Steve Gets A Life

statistics can be counter-intuitive

i paste this here simply because i think its a great example (albeit probably overly-used) on how statistics are useful and sometimes enlightening and counter-intuitive. this was taken from the wikipedia
False positives in a medical test
False positives are a problem in any kind of test: no test is perfect, and sometimes the test will incorrectly report a positive result. For example, if a test for a particular disease is performed on a patient, then there is a chance (usually small) that the test will return a postive result even if the patient does not have the disease. The problem lies, however, not just in the chance of a false positive prior to testing, but determining the chance that a positive result is in fact a false positive. As we will demonstrate, using Bayes' theorem, if a condition is rare, then the majority of positive results may be false positives, even if the test for that condition is (otherwise) reasonably accurate.
Suppose that a test for a particular disease has a very high success rate:

• if a tested patient has the disease, the test accurately reports this, a 'positive', 99% of the time (or, with probability 0.99), and
  
• if a tested patient does not have the disease, the test accurately reports that, a 'negative', 95% of the time (i.e. with probability 0.95).

Suppose also, however, that only 0.1% of the population have that disease (i.e. with probability 0.001). We now have all the information required to use Bayes' theorem to calculate the probability that, given the test was positive, that it is a false positive.
Let A be the event that the patient has the disease, and B be the event that the test returns a positive result. Then, using the second form of Bayes' theorem (above), the probability of a true positive is

and hence the probability of a false positive is about  (1 - 0.019) = 0.981.
Despite the apparent high accuracy of the test, the incidence of the disease is so low (one in a thousand) that the vast majority of patients who test positive (98 in a hundred) do not have the disease. (Nonetheless, this is 20 times the proportion before we knew the outcome of the test! The test is not useless, and re-testing may improve the reliability of the result.) In this case, Bayes' theorem helps show that the accuracy of tests for rare conditions must be very high in order to produce reliable results from a single test, due to the possibility of false positives.

Posted by Steve on March 12, 2004 04:44 PM