Chapter 8: The Normal Distribution

Instead, probabilities for continuous variables are calculated over intervals and represented graphically through a probability density function, a curve whose total area underneath always equals one. The normal distribution, also called the Gaussian distribution, is then introduced as a symmetric, bell-shaped distribution characterized by its mean and variance. A fundamental property of the normal distribution is that fixed percentages of data consistently fall within standard deviations from the mean: approximately 68 percent within one standard deviation, 95 percent within two, and 99.7 percent within three, regardless of the specific mean or standard deviation values. To simplify calculations across the infinite variety of possible normal distributions, statisticians employ standardization, transforming any normal variable into the standard normal variable with mean zero and standard deviation one through the z-score formula. The chapter explains how to use standard normal tables to determine probabilities and applies symmetry properties to find probabilities for negative z-scores. Finally, the chapter demonstrates how the binomial distribution, a discrete probability model, can be approximated using the normal distribution under appropriate conditions, specifically when both np and nq exceed five. This approximation technique requires applying continuity corrections to account for the difference between discrete and continuous representations, allowing statisticians to solve otherwise computationally intensive binomial problems efficiently using normal distribution methods.