4.3 The Normal Distribution

Content

• 4.3.1 The Standard Normal Distribution
• 4.3.2 Percentiles of the Standard Normal Distribution
• 4.3.3 Notation for Critical Values
• 4.3.4 Nonstandard Normal Distributions
• 4.3.5 Percentiles of an Arbitrary Normal Distribution
• 4.3.6 The Normal Distribution and Discrete Populations
• 4.3.7 Approximating the Binomial Distribution
• EXERCISES Section 4.3 (28–58)

Introduction

The normal distribution is the most important one in all of probability and statistics. Many numerical populations have distributions that can be fit very closely by an appropriate normal curve. Examples include heights, weights, and other physical characteristics (the famous 1903 Biometrika article “On the Laws of Inheritance in Man” discussed many examples of this sort), measurement errors in scientific experiments, anthropometric measurements on fossils, reaction times in psychological experiments, measurements of intelligence and aptitude, scores on various tests, and numerous economic measures and indicators. In addition, even when individual variables themselves are not normally distributed, sums and averages of the variables will under suitable conditions have approximately a normal distribution; this is the content of the Central Limit Theorem discussed in the next chapter.

normal distribution

A continuous rv $X$ is said to have a normal distribution with parameters $\mu$ and $\sigma$ (or $\mu$ and ${\sigma }^2$ ), where $-\infty <\mu <\infty$ and $0<\sigma$ , if the pdf of $X$ is

$$f\left(x;\mu ,\sigma \right)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-{\left(x-\mu \right)}^2/\left(2{\sigma }^2\right)}\text{\hspace{1em}(4.3)}$$

Again $e$ denotes the base of the natural logarithm system and equals approximately 2.71828, and $\pi$ represents the familiar mathematical constant with approximate value 3.14159. The statement that $X$ is normally distributed with parameters $\mu$ and ${\sigma }^2$ is often abbreviated $X\sim N\left(\mu ,{\sigma }^2\right)$ .

Clearly $f\left(x;\mu ,\sigma \right)\ge 0$ , but a somewhat complicated calculus argument must be used to verify that ${\int }_{-\infty }^{\infty }f\left(x;\mu ,\sigma \right)dx=1$ . It can be shown that $E\left(X\right)=\mu$ and $V\left(X\right)={\sigma }^2$ , so the parameters are the mean and the standard deviation of $X$ . Figure 4.13 presents graphs of $f\left(x;\mu ,\sigma \right)$ for several different $\left(\mu ,\sigma \right)$ pairs. Each density curve is symmetric about $\mu$ and bell-shaped, so the center of the bell (point of symmetry) is both the mean of the distribution and the median. The mean $\mu$ is a location parameter, since changing its value rigidly shifts the density curve to one side or the other; $\sigma$ is referred to as a scale parameter, because changing its value stretches or compresses the curve horizontally without changing the basic shape. The inflection points of a normal curve (points at which the curve changes from turning downward to turning upward) occur at $\mu -\sigma$ and $\mu +\sigma$ . Thus the value of $\sigma$ can be visualized as the distance from the mean to these inflection points. A large value of $\sigma$ corresponds to a density curve that is quite spread out about $\mu$ , whereas a small value yields a highly concentrated curve. The larger the value of $\sigma$ , the more likely it is that a value of $X$ far from the mean may be observed.

Figure 4.13 (a) Two different normal density curves (b) Visualizing $\mu$ and $\sigma$ for a normal distribution