The number of tries required to win in international rugby sevens

3.1Evaluating P[T]

P is the probability of scoring T tries in the competition. Data from Table 1 are used to calculate P[T] for each competition and the average of all three competitions by taking the value in the Total column for try values in each year and dividing by the total number of team performances in that year. The results are shown in Table 2 and plotted in Fig. 1. The distributions are quite similar with, perhaps, a slight shifting to the right (an increased number of higher scoring tries were scored) through time. Data from the 2016–17 competition may have exhibited too few six and seven try performances relative to the other two years. The shape of the distributions suggests a Poisson process may be at work despite the slight over-dispersion seen in the data.

Fig. 1

The proportion of the number of tries scored in each competition. The data are quite consistent year to year with equivalent modal values of 3. The mean value of the number of tries scored and the standard deviation and variance of the number of tries scored both increased slightly through time.

The number of tries scored in a rugby sevens match is an example of “count data.” The distribution of count data may be expected to be described by the Poisson distribution provided that the elements being counted occur independently and with a constant probability of occurrence at any given time (Gelman et al., 2013).

The Poisson distribution is given (in the format of this study) by the following equation (Forbes et al., 2011:p.152):

Researchers in goal scoring sports have sometimes modeled score distributions as a Poisson distribution. Maher (1982) noted that many investigations have found that the distribution of soccer scores from English domestic competitions is described by the Poisson distribution while noting that the distribution may be better described by the closely-related negative binomial distribution. His study, however, shows that the Poisson distribution gives a reasonably good fit to the data with some slight systematic differences. Croucher (2002) concluded that the negative binomial distribution yields a better fit than the Poisson but requires much more data collection and calculation. Greenhough et al. (2002) concluded from worldwide soccer scoring data that neither the Poisson nor the negative binomial distributions described the distribution in the extreme values as the observed score distribution is too “heavy-tailed” (too many high scores). They prefer extremal statistics as a better fit but note that the Poisson or negative binomial distributions are sufficient models for English soccer. Popular articles written for the sports betting industry (see Naoumis, 2019, for instance) often assume that scores are distributed according to the Poisson distribution but with little (if any) rigorous justification.

In ice hockey, Mullett (1977) showed that the distribution of the number of goals scored in the National Hockey League follows a Poisson distribution. In a fairly recent study on try scoring in rugby league, Tonkes (2016) observed that the distribution of tries scored in all 201 matches from the 2015 Australian National Rugby League competition are well represented by a Poisson distribution. Jones (2019:p.215) speculated that, while the total score in rugby may not be distributed as a Poisson due to the differing number of points for different ways of scoring, the number of tries (and other ways of scoring such as penalty goals and dropped goals) could be distributed as a Poisson distribution.

There clearly is some difference of opinion regarding how well the Poisson distribution describes soccer and other goal scoring sports. The deviations from a Poisson distribution that have been observed seem minor and from the standpoint of practicality may not be important. We can test this assumption for the current data and draw conclusions regarding the propriety of using the Poisson distribution to describe try scoring in sevens rugby.

As part of this study we test how well the observed distributions approach a Poisson distribution using a Chi Square Goodness of Fit Test (Sokal & Rohlf, 1969). The result for the 2016–2017 competition is shown as an example calculation in Table 3. In this Table the data for seven tries and greater are pooled together to ensure that there are at least five expected occurrences in each try category (Sokal & Rohlf, 1969:p.568; Healey, 2005:p.295). For six degrees of freedom, the calculated chi square value of 7.570 has a p-value of 0.271. The coefficient of determination (r²) (Healey, 2005:p.404) comparing these two distributions was found to be 0.9890 meaning that the Poisson distribution explains 98.9% of the variance in the data. We conclude that the observed distribution is not significantly different from a Poisson distribution despite the slight over-dispersion.

Figure 2 compares the distribution of try data from the 2016–17 competition with a Poisson distribution with the same mean. Based upon the chi square goodness of fit analysis the differences between the two curves are attributable to sampling and are not significant. The overall agreement between the two distributions is apparent.

Fig. 2

Distribution of the count of the number of tries scored in the pool round of the 2016–2017 competition compared with a Poisson distribution with the same mean (λ= 2.9104). The p-value of the Chi Square Goodness of Fit is 0.271 and the Coefficient of Determination (r²) = 0.9890.

Table 4 provides a summary of the results for the Chi Square Goodness of Fit Test for all three years and the averaged data. As with the 2016–17 data, the distributions of the number of tries scored in the other two competition years do not differ significantly from a Poisson distribution – p-values are all well in excess of 0.05 and the values of r² are all in the vicinity of 0.98 or 0.99. We conclude that we can be confident that the distributions of the number of tries scored in each year of the Sevens World Series we investigated closely follow a Poisson distribution with a mean value (λ) equal to the observed mean for the competition. Also, the distribution of the averaged data is not significantly different from a Poisson. This is to be expected as a result of the Additive Property of the Poisson distribution (Jones, 2019:p.212).

These results suggest that the distribution of the number of tries scored within each competition conform to a Poisson distribution. This result may be disconcerting to some. Panditt (2016) argues that the conformance of any real distribution to a Poisson distribution could be interpreted as evidence that the occurrence of each event is random and not the result of intent or design. Such a conclusion might trouble us – despite the consistency of the Laws under which the game is played; despite the consistent application of these Laws by Referees; despite the time spent by coaches training and developing their teams; despite the decisions and actions of the players – scoring tries in rugby sevens would seem to be as random as Prussian cavalry officers dying from horse kicks or the location of the impacts of V-1 flying bombs during World War Two.

In reality, the apparent randomness is due to our focus on the collective probability of the number of tries scored in sevens rugby. Each individual try has an identifiable set of predisposing factors illuminated by the studies of performance analysis in sevens rugby. Each try is scored in a more deterministic manner that we can observe and possibly predict given the opponents or the status of the game. It is only when we look at all of the matches and all of tries scored within a competition that the random nature appears (Panditt, 2016).

3.2Evaluation of P[W]

We can calculate the probability of winning a match P[W] for each competition directly from the data in Table 1. This probability equals the total number of winning performances divided by the total number of matches (240). In each competition the value of P[W] is less than 0.5 due to the existence of tied scores in the pool rounds. The values of P[W] by competition year and the average are given in Table 5. The number of ties per competition is seen to decrease through time and P[W] increased slightly. The probability of winning was fairly stable equaling between 0.48 and 0.49 in each competition and in the overall average value.

3.3Evaluation of P[T|W]

The likelihood function given by P[T|W] – the probability of scoring T tries given that a team has won – is the third value we compute from the observed data. The likelihood provides all of the information contained in the data regarding the relationship between try scoring by the winning teams (Box & Tiao, 1973:p.10). The distributions of P [T|W] for all three of the competitions are given in Table 6 and shown in Fig. 3. This figure shows that very few winning teams scored fewer than two tries. The most likely number of tries scored by the winning team was 4. The distributions show that the range of tries scored by winning teams still spans the range of zero to nine tries – the same as for the number of tries scored in the competitions as a whole. The distributions of P [T|W] and P[T] have different shapes with different means and variance. The P [T|W] distribution is more peaked than the P[T] distribution and the means and the modes of the distributions have been shifted to the right to higher values. While still resembling a Poisson distribution, detailed analysis of the P [T|W] distributions has revealed that the true form of the observed distributions is better described by the related Gamma distribution. Jones (2019:pp.224-226) illustrates how the Poisson distribution and the Gamma distribution are complementary functions.

Fig. 3

The proportion of the number of tries scored by the winning team in each competition (P[T|W]). Note the failure of any team scoring zero tries to win. In the 2016–2017 competition no team won when scoring one try. The mean value of the number for tries scored by the winning team and the standard deviation and variance of tries scored by the winning team both increase slightly through time. The modal value in each competition was 4.

The equation for the Gamma distribution is given by (Gelman et al., 2013:p.578) as (written in the format of this study):

T = the number of tries scored by the winning teams,

α=x¯2s2 is the “shape factor” where x¯ is the observed mean number of tries scored by the winning team and s² is the variance,

β=s2x¯ is the “scale factor”

and Γ (α) is the Gamma function of the value α (Artin, 1964).

The Gamma function is a complicated expression equal to (in the format of this study):

We note that the Gamma distribution is a continuous distribution while P [T|W] is a discrete distribution. We find that if we evaluate the values of the Gamma distribution only at the discrete integer values of T in computing P [T|W] we observe that the values of P [T|W] sum to 1 as required of a true probability. Hence, in this study the Gamma distribution reported is actually a discrete distribution represented by the integer values of the continuous Gamma distribution. There is, of course, a risk of this approach as we would expect that the change from a continuous to a discrete function should add another source of variance. In the end the data will tell us of this discrete approach is satisfactory or not.

We also note that if α > 1 (i.e. – the square of the mean exceeds the variance) the value of the Gamma distribution equals zero when T = 0. This requires that any team scoring zero tries cannot win if the distribution of P [T|W] is truly a Gamma distribution. While it is possible for a team to win by scoring only penalty goals or dropped goals in rugby, scoring in these ways is highly unusual in the seven-a-side game. The data used in this study show that teams did not win without scoring at least one try.

The three distributions of P [T|W] (Table 6 and Fig. 3) are quite similar in appearance. There is evidence that the data from the 2016–2017 competition tend towards smaller values of T and the data for the 2018–2019 competition tends towards higher values of T. The mean values of T shown in Table 1 confirm that the average number of tries scored by the winning teams was not stationary through time and increased in each competition. In all three competitions no team won when scoring zero tries and any team scoring one try is unlikely to win.

We can perform a Chi Square Goodness of Fit Test between P [T|W] and the Gamma distribution with parameters computed from the observed mean and variance within the data set of one competition. An example calculation is provided for the 2016–17 competition in Table 7 and plotted in Fig. 4 showing that the distribution of tries scored by the winning team essentially follows a Gamma distribution. Here the p-value was 0.390 and the r² value between the data and the Gamma distribution was 0.9875. The comparison between the two distributions provided in Fig. 4 shows that the agreement is striking.

Fig. 4

Comparison between P[T|W] and a Gamma distribution for the 2016–17 competition with parameters computed from the data in Table 1 and given in Table 7. The p-value of the Chi Square Goodness of Fit is 0.390 and r² is 0.9875.

Table 8 summarizes the Chi Square Goodness of Fit Tests for all three competitions and the averaged data. For the three competitions the p-values were in excess of 0.390 and the r² values ranged from about 0.97 to 0.99. For each competition year the values of P [T|W] are consistent with a Gamma distribution. Table 9 is a summary of the parameters used in computing the Gamma distributions. For these three years, the values of α range from 8.519 to 9.993 and β values range from 0.452 to 0.499. Additional research is needed to test any universality associated with these values of α and β but their relative consistency over three years’ of Sevens World Series competition suggests there may be a common process at work.

As a check on the veracity of this result a Monte Carlo-type experiment was performed using 5,000 paired samples from a Poisson distribution with mean of 3. As shown in the Appendix to this study, the 10,000 variates are found, as expected, to be distributed as a Poisson distribution, and the distribution of the 4,208 “winners” of the comparisons between the samples were seen to be well described by the Gamma distribution with parameters computed from the winners of the sample pairs. The theoretical number of equal-try matches in this example is found to result in 16.7% ties and the sample of 5,000 “winners” during this experiment resulted in 15.8% ties (792 values). These compare well to the previously mentioned World Rugby (2016) data for the 2015–16 competition where the winning team scored more tries in 88% of the matches and in the remaining 12% of the matches the number of tries scored was equal. The difference between the theoretical value for the number of equal-try matches and the observed value are not significant at the 95% level.

We conclude that the distributions of P [T|W] in these data are consistent with a discrete distribution represented by the integer values of a continuous Gamma distribution. We believe that this is a novel point that has not been demonstrated previously in the literature. We note that there is no compelling a priori reason for this to occur. Gamma distributions are typically used to describe such phenomena as the time of the n-th occurrence of several events distributed as a Poisson process; amounts of rainfall over an area; the size of loan defaults and insurance claims; the flow of items through manufacturing or distribution facilities and processes; the demand on web servers; the timing of disk drive failures; and the demand on telecommunications centers (Jones, 2019, p. 227). The distribution of the number of tries scored by the winning team in a pairwise competition where the number of tries scored is a Poisson process is not similar to any of these processes.

The results of this study, including the Monte Carlo simulation, strongly suggest that the distribution of P [T|W] truly follows the Gamma distribution. Further research is necessary to discern why this is so.

3.4Evaluating P [W|T]

We are now in a position to use Bayes’ Theorem to calculate P [W|T] – the probability of winning a match in the World Rugby Sevens World Series given the number of tries scored by a team. We note that while P [W|T] can be computed accurately in this manner, it turns out that there is a simpler and mathematically equivalent way to get there (Stone, 2013:p.70). P [W|T] can be computed directly from the count data given in Table 1. Within any given competition, P [W|T] is equal to the number of times the winning team scored T tries divided by the total number of times that T tries were scored.

Table 10 shows the values of P [W|T] for each competition and these are plotted in Fig. 5. We see that the distribution is a step-like pattern rising from a probability of zero when no tries are scored to a probability of 1.00 when five tries are scored and is constant at 1.00 thereafter to the maximum observed number of tries. The winning probability from scoring only one try is quite bleak, reaching an average value of fewer than two wins in each 100 matches. Scoring two tries raises the winning probability from 10% to 20%. With three tries a team is close to winning 50% of its matches (43% to 52%). Four tries resulted in a fairly high certainty of a win, ranging from 91% to 93% of the time. Five or more tries resulted in certain victory in these competitions.

Fig. 5

The probability of winning given that a team scores T tries (P[W|T]) for three competitions. Teams that scored one try or did not score a try had essentially no chance of winning. The 50–50 point for winning was about 3 tries, and teams scoring five or more tries were certain to win in these competitions.

These results might inform coaching decisions regarding teams in the Sevens World Series. To remain competitive teams evidently must have the consistent ability to score three or more tries per match. Without this there is no pathway towards success in these competitions (van Rooyen, 2015). Even with three tries per match and the resultant 50% winning probability it is unlikely that a team will consistently finish in the top two places in pool play. If a team fails to finish in the top two of their pool and they eventually win the Challenge Trophy (the knock-out round for the third and fourth place finishers in each pool) the maximum number of points they can acquire is only 36% of the points the tournament champion receives (van Rooyen, 2015). Competitive teams must routinely finish in the top two of their pool which, according to the values of P [W|T] developed in this study, requires routinely scoring more than three tries per match.

Teams can be quite certain of winning by scoring four tries in a match and virtually certain of winning by scoring five or more tries. There is no additional benefit to be accrued (based on the data used in this study) from scoring more than five tries. While the observed distribution of P accounts for the probability of scoring up to nine tries in a match – and there have been 145 occurrences of six or more tries in a match in 1,440 team performances (an observed probability of 0.10) – from a practical perspective the effort required to score more than five tries is essentially wasted. We strongly suspect that in some future Sevens World Series matches a team will score five or more tries and lose the match but this has not been observed in the last three years. At present there is no evidence that “running-up the score” provides tangible benefits in terms of winning matches in exchange for the additional effort. There may, however, be real psychological benefits over future opponents from scoring more than five tries in a match. Coaches must weigh the costs and benefits.

Finally, we believe that most experienced rugby sevens coaches sense much of the results of this study intuitively. It should surprise no high level sevens rugby coach that their team needs to score three or more tries in a match to be competitive, provided that the team’s technical and tactical abilities (particularly in defense) are good enough. The benefit of this study is that we can now address this issue more quantitatively. Two tries does not just result in losing a rugby sevens match, two tries loses the match for a team approximately 80% of the time. Four tries do not just generally ensure a victory, four tries is sufficient to win 90% of the time. This quantification of the game and the education of coaches who can articulate and interpret the probabilities will allow them to make better decisions on how they should prepare their teams.