The Numbers Behind NUMB3RS Read Online Free PDF

three attempted murders. Because the VA is a federal facility, the trial would be in a federal court rather than a state court, and subject to federal laws. A significant consequence of this fact for Gilbert was that although Massachusetts does not have a death penalty, federal law does, and that is what the prosecutor asked for.
    STATISTICS IN THE COURTROOM?
    An interesting feature of this case is that the federal trial judge ruled in pretrial deliberations that the statistical evidence should not be presented in court. In making his ruling, the judge took note of a submission by a second statistician brought into the case, George Cobb of Mount Holyoke College.
    Cobb and Gehlbach did not disagree on any of the statistical analysis. (In fact, they ended up writing a joint article about the case.) Rather, their roles were different, and they were addressing different issues. Gehlbach’s task was to use statistics to determine if there were reasonable grounds to suspect Gilbert of multiple murder. More specifically, he carried out an analysis that showed that the increased numbers of deaths at the hospital during the shifts when Gilbert was on duty could not have arisen due to chance variation. That was sufficient to cast suspicion on Gilbert as the cause of the increase, but not at all enough to prove that she did cause the increase. What Cobb argued was that the establishment of a statistical relationship does not explain the cause of that relationship. The judge in the case accepted this argument, since the purpose of the trial was not to decide if there were grounds to make Gilbert a suspect—the grand jury and the state attorney’s office had done that. Rather, the job before the court was to determine whether or not Gilbert caused the deaths in question. His reason for excluding the statistical evidence was that, as experiences in previous court cases had demonstrated, jurors not well versed in statistical reasoning—and that would be almost all jurors—typically have great difficulty appreciating why odds of 1 in 100 million against the suspicious deaths occurring by chance does not imply that the odds that Gilbert did not kill the patients are likewise 1 in 100 million. The original odds could be caused by something else.
    Cobb illustrated the distinction by means of a famous example from the long struggle physicians and scientists had in overcoming the powerful tobacco lobby to convince governments and the public that cigarette smoking causes lung cancer. Table 2 shows the mortality rates for three categories of people: nonsmokers, cigarette smokers, and cigar and pipe smokers.
    Table 2. Mortality rates per 1,000 people per year.

    At first glance, the figures in Table 2 seem to indicate that cigarette smoking is not dangerous but pipe and cigar smoking are. However, this is not the case. There is a crucial variable lurking behind the data that the numbers themselves do not indicate: age. The average age of the nonsmokers was 54.9, the average age of the cigarette smokers was 50.5, and the average age of the cigar and pipe smokers was 65.9. Using statistical techniques to make allowance for the age differences, statisticians were able to adjust the figures to produce Table 3.
    Table 3. Mortality rates per 1,000 people per year, adjusted for age.

    Now a very different pattern emerges, indicating that cigarette smoking is highly dangerous.
    Whenever a calculation of probabilities is made based on observational data, the most that can generally be concluded is that there is a correlation between two or more factors. That can mean enough to spur further investigation, but on its own it does not establish causation. There is always the possibility of a hidden variable that lies behind the correlation.
    When a study is made of, say, the effectiveness or safety of a new drug or medical procedure, statisticians handle the problem of hidden parameters by relying not on observational data, but instead by