Is God a Mathematician_ - Mario Livio [68]
Facts and Forecasts
Scientists who try to decipher the evolution of the universe usually try to attack the problem from both ends. There are those who start from the tiniest fluctuations in the cosmic fabric in the primordial universe, and there are those who study every detail in the current state of the universe. The former use large computer simulations in an attempt to evolve the universe forward. The latter engage in the detective-style work of trying to deduce the universe’s past from a multitude of facts about its present state. Probability theory and statistics are related in a similar fashion. In probability theory the variables and the initial state are known, and the goal is to predict the most likely end result. In statistics the outcome is known, but the past causes are uncertain.
Let’s examine a simple example of how the two fields supplement each other and meet, so to speak, in the middle. We can start from the fact that statistical studies show that the measurements of a large variety of physical quantities and even of many human characteristics are distributed according to the normal frequency curve. More precisely, the normal curve is not a single curve, but rather a family of curves, all describable by the same general function, and all being fully characterized by just two mathematical quantities. The first of these quantities—the mean—is the central value about which the distribution is symmetric. The actual value of the mean depends, of course, on the type of variable being measured (e.g., weight, height, or IQ). Even for the same variable, the mean may be different for different populations. For instance, the mean of the heights of men in Sweden is probably different from the mean of the heights of men in Peru. The second quantity that defines the normal curve is known as the standard deviation. This is a measure of how closely the data are clustered around the mean value. In figure 36, the normal curve (a) has the largest standard deviation, because the values are more widely dispersed. Here, however, comes an interesting fact. By using integral calculus to calculate areas under the curve, one can prove mathematically that irrespective of the values of the mean or the standard deviation, 68.2 percent of the data lie within the values encompassed by one standard deviation on either side of the mean (as in figure 37). In other words, if the mean IQ of a certain (large) population is 100, and the standard deviation is 15, then 68.2 percent of the people in that population have IQ values between 85 and 115. Furthermore, for all the normal frequency curves, 95.4 percent of all the cases lie within two standard deviations of the mean, and 99.7 percent of the data lie within three standard deviations on either side of the mean (figure 37). This implies that in the above example, 95.4 percent of the population have IQ values between 70 and 130, and 99.7 percent have values between 55 and 145.
Figure 36
Suppose now that we want to predict what the probability would be for a person chosen at random from that population to have an IQ value between 85 and 100. Figure 37 tells us that the probability would be 0.341 (or 34.1 percent), since according to the laws of probability, the probability is simply the number of favorable outcomes divided by the total number of possibilities. Or we could be interested in finding out what the probability is for someone (chosen at random) to have an IQ value higher than 130 in that population. A glance at figure 37 reveals that the probability is only about 0.022, or 2.2 percent. Much in the same way, using the properties of the normal distribution and the tool of integral calculus (to calculate areas), one can calculate the probability of the IQ value being in any given range. In other words, probability theory and its complementary helpmate, statistics, combine to give us the answer.