Mathematical analysis reveals that for players with good control, using an unorthodox underarm technique gives better odds of scoring

It might invite ridicule, but it gets results. A scientific analysis has concluded that using a “granny style” underarm technique is the optimal way to take a free throw in basketball.

Adopting the unorthodox strategy could result in marginal gains for professional players, the research suggests. And, as sporting doctrine goes, marginal gains can lead to remarkable results.

Madhusudhan Venkadesan, who led the work at Yale University, said: “Our mathematical analysis shows that if the thrower is capable of controlling the release angle and speed well, the underarm throw is slightly better for a basketball free throw.”

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However, it remains to be seen whether science will prove more persuasive than professional advocates of the underarm style.

The retired NBA player Rick Barry, a pioneer of the underarm free throw, was one of the most effective shooters of all time and when he retired in 1980 his 90% free throw record ranked first in NBA history. But he struggled to convince his teammates due to the inescapable fact that shooting underarm “makes you look like a sissy,” Barry said.

Venkadesan acknowledges that it is a difficult case to make.

“One suspects there are social and cultural reasons you don’t see that practised too often,” he said. “So what if some call it the ‘granny throw’? What matters is that the ball goes through the hoop! Rick Barry’s record does support the underarm throw.”

The study, published in the journal Royal Society Open Science, considered the chances of the ball being on target, depending on the style, speed and accuracy of a throw.

It found that if the player is capable of controlling the release angle and speed well, the underarm throw has slightly better odds of going in. But for amateurs who have only crude control, the release of the ball overarm is safer, sparing casual players the dilemma of choosing style or results.

An important factor in comparing the two strategies was how the ball approaches its target. When the ball approaches the net from directly above, as in a typical underarm throw, the cross-section of the target is large from the ball’s vantage point. This is good, as it means that if a throw is close to being exactly on target it has a very high chance of going in.

However, in trying to achieve this straight down entry, the amateur risks lobbing the ball extremely high due to their mediocre control. In this scenario, a small error in the timing of the release can cause the ball to grossly overshoot or undershoot the hoop.

So the overarm shot, where the ball “sees” a smaller cross-section of the hoop, but is less likely to go wildly off course, is a more conservative strategy.

“This competition between the entry angle and speed underlies both the speed-accuracy trade-off and the relative accuracy of one style versus another,” said Venkadesan.

For the professional player, the analysis predicts, this trade-off is finely balanced and probably within the margins of error of the model, which did not consider the backboard.

Barry, no doubt, would view the findings as confirmation of what he has argued all along. “From the physics standpoint, it’s a much better way to shoot,” he told the author Malcolm Gladwell in a recent interview. “You have a little bit more margin for error than when you shoot overhand.”