0.278 0.436 0.1044 0.0319 0.0073 0.40: 0.60 130251 0.018 102191 20059 5277 902 3084 13328 9638 971 0.287 0.447 0.1023 0.0740 0.0075 0.60: 0.80 144497 0.020 96845 18493 4948 745 2724 12691 29519 994 0.278 0.429 0.0878 0.2043 0.0069 0.80: 1.00 24169 0.003 13248 2140 526 92 215 1741 8697 119 0.224 0.327 0.0720 0.3598 0.0049 Totals 7067225 6288580 1149851 289078 40877 167762 556376 55346 45783 0.262 0.401 0.0787 0.0078 0.0065

Table 4. 1956 to 2007 ML Runs in First IBB Situations in Inning SA Range TEMP BA SA BB IBB IWFR RISN NISN RN/N RISG NISG RG/N T-TEST <0.40 4870 0.255 0.351 187334 15983 0.087 67959 97989 0.694 6525 8015 0.814 0.0000 0.40:0.45 2765 0.274 0.425 126030 11764 0.099 52728 67992 0.776 5948 6266 0.949 0.0000 0.45:0.50 1942 0.284 0.472 100181 10333 0.110 44198 53646 0.824 5408 5583 0.969 0.0000 0.50:0.55 962 0.294 0.522 56960 6417 0.126 25297 29391 0.861 3820 3656 1.045 0.0000 0.55:0.60 380 0.306 0.572 26211 3644 0.166 11144 12149 0.917 2082 2021 1.030 0.0002 0.60:0.65 135 0.317 0.621 10659 1677 0.204 4405 4445 0.991 1014 980 1.035 0.3419 0.65:0.70 32 0.333 0.673 2977 579 0.269 1068 999 1.069 306 330 0.927 0.0835 >0.70 15 0.347 0.749 1824 474 0.395 435 387 1.124 203 222 0.914 0.0643 11101 0.272 0.422 512176 50871 207234 266998 0.776 25306 27073 0.935 0.0000

Table 5. Hitting in IBB Situations, SA >= 0.700 RANK PLAYER TM YR AB BA SA HR BB IBB IWFR RISN NISN RN/N RISG NISG RG/N T-TEST 1 Barry Bonds SFN 01 476 0.328 0.863 73 177 35 0.479 23 24 0.958 18 18 1.000 0.9098 2 Barry Bonds SFN 04 373 0.362 0.812 45 232 120 0.763 16 14 1.143 32 39 0.821 0.4383 3 Barry Bonds SFN 02 403 0.370 0.799 46 198 68 0.724 10 15 0.667 16 33 0.485 0.5321 4 Mark McGwire SLN 98 509 0.299 0.752 70 162 28 0.327 49 31 1.581 8 11 0.727 0.2070 5 Jeff Bagwell HOU 94 400 0.367 0.750 39 65 14 0.162 54 41 1.317 11 8 1.375 0.9205 6 Barry Bonds SFN 03 390 0.341 0.749 45 148 61 0.772 9 12 0.750 31 30 1.033 0.5589 7 Mark McGwire SLN 00 236 0.305 0.746 32 76 12 0.296 6 13 0.462 9 4 2.250 0.0084 8 Sammy Sosa CHN 01 577 0.328 0.737 64 116 37 0.448 31 30 1.033 23 22 1.045 0.9767 9 Ted Williams BOS 57 420 0.388 0.731 38 120 34 0.641 13 12 1.083 18 17 1.059 0.9577 10 Mark McGwire OAK 96 423 0.312 0.730 52 116 16 0.268 33 29 1.138 4 10 0.400 0.1515 11 Frank Thomas CHA 94 399 0.353 0.729 38 109 12 0.244 27 27 1.000 9 11 0.818 0.6393 12 Larry Walker COL 97 568 0.366 0.720 49 78 14 0.213 55 42 1.310 8 8 1.000 0.5627 13 Albert Belle CLE 94 412 0.357 0.714 36 58 9 0.161 35 31 1.129 3 4 0.750 0.6010 14 Larry Walker COL 99 438 0.379 0.710 37 57 8 0.182 37 30 1.233 9 4 2.250 0.2246 15 Mickey Mantle NYA 56 533 0.353 0.705 52 112 6 0.115 37 36 1.028 4 3 1.333 0.5857 Totals 6557 0.347 0.749 716 1824 474 0.395 435 387 1.124 203 222 0.914 0.0643

Table 6. Hitting in IBB Situations: Top 15 by IWFR RANK PLAYER TM YR AB BA SA HR BB IBB IWFR RISN NISN RN/N RISG NISG RG/N T-TEST 1 Barry Bonds SFN 03 390 0.341 0.749 45 148 61 0.772 9 12 0.750 31 30 1.033 0.5589 2 Barry Bonds SFN 04 373 0.362 0.812 45 232 120 0.763 16 14 1.143 32 39 0.821 0.4383 3 Barry Bonds SFN 02 403 0.370 0.799 46 198 68 0.724 10 15 0.667 16 33 0.485 0.5321 4 Albert Pujols SLN 06 535 0.331 0.671 49 92 28 0.647 20 10 2.000 21 17 1.235 0.1653 5 Ted Williams BOS 57 420 0.388 0.731 38 120 34 0.641 13 12 1.083 18 17 1.059 0.9577 6 Willie McCovey SFN 69 491 0.320 0.656 45 121 45 0.632 16 14 1.143 18 24 0.750 0.3644 7 Barry Bonds SFN 07 340 0.276 0.565 28 132 43 0.603 17 18 0.944 15 25 0.600 0.2713 8 Willie Mays NY1 57 585 0.333 0.626 35 77 16 0.560 6 6 1.000 22 10 2.200 0.2373 9 Adolfo Phillips CHN 67 448 0.268 0.458 17 80 29 0.545 6 12 0.500 5 11 0.455 0.8941 10 Gary Sheffield FLO 94 322 0.276 0.584 27 51 11 0.526 8 8 1.000 10 4 2.500 0.0897 11 Willie McCovey SFN 70 495 0.289 0.612 39 137 40 0.522 24 26 0.923 19 29 0.655 0.4062 12 Ichiro Suzuki SEA 02 647 0.321 0.425 8 68 27 0.520 4 15 0.267 12 19 0.632 0.2440 13 Tony Pena SLN 89 424 0.259 0.337 4 35 19 0.514 7 9 0.778 9 11 0.818 0.9463 14 Tony Gwynn SDN 87 589 0.370 0.511 7 82 26 0.510 20 21 0.952 28 22 1.273 0.5482 15 Barry Bonds SFN 93 539 0.336 0.677 46 126 43 0.507 19 19 1.000 27 23 1.174 0.6723 Totals 7001 0.325 0.606 479 1699 610 0.610 195 211 0.924 283 314 0.901 0.8419

Table 7. Barry Bonds Career Hitting RANK PLAYER TM YR AB BA SA HR BB IBB IWFR RISN NISN RN/N RISG NISG RG/N T-TEST 1 Barry Bonds PIT 86 413 0.223 0.416 16 65 2 0.062 10 22 0.455 0 0 0.000 2 Barry Bonds PIT 87 551 0.261 0.492 25 54 3 0.086 9 23 0.391 0 0 0.000 3 Barry Bonds PIT 88 538 0.283 0.491 24 72 14 0.297 17 24 0.708 10 10 1.000 0.4876 4 Barry Bonds PIT 89 580 0.248 0.426 19 93 22 0.375 13 25 0.520 7 11 0.636 0.7453 5 Barry Bonds PIT 90 519 0.301 0.565 33 93 15 0.185 29 35 0.829 7 6 1.167 0.4534 6 Barry Bonds PIT 91 510 0.292 0.514 25 107 25 0.297 40 32 1.250 5 13 0.385 0.0620 7 Barry Bonds PIT 92 473 0.311 0.624 34 127 32 0.368 25 29 0.862 19 18 1.056 0.5742 8 Barry Bonds SFN 93 539 0.336 0.677 46 126 43 0.507 19 19 1.000 27 23 1.174 0.6723 9 Barry Bonds SFN 94 391 0.312 0.647 37 74 18 0.316 32 34 0.941 17 12 1.417 0.3449 10 Barry Bonds SFN 95 506 0.294 0.577 33 120 22 0.346 18 27 0.667 8 12 0.667 1.0000 11 Barry Bonds SFN 96 517 0.308 0.615 42 151 30 0.443 23 28 0.821 20 21 0.952 0.7212 12 Barry Bonds SFN 97 532 0.291 0.585 40 145 34 0.423 30 32 0.938 17 23 0.739 0.5764 13 Barry Bonds SFN 98 552 0.303 0.609 37 130 29 0.312 40 46 0.870 17 17 1.000 0.6954 14 Barry Bonds SFN 99 355 0.262 0.617 34 73 9 0.286 13 15 0.867 9 7 1.286 0.5204 15 Barry Bonds SFN 00 480 0.306 0.688 49 117 22 0.389 25 29 0.862 19 15 1.267 0.2123 16 Barry Bonds SFN 01 476 0.328 0.863 73 177 35 0.479 23 24 0.958 18 18 1.000 0.9098 17 Barry Bonds SFN 02 403 0.370 0.799 46 198 68 0.724 10 15 0.667 16 33 0.485 0.5321 18 Barry Bonds SFN 03 390 0.341 0.749 45 148 61 0.772 9 12 0.750 31 30 1.033 0.5589 19 Barry Bonds SFN 04 373 0.362 0.812 45 232 120 0.763 16 14 1.143 32 39 0.821 0.4383 20 Barry Bonds SFN 05 42 0.286 0.667 5 9 3 0.600 1 1 1.000 0 1 0.000 21 Barry Bonds SFN 06 367 0.270 0.545 26 115 38 0.481 21 21 1.000 15 14 1.071 0.8757 22 Barry Bonds SFN 07 340 0.276 0.565 28 132 43 0.603 17 18 0.944 15 25 0.600 0.2713 Totals 9847 0.298 0.607 762 2558 688 0.426 440 525 0.838 309 348 0.888 0.5416

Table 8. 1956 to 2007 ML Runs after first IBB in Inning by IBB NN Value No IBB IBB Given NN RANGE NFN RFN RFN/N STD NFY RFY RFY/N STD T-TEST -0.20: 0.00 13932 10604 0.761 1.101 108 91 0.843 1.320 0.4444 0.00: 0.20 80156 58566 0.731 1.088 1038 854 0.823 1.317 0.0069 0.20: 0.40 86872 59541 0.685 1.082 2671 2299 0.861 1.306 0.0000 0.40: 0.60 78438 61751 0.787 1.164 5810 5475 0.942 1.387 0.0000 0.60: 0.80 70188 55220 0.787 1.143 16579 15871 0.957 1.400 0.0000 0.80: 1.00 9652 6621 0.686 1.078 5300 4131 0.779 1.227 0.0000 Totals 339238 252303 0.744 1.117 31506 28721 0.912 1.361 0.0000

Table 9. Runs from first IBB situation by Inning State OUTS BASES RISN NISN RN/N RISG NISG RG/N T-TEST 0 --- 0 0 0 1-- 0 0 0 -2- 13274 12408 1.070 504 517 0.975 0.0931 0 12- 53 38 1.395 0 --3 8169 6016 1.358 169 163 1.037 0.0009 0 1-3 2905 1664 1.746 115 131 0.878 0.0000 0 -23 31122 15395 2.022 2447 1165 2.100 0.0912 0 123 0 1 0.000 1 --- 0 0 1 1-- 0 0 1 -2- 51716 68326 0.757 5230 6478 0.807 0.0004 1 12- 0 0 1 --3 22129 22179 0.998 1311 1184 1.107 0.0004 1 1-3 3262 2669 1.222 199 255 0.780 0.0000 1 -23 44360 31863 1.392 12657 8353 1.515 0.0000 1 123 0 1 0.000 2 --- 0 0 2 1-- 0 0 2 -2- 44104 118895 0.371 3209 8484 0.378 0.3973 2 12- 54 270 0.200 2 --3 11675 28067 0.416 412 1153 0.357 0.0114 2 1-3 1502 2985 0.503 45 78 0.577 0.4884 2 -23 17978 28461 0.632 2423 3545 0.683 0.0105 2 123 0 0 tots 252303 339238 0.744 28721 31506 0.912 0.0000 OUTS BASES RISN NISN RN/N RISG NISG RG/N T-TEST all --- 0 0 all 1-- 0 0 all -2- 109094 199629 0.546 8943 15479 0.578 0.0001 all 12- 107 308 0.347 all --3 41973 56262 0.746 1892 2500 0.757 0.5971 all 1-3 7669 7318 1.048 359 464 0.774 0.0000 all -23 93460 75719 1.234 17527 13063 1.342 0.0000 all 123 0 2 0.000 tots 252303 339238 0.744 28721 31506 0.912 0.0000 OUTS BASES RISN NISN RN/N RISG NISG RG/N T-TEST 0 all 55523 35522 1.563 3235 1976 1.637 0.0281 1 all 121467 125038 0.971 19397 16270 1.192 0.0000 2 all 75313 178678 0.422 6089 13260 0.459 0.0000 tots 252303 339238 0.744 28721 31506 0.912 0.0000

Table 10. Average Runs by IBBs and IBB Situations per Inning IBB\IBB Sit 0 1 2 3 4 5 6 7 8 Totals 0: 0.298 1.003 1.324 1.971 2.918 4.039 4.394 7.000 0.459 1: 1.095 1.141 1.533 2.171 2.907 4.118 5.000 8.000 1.412 2: 1.957 1.752 1.854 2.399 3.408 4.786 7.000 6.000 2.176 3: 1.000 3.636 3.875 5.500 8.000 4.114 4: 5.000 5.000 Totals: 0.301 1.019 1.366 2.030 2.939 4.123 4.853 7.400 6.000