How Is Competitive Balance Measured in Football Leagues? Noll-Scully, Concentration and Points Spread
Competitive balance in football is the degree to which league outcomes stay open rather than predictable. Analysts measure it through the spread of points within a season, the concentration of points among a few clubs, and the persistence of rankings over time. No single index covers all three, and the inputs come from standings and results on RubiScore.
This data-study asks one question: how do analysts put a number on how balanced a league is? It covers the idea behind the measures, three families of indices, a hypothetical calculation and the confounders that blur comparisons. Final tables, match results and goal totals, the kind found at https://rubiscore.com, are enough to rebuild the simple versions by hand.
Why Does Competitive Balance Matter?
The idea comes from sports economics. In 1956, Simon Rottenberg argued in the Journal of Political Economy that fans pay to watch contests whose outcome is uncertain. That became the uncertainty of outcome hypothesis. In 1964, Walter Neale separated uncertainty inside one match, which he illustrated with the boxers Joe Louis and Max Schmeling, from uncertainty about the league standings.
The hypothesis is influential but not settled. A 2015 re-examination by Brad Humphreys and Li Zhou found little empirical or theoretical support for Neale's two propositions relative to factors such as loss aversion and a preference for home wins. A SABR study of baseball adds that a league where every result was random would hold little interest, so perfect balance is not the target.
What Can the Data Capture?
Later surveys sort measures by time horizon, following Pawlowski and Nalbantis (2019).
- Match level. How uncertain one result is, usually estimated from win probabilities.
- Season level. How unequal the final table is, through the spread or concentration of points.
- Multi-season level. How much rankings move from one year to the next.
Measures are also ex ante or ex post. Ex ante versions use pre-season strength estimates, such as Elo ratings or betting data. Ex post versions use what happened. Everything below is ex post and needs only points, results and goals.
How Does the Noll-Scully Ratio Measure Points Spread?
The simplest season-level measure is the standard deviation of team points or win percentage. Alone it is hard to read, because even equal teams produce spread by chance, and the amount depends on how many matches are played.
The Noll-Scully ratio, tied to the work of economists Roger Noll (1988) and Gerald Scully (1989), divides the observed standard deviation by the one expected in a league of equally strong teams. Without ties, the idealised standard deviation of win percentage is 0.5 divided by the square root of the number of games. A ratio of 1 means chance alone could explain the spread. Higher values mean more imbalance. In Major League Baseball, ratios have mostly sat between 1.5 and 2.5 since 1960, according to a SABR study (Gordon, 2020).
Football does not fit the original formula, for three reasons.
- Draws. Michie and Oughton (2004) argued that the measure is inadequate for football, and Deb (2021) notes that it was built for US sports where ties almost never occur. Draw shares vary. Deb reports that in 2018/19, 18.7 percent of Premier League matches were drawn, against 23.9 percent in the Bundesliga, 28.4 percent in Serie A and 29 percent in both La Liga and Ligue 1.
- Scoring system. With three points for a win and one for a draw, points and win percentage are different quantities, so the benchmark must be rebuilt for the points system in use.
- Season length. Doria and Nalebuff (2021) showed that the ratio is biased: it inflates measured imbalance in leagues with longer seasons. A 34-match league and a 38-match league are not strictly comparable.
Another route scales observed spread against a completely unbalanced league, as in the National Measure of Seasonal Imbalance introduced by Kelly Goossens in 2006. Because researchers make different choices, ratios from different papers are not directly comparable.
A Hypothetical Calculation
Every figure in this section is hypothetical and not a real league. Take eight teams, Team A to Team H, in a double round-robin. Each plays 14 matches, 56 in all, with three points for a win and one for a draw. Assume 14 matches end level, a 25 percent draw rate. Total points are then 3 × 56 − 14 = 154. The teams finish on 31, 27, 24, 21, 18, 15, 11 and 7 points.
- Spread. The mean is 19.25 points. The standard deviation across the eight teams is about 7.6 points.
- Chance benchmark. If all eight were equal, wins and losses would each occur 37.5 percent of the time and draws 25 percent. One match would be worth 1.375 points on average, with a standard deviation of about 1.32. Over 14 matches, chance alone gives a spread of about 1.32 times the square root of 14, or 4.9 points. This is a simplification.
- Ratio. 7.6 divided by 4.9 is about 1.54. A draw-free benchmark (1.5 times the square root of 14, about 5.6) would give 1.35, so the benchmark choice alone moves the answer.
- Concentration. The squared point shares sum to 0.1445, against 0.125 for equal teams. The top three hold 53.2 percent of points, against 37.5 percent if equal.
- Persistence. Suppose next season's order is B, A, D, E, C, G, H, F. Spearman rank correlation with this season's order is about 0.83. Kendall's tau is about 0.64, because 23 of the 28 pairs keep their order.
What Do Concentration Measures Add?
Concentration measures borrow from industrial economics, where they describe how much of a market a few firms hold. In football the market is the pool of points.
- Herfindahl-Hirschman index. The sum of squared point shares. It equals one divided by the number of teams when points are equal and rises as points pile up at the top. Owen, Ryan and Weatherston (2007) set out its use for sports leagues.
- Top-N share. The share of points held by the best N clubs, compared with N divided by the number of teams. Deb (2021) used the top six, scaled so that a perfectly equal league scores 1.
- Gini coefficient. A general inequality measure. Because league play is zero-sum, simple versions need adjustment (Utt and Fort, 2002).
The achievable range of concentration depends on league size and scoring system (Avila-Cano and co-authors, 2021; Triguero-Ruiz, Owen and Avila-Cano, 2023), so raw values from an 18-team and a 20-team league are not directly comparable.
How Is Persistence Over Time Measured?
A balanced season can sit inside a predictable decade, so multi-season measures track whether rankings move.
- Rank correlation. Spearman or Kendall coefficients between consecutive seasons. Groot (2008) proposed Kendall's coefficient as a dynamic balance measure. Values near 1 mean a stable hierarchy. Values near 0 mean last year's table says little about this year's.
- Champion repeat rate. In a study of eight European leagues between 1959 and 2008, the champion was the same club in two consecutive seasons in 124 of 377 season-to-season comparisons, about one in three (Manasis, Ntzoufras and Reade).
- Distinct champions. Counting different winners, or different top-four clubs, over a fixed window is easy to explain.
- Title-race gaps. The points gap between first and second, or the number of clubs within a set margin of the lead, describes suspense directly.
When Do the Measures Disagree?
Published football studies show that changing the measure can change the story.
- Champions League group stage. Earlier studies using HHI-based measures reported declining balance. Csató and Petróczy (2025) used six alternative indices built on Elo ratings and rank correlations and found no trend between 2003/04 and 2023/24.
- English top flight. Basini and co-authors (2023) applied a Bayesian stochastic block model to win-draw-loss networks for 1978/79 to 2019/20. They reported evidence of a change around the early 2000s, from a reasonably balanced league to a two-tier one.
- Attendance. In a panel of eight European leagues from 1959 to 2008, conventional all-team indices showed no significant long-run link with attendance in the authors' model, while indices weighted toward the title race and European places did.
The same league can look balanced through one lens and top-heavy through another.
What Confounds the Numbers?
No index removes these effects.
- League size and season length. Fewer clubs mean a larger equal share per club, and the Noll-Scully ratio depends on the number of matches.
- Promotion and relegation. In open leagues the membership changes each year, so season-to-season comparisons mix returning clubs with newcomers.
- Financial inequality. Spending gaps shape dominance, but an index records outcomes, not causes.
- European qualification slots. The number and value of places change what the top of the table is worth, which is why weighted indices exist.
- Schedule and rules. Split phases, play-offs, unbalanced schedules and points deductions change totals without changing underlying strength. Trandel and Maxcy (2011) proposed adjusting spread measures for home advantage.
- Sample size. One season gives a noisy estimate. Multi-season averages are steadier.
How Can a Reader Run a Simple Check?
Standings, points, goals and home and away splits are the raw inputs, and platforms such as RubiScore publish them for each competition.
- First, fix the league and a window of seasons, then record each club's final points.
- Second, compute the standard deviation and point shares, and build a chance benchmark from the observed draw rate.
- Third, compare consecutive seasons by rank correlation and count the distinct champions.
Verdict
Competitive balance can be measured, but not with one number. Points-spread ratios show how unequal a season's table is compared with chance. Concentration indices show who holds the points. Persistence measures show whether the same clubs keep winning. Football's draws, three-point wins and varying season lengths mean every adapted ratio needs its benchmark stated.
The caveats are real. Published studies disagree when the measure changes, as the Champions League group stage and English top-flight examples show. The attendance result comes from one panel and one model, and the wider evidence on whether balance drives demand remains mixed. Treat any index as a description of outcomes, not a cause. Use several measures over several seasons, with the same method for every league compared.