With the NBA finals between the Cleveland Cavaliers and Golden State Warriors set to begin on June 4th, punters looking for an edge might be tempted to take the juicy odds bookmakers are offering on away teams. In Britain, Betfair is pricing in a 70% likelihood Golden State wins its games at home, compared with just a 60% chance they prevail on the road. If playing at home really isn’t worth what it used to be, then betting on the visitor in every game should be money in the bank.

But the bookies usually know best, and this series is no exception. The Economist is pleased to report that the Curious Case of the Disappearing Home-Court Advantage has been solved: what looked in January like a vanishing act worthy of Houdini now seems more akin to a brief unexcused absence. According to data provided by David Corby of Basketball-Reference.com, from 1979 to 2014 NBA teams won 62.2% of their regular-season games at home. From January 28th, the day that the first ESPN piece about the decline of home-court advantage appeared, until the end of the regular season on April 15th, home teams won…62.1% of their games (see chart, in which the 2014-15 season has been divided into pre- and post-publication segments). This whiplash-inducing regression to the mean has continued in the playoffs: through the first three rounds, hosts have won precisely 60% of the time. The deviation that was supposed to occur less than once every 1,000 trials by chance turned out to have absolute zero predictive power whatsoever.

Could the authors of the ESPN story have known better? Prediction is difficult, especially about the future, and the evaporation of two-thirds of the historical home-court advantage over a nearly 700-game sample certainly seemed too improbable to ignore. But had they listened to the sage counsel of Nate Silver, their ESPN colleague who wrote a best-seller about how to distinguish true statistical signals from distracting background noise when making forecasts, they could indeed have avoided sending the basketball commentariat on a wild goose chase. Instead, they provided an object lesson in the perils of frequentism, the easily misused approach to statistics that Mr Silver dedicated his book to refuting and which lies at the heart of most of the field’s greatest misses, from medical studies that can’t be replicated to the faulty risk models that brought down the global financial system in 2008.

Frequentism interprets the world through a purely observational lens, basing its confidence in a conclusion on how often events occur and the odds that randomness alone could produce a given result. In the social sciences, a finding is said to be “statistically significant” if this number is below 5% or occasionally 1%; in fields like particle physics, the accepted threshold is one in many millions. There’s nothing wrong with the theory behind frequentism, but the authoritative-sounding heft it offers researchers masks the method’s enormous sensitivity to the parameters they set. It enables statisticians to make virtually anything they want seem mathematically airtight. For example, what were the odds that you would read this article at precisely this minute on this day—one in 525,600 (the number of minutes in a year)? Multiply that by the chances that your parents would meet and get together to create you in the first place—roughly one in three billion—and the event occurring in your life this very second suddenly seems utterly impossible.

So there is reason to be wary when researchers bandy about infinitesimally small probabilities as proof of their findings, and to withhold sympathy when forecasters blame their errors on unprecedented bad luck. In 2007 David Viniar, the former chief financial officer at Goldman Sachs, tried to explain away the poor performance of one of the bank’s hedge funds, saying “we were seeing things that were 25-standard-deviation moves several days in a row”. However, the odds of a 25-standard-deviation event are comparable to winning the lottery 21 or 22 times in a row: the fact that it happened even once proved that the model used to estimate those probabilities was gravely flawed.

Mr Haberstroh and Mr Ilardi seem to have succumbed to a similar fallacy. Although three standard deviations are a far cry from 25, and such events do happen from time to time, they should be extraordinarily rare. In fact, shifts in home-court advantage of the magnitude they identified (about one point per hundred possessions) have occurred multiple times in the past 20 seasons alone. From 1994 to 1995 the home edge fell by 0.9 points per 100 possessions, and from 1997 to 1999 it rose by 1.0 points. By the authors’ interpretation, both of those moves were so unlikely that they almost surely represented a fundamental, enduring change in home teams’ expected win rate. In fact, those fluctuations occurred during the most stable period of home-court advantage in the league’s history: two decades in which a five-year moving average of the home victory rate barely budged.

None of this means that the “true” home-court edge is set in stone, or that all deviations from the prior norm will regress back to the established mean. On the contrary, numerous changes in the ratio over the years have stuck: for example, NBA teams won 65% of their regular-season home games from 1975-1992, compared with just 60% since then. And despite home teams’ strong run during the final months of the season, their overall 57.4% win percentage 2014-15 was indeed the lowest in the league’s history. Is there any reason to believe that some portion of this decline may turn out to be permanent as well?

Answering that question requires moving beyond a pure frequentist analysis of home win percentages alone, and trying to determine whether there are any fundamental changes in the league that might be behind the drop. Mr Haberstroh and Mr Ilardi offer three potential candidates (though they made no effort to test any of them). The first, and probably the weakest, is that teams’ greedy forsaking of rabid working-class fans in order to sell high-priced luxury boxes to big companies has led to a decline in the crowd’s enthusiasm, since corporate suits are too busy toying with their iPhones or hammering out deals to bother cheering. If this effect is real, detecting it would be fairly straightforward. The NBA has historically monitored decibel readings at every game to prevent teams from cranking up the sound system with distracting noise while their opponents have the ball. That data could easily be used to find out whether sound levels have drifted down over time, or whether teams in louder arenas benefit from a bigger home-court advantage than those in quieter ones. However, at the anecdotal level, most noise-related basketball coverage in recent years has focused on the volume being toohigh, not the other way around.

A potentially more compelling argument is that travel conditions have improved. The last time home-court advantage really did shrink for good, in the early 1990s, it coincided with a league-wide shift from commercial flights—a particular hardship for seven-foot (2.15-metre) tall basketball players—to chartered planes, and from budget hotels with burgers-and-fries room service to palatial suites at five-star properties. Mr Haberstroh and Mr Ilardi note that teams have continued to find ways to improve their players’ performance on the road through biometric methods. If scientifically engineered diets, sleep schedules and training regimens can meaningfully reduce wear and tear, that would indeed level the playing field between home and visiting clubs. But if such gains do exist, they are likely to be both difficult to measure and incremental in pace—a poor match for an abrupt decline in home-court advantage during the past few seasons.

The authors’ most promising hypothesis is that home-court advantage has shrunk because of the changing style of play in the NBA. Fouls are called by referees, and referees in almost every sport have historically demonstrated a notorious home-team bias. As a result, the more that players come into physical contact with each other, the more opportunities referees have to slant their judgment calls to please the crowd.

Those chances have decreased in recent years. As the league has shifted towards a more open game relying on long-distance shooting, players have been increasingly spaced out on the floor, making them less likely to bump into each other. As a result, free throws—the uncontested shots granted to offensive players when opponents make illegal contact with them—have become less important. As recently as 2005-06, 20.2% of points were scored on free throws; last year that figure was 17.1%. Holding all else equal, that shift should lead to a small decrease in home-court advantage.

Sure enough, there has indeed been a moderately strong relationship during the past 30 seasons between successful free throws as a share of total points and home-court advantage. Each additional percentage point of overall scoring that comes at the free-throw line has been associated with an extra 1.85 percentage points of win rate for home teams (a correlation that would occur by chance about once every 3,000 times, in case any frequentists are wondering). The past four seasons have been the four lowest since 1975 in the category of made free throws divided by points scored, and three of the four lowest in home-court advantage. The case is not airtight—in past seasons, referees have granted almost identical numbers of free throws to home and away teams, without a corresponding dip in home-court advantage—but it is at least a credible feather on the scale.

Given this plausible structural explanation, it may well turn out to be true that home-court advantage in the NBA has narrowed slightly, from the 1993-2011 average of 60.3% to a figure around the 2011-15 mean of 58.8%. If nothing else, some reduction is likely in the future simply because of the campaign by Adam Silver, who became the NBA’s commissioner last year, to reduce bias by referees—a problem that has been in the spotlight ever since Tim Donaghy, a former referee, admitted to betting on games in 2007. But even if the 2014-15 numbers are the new normal, a decline of 1.5 percentage points is hardly the “#HomeCourtDisadvantage” that Mr Haberstroh and Mr Ilardi suggested as a Twitter hashtag for discussion of their work. A shift of such modest magnitude will hardly cause teams to reduce their efforts to win in the regular season—among the rewards for clubs with the best regular-season records is home-court advantage in the playoffs—or bettors to adjust their wagers in favour of the visitors.

To their partial credit, the authors did post a third article in early March noting that home teams had performed exceedingly well during the previous month (though unlike the first two, this one was buried behind ESPN’s paywall). Moreover, they anticipated likely criticisms of their findings, writing that “you’re probably thinking that this might be a random blip attributable to statistical noise”. They even linked to a blog post that outlines the pitfalls of odds calculations about seemingly improbable events.

But in the end, Mr Haberstroh and Mr Ilardi still couldn’t resist the temptation to build narratives based on (probably) random numbers, even when doing so required them to contradict directly the thesis they published just one month before. Narrowing down their sample even further to a sub-set of games they deemed to be particularly close within a 30-day window, they wrote that “the recent dramatic increase in clutch winning percentage is statistically significant and highly unlikely to be due to chance variation...The home team’s dominance over the past month was so off-the-charts that almost certainly something has to be driving it.” In other words, they now believe that even though home-court advantage is shrinking overall, it has simultaneously grown stronger than ever in games that happen to conclude within an arbitrary scoring margin.

Goldman Sachs probably has no trouble finding investors to buy into its supposedly unlucky hedge funds. But if any of their salespeople happen to read this piece, we’d like to suggest two writers that might be worth a call.