Ye Shiwen: Can statistics explain her win?
When 16-year-old swimmer Ye Shiwen set a new world record for the 400m medley, eyebrows were raised about the margin of her victory, despite her passing all the drugs tests. But does a closer look at the figures really reveal anything unusual?
There was Olympic controversy last week when Ye Shiwen, a young Chinese swimmer, won the 400m individual women's medley in fine style.
Her blistering final 50m was faster than American male swimmer Ryan Lochte's final 50m in the men's event, despite him swimming one of the fastest overall times in history.
John Leonard, the executive director of the American Swimming Coaches Association - but not a member of the US Olympic team - called the performance "disturbing", forcing the swimmer to deny that she had used performance-enhancing drugs.
Ye Shiwen has never failed a drugs test and says the criticism is just "sour grapes".
So can the various elements to her record-breaking swim be simply explained?
1. She beat her personal best by five seconds
Ye Shiwen is very young and still growing.
She's 12cm taller than she was two years ago, when she was 14, and so you would expect her times to improve.
The Australian swimming legend, Ian Thorpe, said this week that he beat his personal best by five seconds when he was a teenager.
One of Ye's fellow swimmers in the 400m individual medley was Australian Stephanie Rice.
Rice beat her personal best in 2008 by six seconds when she set a world record.
Sport scientists say that during a teenage growth spurt, there is a release of hormones that can suddenly increase the powers of endurance.
2. She 'smashed' the world record
Ye set a new world record time of four minutes 28.43 seconds, beating the previous mark by more than a second.
But to take the case of Stephanie Rice, again, she beat the world record by a wider margin back in 2008.
It's impressive but it's not that remarkable.
3. She swam faster than Ryan Lochte
Lochte was 23 seconds faster than Ye overall. She only swam faster than him in the final stage - freestyle.
The comparison with Lochte just isn't that telling. It's not the first time that Lochte has been slower than a woman over the last leg of that race. In Beijing in 2008 when he won bronze, he was slower than the Italian Alessia Fillipi - by more than half a second - and she only came fifth in her own race.
Lochte simply paced himself over the race very differently to Ye Shiwen.
Dr Ross Tucker from the Sports Science Institute at the University of Cape Town warns against reading too much into the comparison with Lochte.
"Lochte didn't swim [the last leg] as fast as some of the other men in that same race. Ye's performance compared to the best men for that leg was maybe not that impressive," he says.
Tucker points out that Rebecca Adlington swam faster than both Lochte and Ye in the final leg of the 800m freestyle at the world championships last year.
"The point is," he says, "that analysing performance and trying to prove doping is a futile task."
He does confirm, however, that Ye Shiwen's performance was exceptional.
"At the last world championships, the top five swimmers of the 400m individual medley are doing the final 100m freestyle about 18-20% slower than the 100 freestyle world record - whereas Ye Shiwen was about 12% slower."
Are there other measures we can apply?
There is another measure one can look at which suggests the race was unusual. Reza Noubary, a maths professor at Bloomsburg University in Pennsylvania, has applied "threshold theory" to world records in the past.
He takes the previous best performances and the amount of time that elapses between them and establishes what records are possible. He looked at the 10 best times for the 400m individual medley before Ye Shiwen beat the world record and came up with an estimate - with 90% confidence.
"Based on these 10 measurements," he says, "the lower band should be 4:29.01. She did it 4:28.43. So in that sense you may call that exceptional."
Ye Shiwen is extraordinary but sometimes we get extraordinary athletes. When Professor Noubary ran the data on the men's 100m track race - before Usain Bolt first broke the world record in 2009 - he said with 90% confidence that the next record wouldn't exceed 9.62 seconds. Bolt's record stands at 9.58 seconds.
There is a cloud that has hung over Chinese swimming ever since 32 of their swimmers tested positive in the 1990s.
And only seven months ago, another teenage swimmer tested positive for the drug EPO.
But none of that means we should doubt Ye Shiwen's integrity, and there is certainly no statistical smoking gun.