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These charts clearly show how some Olympic swimmers may have got

A few years ago, researchers from Indiana University discovered a disturbing pattern at the 2013 Swimming World Championships in Barcelona. According to the lap-time data, athletes assigned to the outer lanes of the pool were consistently swimming faster in one direction than the other.

A mysterious force seemed to be interfering with the competition. But before the researchers could investigate in person, the pool — a temporary facility constructed just for that event — was torn down.

The same strange problem cropped up at the Rio Olympics, and nobody can explain why.

Two independent statistical analyses show discrepancies large enough to cast doubt on the results of some races. The problem may have even cost swimmers medals.

The following chart — from Eastern Michigan University professor Andrew Cornett, and Indiana University's Christopher Brammer and Joel Stager — illustrates the lap time discrepancies in the Rio pool using data from the longer-distance races. Since competitors in those events make several laps back and forth along the 50-meter-long pool, the researchers could compare swimmers against themselves.

Each dot represents a swimmer in one of the 800-meter and 1500-meter races. The vertical position of the dot shows whether the athlete swam faster in one direction or the other.

In a perfectly fair pool, all of the dots should be near the zero line.

In Rio, athletes in the middle lanes were relatively unaffected. But on the lower-numbered side of the pool, competitors were about 0.4 seconds faster swimming away from the starting blocks.

The higher-numbered side of the pool suffered from the opposite problem. Swimmers were about 0.2 seconds slower in their outgoing laps compared to their return laps. (In all of this, the researchers ignored the first and last laps because those are strategic parts of the race.)

To understand the pattern of lane bias, consider the example of Norwegian swimmer Henrik Christiansen, who is marked on the chart in pink. In his preliminary heat for the 1500-meter race, Christiansen was assigned to Lane 2. In that race, his outgoing laps were 0.28 seconds faster than his incoming laps, on average.

When Christiansen made it through to the final, he was assigned to the other side of the pool. In Lane 8, Christiansen swam a completely different race. His outgoing laps were slower than his incoming laps.

Because most Olympic events require athletes to swim an even number of laps, the effects of any currents in the water should have somewhat canceled out. Those who got a boost swimming in one direction would have to fight the same force when they turned around.

The one exception is the 50-meter freestyle, which only covers a single length of the pool.

Barry Revzin, a data analyst based in Chicago who used to swim competitively at MIT, found compelling evidence of the lane effect by comparing swimmers' times as they rose through the heats and semi-finals. He took advantage of the fact that athletes were typically assigned different lanes as they progressed through the competition.

On average, Revzin found that swimmers raced faster when they moved to higher-numbered lanes, and slower when they moved to lower-numbered lanes.

By Revzin's estimates, shifting two lanes to the right in the Rio pool would have given a swimmer a boost of about 1 centimeter per second. In a 50-meter freestyle race at Olympic pace, that translates to a difference of about one-tenth of a second.