Skeptoid recently did an episode on Ganzfeld Experiments. These are parapsychology experiments looking for telepathy. A sender concentrates for 30 minutes on a target object, while the receiver wears headphones that play white noise and can only see uniform red light. Then the receiver tries to pick out the target object from a line-up based on their impressions. The theory behind the experiments is that any psi abilities we have are washed out by everyday noise. The receiver is placed under conditions of reduced noise so that psi abilities may emerge.
Ganzfeld Experiments were able to find some small effects, one review showing a success rate of 30% (which is significantly higher than the 25% success rate from chance alone). But there are numerous biases compromising the results, such as sensory leakage, the file drawer effect, and poor statistical analysis. Ultimately, researchers seem to have seem to lost interest, and there have not been further attempts to replicate.
In this discussion, I was struck by the gulf between parapsychology claims and psychic claims.
In the Ganzfeld experiments, there's an effect of 5% increased success rate, and that's if you believe the methodology is sound. It's also under perfect conditions with as little noise as possible. I suspect that few psychics make specific claims about their success rates, but I gather that they're supposed to be right most of the time, with just enough exceptions that it doesn't hurt them when you point out a few past errors. They're obviously claiming something better than 5%.
If it were only a 5% difference, I don't think we'd even be able to tell. Are you able to estimate the probabilities of everyday occurrences in your life with precision of 5%? What's the probability that there are enough seats when I get on the bus every morning? What's the probability that I'm the first person in the office? What's the probability that it will be warm enough for me to unzip my jacket?
If the parapsychology researchers are honest, then they have to admit that the small effects they observe have nothing to do with the paranormal perceived by most people. This also goes for experiments on the efficacy of prayer.
FYI: Statistics on belief in paranormal.