Nineteen out of 20 people shot in the head die from their injuries. But Gabrielle Giffords could be among the lucky ones.

The doctor treating her said she was in critical condition after a two-hour operation but that he was "very optimistic" about her recovery.

It may be days before she is out of danger. National statistics show that only 5 per cent of people survive after being shot in the head. But Ms Giffords's chances of survival appear better than even. She was responding to simple commands last night, such as holding up two fingers when asked to do so.

That is the first positive sign that will have encouraged her doctors. Patients who are responsive on admission do better in the long term than those who are in a deep coma.

The second positive sign is that the bullet fired by Jared Lee Loughner passed directly through her head and did not become lodged within it. Less tissue damage results when the bullet exits the skull.

But the third, and most important, factor is the trajectory the bullet followed. In 2008, a Chinese food deliveryman, Fengwang Chen, 31, was shot in the head by thugs trying to steal an order from him in New York. The bullet entered his skull behind his ear and lodged in his jaw, missing his brain. He survived.

Paul Vesta, director of neurocritical care at the Ronald Reagan UCLA Medical Centre in Los Angeles, said there were a number of scenarios that could make it possible to survive a gunshot to the brain. He said: "If it's a glancing blow that injures the skull and a small amount of brain and doesn't go directly through the whole brain is one case. People can also survive with parts of the brain missing."

Even if Ms Giffords survives, she may be brain damaged. But those shot in the front of the head have a better chance than those shot from the side, because a bullet travelling from front to back generally destroys just one of the brain's two hemispheres.

John Boockvar, an associate professor of neurological surgery at New York-Presbyterian/Weill Cornell Medical Centre, said: "If the shooter is shorter and the bullet goes just through the frontal lobe, there could be near normal recovery."