Just how great a tennis player is Serena Williams?
The next women's champion of the US Open will be decided this weekend. World number one Serena Williams is the heavy favourite for the title.
Still, Williams has already made history at the tournament. By the time she reached the quarter-final and earned her 308th major win, she had surpassed Roger Federer's record for most matches won at Grand Slams by any male or female player in tennis history.
If Williams were to win the whole thing, it would put her Grand Slam trophy tally at 23. This would eclipse Steffi Graf and give Williams the record of most women's Grand Slam titles in the Open Era.
In today's game, Grand Slam titles, the four most prestigious titles in professional tennis, are the standard by which the greatness of a tennis player is measured. Only three women other than Williams and Graf have earned more than ten singles titles during the Open Era: rivals Chris Evert and Martina Navratilova, who each earned 18 titles in their careers, and Billie Jean King, who won 12.
The fact that Williams has won 22 titles and is still not only playing but continues to be the favourite at every Grand Slam she enters has led more than one commentator to declare her the "greatest of all time".
But are Grand Slam titles the best way to measure a tennis player's greatness?
As a summary of career achievement, the exclusive focus on majors won has some serious drawbacks. It ignores all other tournament results, which make up the majority of a professional player's season. Major counts also fail to capture two aspects of Williams' career that many believe make it especially noteworthy: the magnitude and the longevity of her dominance in the sport.
A more comprehensive measure of greatness is a player's career Elo trajectory. An Elo rating is a statistical approach for rating athlete quality and predicting how an athlete will fare against a specific opponent.
Versions of Elo rating systems exist for most of the major professional sports around the world, and a tennis Elo system is currently being implemented by the stat-crunchers at FiveThirtyEight for forecasting the outcomes of the US Open.
A tennis Elo rating is a continuously updated number that moves in step with the peaks and troughs of a player's career, acting as a kind of stock ticker whose sole job it is to index the quality of a player's game.
Unlike official player rankings, Elo ratings are based on a statistical model that captures how a player has performed above or below expectation. A player's rating updates with every match, each time taking into account the ability of the player's opponent, so that players earn more points for beating tougher opponents.
In this way, Elo ratings adjust for the competitive context in which a player plays, which make cross-era comparisons more meaningful than title counts.
When we look at the career Elo trajectories of the ten women with the most Grand Slam titles in the Open Era, we see several ways in which Serena Williams' career stands out. She is the only women in this group after Martina Navratilova to have earned an Elo rating greater than 2,400 after turning 34. And this isn't the only indicator of the longevity of her dominance.
While quality for most top players hits a peak then gradually declines, Serena Williams has had two peaks in her career: one at the young age of 21 when her Elo hit a high of 2,578 and another at 33 when it rose to 2,486.
So, although Monica Seles, Graf, and Navratilova achieved higher individual Elo ratings in their career, Serena Williams has been the only player to have returned to being the most dominant player in the sport in her 30s.
We can look at the number of years the players held the highest Elo rating and, when they had the highest rating, how great a lead they had over the next strongest player in that year (in the figure below). This analysis shows us that only two women since the early 1970s have had as many as eight years as the most highly rated player: Serena Williams and Graf.
Graf's period extended over 11 years (1988 to 1999), while Serena Williams' has lasted 13 years (2002 to 2015), and is still going. Even more impressive is that the typical gap between Serena Williams' peak rating in the years she had the number-one Elo rank has been 80 points, while Graf's was only 60.
But most impressive is that Serena Williams' dominance has been best in her 30s. Williams has been the most highly rated player according to the Elo system every year since she turned 30 and has had an average edge over the second-best player of 150 points. This means that, in her 30s, Williams has been a nearly three-to-one favourite over the next best player.
In other words, during Serena Williams' current period of dominance, she has been unrivalled. No other player has consistently tested her and she has rarely, if ever, been outplayed. This is another way her career has been remarkable among the greats of the game.
Past champions like Evert and Graf had long, heated rivalries with Navratilova and Seles respectively that pushed all of them to a higher level. Serena Williams, on the other hand, has played some of her best tennis when her only rival has been herself.
Serena Williams turns 35 later this month. It's an age at which most professional tennis players have retired or seen a major decline in their performances, but her reign over the sport continues. While pundits may debate how to measure her greatness for years to come, the distinction of what she has already accomplished is inarguable.