With slow motion replays from every conceivable angle, tactical cameras stationed in the rafters and Hawk–Eye watching the goal line, it seems no action can take place on the pitch without our knowing about it. A tremendous number of these events, simultaneous and successive, cohere into a football match in a fast, confusing, and seemingly random manner. No matter how omniscient our view of the 22 players making and checking runs, the hundreds of passes, or the shots and saves, it’s hard to explain the connections between them, exactly how or why a two-goal lead was lost. We certainly form impressions and make arguments about a match; with few exceptions, however, multiple narratives will fit the result. Some explanations are drawn from the pundit’s list of usual suspects that includes a failure to kill the game and a “weak mentality”, others emphasise formations and the tactical battles they engender. Given the perfect view we all share, perhaps the presence of so many plausible takes should strike us as puzzling, to say nothing of the difficulty of choosing between them.

Football analytics has made a brash entry into this scene in the past few years, promising to correct long-standing misconceptions about the game by using the panacea of quantification. This new empirical drive may resolve some of our problems with understanding football that are not unique to the sport, nor to sport in general; problems that emerge from our modest memories and cognitive abilities, our partisan tendencies. In some scientific inquiries, we’ve managed to overcome these limitations by making well-calibrated observations and testing their significance using statistics, building a groundwork for understanding physics or the climate. Other aspects of the world have not proved as amenable to quantification, chief among them those involving human beings. I’m going to consider football analytics to be a scientific investigation into football, asking whether the rigid methods of these analyticos can capture the complexity unique to football. Are these numbers and charts helping us understand the game? Or do they just look cool in our Twitter feeds while accomplishing little else?

Analytical techniques systematically track and record events on the pitch in a way that makes the data directly comparable across games and leagues. The metrics, as defined, are applied to every shot, pass, and take-on in every league game, allowing us to think through a vast amount of information all at once, where before our minds were restricted to a few of the most salient moments. Analytical tools are multiplying, quickly and in interesting ways. As expected goals (ExGs ) gain mainstream fame, several new analytical metrics are emerging from a crowded tunnel. Many are similar to ExG, approximating the likelihood that an assist leads to a goal (ExA) or a pass finds the intended target (ExP). Others are quite different: the ‘pass maps’ popularised by Sander Ijtsma (better known as @11Tegen11) have nothing to do with ExP, nor with ‘pass angle maps’. Measures now exist for most footballing concepts, and more are invented almost daily.

In fact, we are nearing the point at which a common name for all of analytics misleads by obscuring important differences. Although people practicing analytics share a common disposition to gobble up whatever data Opta makes available, the analysis that comes out uses increasingly diverse assumptions. This is reasonable: analytics is fairly new, so it makes sense to try many different approaches. At the same time, a proliferation of models with incompatible assumptions makes it difficult to fit all the knowledge gained back together again.

ExG and ExA, in essence, count up the number of times an event of a certain type occurs. One can do this sort of analysis on anything, so long as the occurrences that must be counted are clearly defined. At the simplest level, keeping track of plain old shots is an instance of this type of analysis. You can do all the same sorts of things with shots as we see with ExGs on analytics Twitter — average them across a season, rank teams by shots taken or plot them on graphs with a dot for each team. However, concerns about inconsistencies within shot counts have stimulated a desire to replace them with ExG as a measure of offensive production. Shots don’t differentiate between very good opportunities in the penalty box and a once in a lifetime pop from 40 yards: both count as one. This means the tally has the capacity to mislead; we might underrate a team that takes few shots from very good areas relative to one that takes many shots from distance.

To alleviate this problem, analyticos have introduced models meant to approximate how many times, out of a hundred, a player would be expected to score a given attempt. Factors such as the type of shot, its angle and distance from goal, the type of assist, and many more are plugged into an equation that returns a decimal value between 0 and 1, which stands for the percentage chance of a goal according to the model. An attempt from six-yards out is worth far more ExG points than one from distance, since players score from close range much more often. The mathematical definition of ExGs projects a new countable entity onto the pitch, which can be statistically analysed. The wide range of information about each shot that is taken into account forms entities with great homogeneity, in that attempts are recorded in a way reflects their quality. The consistency of this construction increases the power of any generalisations or predictions one might make on their basis.

The quality of a chance, though, is hideously difficult to evaluate no matter how many variables one includes, simply because the football’s complexity outstrips our ability to measure it. For instance, important factors such as defensive pressure and the position of the goalkeeper aren’t directly measured. Instead, other data must be used as ‘indicators’ of defensive pressure, but these are abstractions of complex states of affairs that amount to educated, yet fallible, guesses. Say we get the locations of all the players, how about the centre of gravity of the striker –balanced over the ball or leaning back? Or the goalkeepers? These may seem like pedantic worries, but they stem from football’s nature as a continuous game, without clear boundaries between a shot and the context which produced it. Relevant data will always be left out.

Simplification is indispensable in many sciences and by no means invalidates the results of study, but calls for the injection of caution when interpreting them. ExGs are very useful – they keep track of things in a revealing way, both over the course of a game and a season – but one must remember that the knowledge comes with misrepresentations hiding in the stacked equations, making them hard to account for. Distortions that lead to strange results, such as the contrast between how well ExGs predict how many goals a team will score and how poorly ExGs-conceded predict goals against. One would think the two go together, but they don’t. The danger is that advanced statistical models still obscure important differences between attempts, just less transparently than shot counts. These statistics are powerful, but their imperfections make them less than genies revealing the ‘true’ strength of a team.

More radical ways of stripping away complexity than the abstractions of statistical models are creeping into football analytics, more like fiction than simplification. Passmaps, popularised on Twitter by @11tegen11, are graphs that show the network teammates form by passing to one another, making patterns of midfield passing and wing-play apparent. The Swiss mathematician Leonhard Euler first developed graphs like these in his famous solution of the Seven Bridges of Königsberg problem. He created a bare-bones representation of the city using only dots, or ‘nodes’, for landmasses and lines for the bridges that connect them. Dispensing with all irrelevant features such as distance or the shape of islands made it easier prove there was no route around Königsberg that crossed each bridge only once. The simplicity of Euler’s graph has encouraged applications in a myriad of different contexts in which individual things, of some sort, are ‘connected’ to one another, in some sense.

Using this formalism on passing in football is almost trivially easy. One just needs to imagine each team’s players to be 11 nodes, then monitor their connections to one another by recording who passes to whom. Pass attempts and completions are widely available, so a network can be generated that puts weightier lines between players who passed to one another a lot, and none between those who never connected. The same algorithm that powers Google search, ‘page rank’, can assess a player’s importance to her team’s network. You can just as easily assess centrality by looking at the graph itself, nodes with many arrows in and out are most influential.

Since network graphs have no regard for distance or location, the dots representing each player could be arranged in any configuration so long as the links (or lack thereof) between any two dots are not altered. The person constructing the passing network could set them up to minimise lines crossing over one another for ease of viewing. Such an arrangement might place a striker closer to a centre-back than the attacking midfielder, requiring some mental gymnastics to project the meaning of the graph onto the game being described. @11Tegen11 and others have settled on arranging the dots to match the average position of players’ touches, a reasonable enough principle for spatialising disembodied nodes. Unfortunately, the fluidity of player roles causes problems. As  Jan Mullenberg observes in his article ‘The Sense and Nonsense of a Passmap’, if “Bale plays during 90 minutes on both the left side and the right side of the field… you will get a position somewhere in the middle of the field,” even if he never touched the ball there. When a couple of players do this, their dots end up stacked on top of one another, completely obscuring the lines between them — the most relevant part of the graph. Passmaps will often look strange, and sometimes uninformative, no matter what procedure one uses because positional data is slapped on top of a graph that treats players in purely conceptual space.

Even with this limitation, passmaps are a conceptually sound tool because of the coincidence of high-quality, available, data (we know exactly who passed to whom and how many times) and a mathematical formalism that makes perfect sense to apply to that data. Although players do have a spatial location and passes travel a certain length, a team is a collection of 11 different individuals who form connections to one another by passing. Network analysis only runs into problems when the idealised treatment of passing behaviour is forced into contact with physical reality.

If we want to think about players occupying definite positions in the space of the pitch, geometry seems a promising tool. We often hear of players creating triangles or about midfield destroyers patrolling a zone, talk that uses the language of shapes and lines. Geometric analysis is difficult without positional data on all players because we can’t see the shapes they form without it.

Fortunately, this data will likely become more accessible as tracking technology, which already exists, is rolled out. Even Fifa is on board. With access to this data, geometric analyses sketched in academia could enter widespread use. An example of this is the ‘team centroid’ method, which measures positional advantage using “the geometric centre of the positions of all players from a team,” as described by Robert Rein and Daniel Mammert. Another is the ‘convex hull’ method for drawing a line that surrounds a team in order to determine surface area they cover collectively. When we quantify space in these ways, it enables the verification, and refinement, of observations about how ‘compact’ a defensive block is or how well a team ‘stretches play’.

A particularly interesting geometrical method with relevance to football is the Voronoi diagram, which has been applied to fields as diverse as cell biology and ecological models of forest growth. In a footballing context, these diagrams divide the pitch using an intuitive principle: each player controls a zone that contains “the area of the pitch that is closer to that player than to any other,” according to Michael Sumpter. The pitch disintegrates into a tessellated mass of interlocking shapes, with a player in the centre of each. If a loose ball breaks into a player’s zone, one can expect him to retrieve it, being closest to it. It’s more contentious on the borders of the zones, where the 50/50 balls invite collision. When the zones of teammates touch, they have excellent control over that space and can safely play passes, but an opposing zone in the way forecloses that possibility. With continuous locations, we can animate these diagrams to see how spatial control changes moves with time, how passing lanes are opened by off the ball runs and defensive counter-movements close gaps.

A geometric understanding of football also has its limitations based on its internal constraints. It requires looking straight down on the pitch, with only the tops of players’ heads visible. The flat plane and points are the only materials that geometry, of the two-dimensional variety at least, can admit. As such, these constructs are all presuppose Euclid’s axioms, including the fifth that states two parallel lines will continue straight on forever, never curving towards or away from one another. Luka Modrić doesn’t obey this assumption, bending and twisting the ball’s path as he does.  A dainty chip over a defender makes a mockery of his Voronoi zone. Even if the pitch and players were perfectly described, the ball would regularly shatter the assumptions that make the geometric description possible.

Each of the metrics and statistical constructs I’ve mentioned can contribute, in some way, to a greater understanding of football. The methods regularise play in a way that makes previously hidden patterns pop out. They enable a long-term view of a team’s performance within a few numbers. We can see passing networks that are impossible to notice without assistance. To deliver these advances, each technique needs to stretch and simplify reality for reasons of mathematical necessity or data availability, thereby always raising the possibility that these patterns may be the result of what the test happens to be testing, as well as worries about what’s missed. Even the most ardent analytico admits that, although there are many illuminating stats and graphs, none are perfect representations of what happens on the pitch. What’s necessary is the ability to distinguish insightful uses of analytics from misleading ones.

Perhaps integrating a whole bunch of imperfect stats would paint a truer picture than any in isolation, overcoming doubts about how truly they represent the game. Cobbling together two metrics might overcome some of the limitations of each, but could also amplify them. If they use different methods to measure the same thing, similar results would bolster claims to adequacy, but divergence would call both into question. In either case, it requires judgment to determine which we ought to prefer. Most often, no comparative tests will be possible, since the methods are evaluating different things using different mathematical assumptions. There are no further mathematics for easily combining or comparing statistical, network and geometric paradigms.

Fitting them together, in light of the particular issues football presents, is the job of analysis again, yes, but one that uses human observation and experience to make the lessons of incommensurable formalisms cohere into a narrative. The range of analytic measures posted on Twitter all suggest reasons for why a match played out as it did, some of which tell similar stories while others raise tension. The empirical grounding of these measures does not make the task of applying them to a match, singly or in concert, any easier. Trying to use the same sort of mathematical techniques to assess their usefulness digs deeper into rabbit holes of assumptions and abstractions. Those who construct the metrics might say they are true to the extent they are predicative; unfortunately, matters have never been that simple in any inquiry. Most analytic measures don’t predict, they describe. Those that predict may do so by touching only irrelevant variables that co-vary with what is actually important. They may predict by the sheer force of the convoluted mathematics they contain, à la Ptolemaic geocentrism. Most importantly, even prediction based on relevant variables does not guarantee any understanding of their significance.

Analytics cannot be used on their own, as though they are isolated grains of data-driven insight. They must be used together with all the older ways of reading a football match usually disparaged by analyticos. Those who attend to the demeanour of a team and assess their psychological states have a place: the way a team responds to adversity can’t be measured but I’m sure it affects finishing rates. Former players might even have some insight into those situations, since, you know, they’ve been there before. Older paper and pencil diagrams of tactical battles can be as illuminating as a data-driven passmaps, if not more so. The best analysis uses all available experience, knowledge and data, not just one or two of those. The inattention of traditional pundits to newer measures deserves criticism, and analyticos fill their boots (with rather too much relish). Paul Riley, writing on Statsbomb, is condescending towards “real football guys” for paying attention to actual scorelines, then goes on to make the ridiculous claim that ExG is an “actual real test” of the “strength, heart, and spine” of a team all at once. Such folly emerges when criticism is doled out without any reflexivity, rooted in a faith in mathematics that assumes all useful perspectives can be put into its terms. We should “forget all the other nonsense.” I’d remember that the nature of analytical techniques, partial and limited as they are, guarantees the continued existence of alternative, productive, ways of watching football. There’s no need for two cultures, they’re both worse for it.