Too Many Intentional Bases on Balls?

John F. Jarvis

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