This formula gives the probability, Y, of each value of X, as illustrated in Figure 0.1 where X is women's heights. In this formula, it denotes the ratio of the circumference of a circle to its diameter, and e denotes the base for natural logarithms.
This formula shows that all normal distributions have the same overall shape; they differ only in their mean, μ, and standard deviation, σ. The multiplier, 1/σ √2π, makes the total area under the normal curve equal to 1, as it must to be a probability distribution.
For the curious, this formula contains three famous numbers. Two are π and e, workhorse numbers that pop up everywhere in mathematics. Both are transcendental numbers, so called because they are not the roots of any polynomial equation with rational coefficients. √2 is an irrational number—not expressible as the ratio of two whole numbers—and hence not really a number to the ancient Greeks. The discovery that √2 was irrational caused a crisis in the Pythagorean religion analogous to, although lesser than, the crisis in the Catholic church caused by Galileo's discoveries.