Normal Distribution

The normal distribution is a hypothetical symmetrical distribution used to make comparisons among scores or to make other kinds of statistical decisions. The shape of this distribution is often referred to as “bell shaped” or colloquially called the “bell curve.” This shape implies that the majority of scores lie close to the center of the distribution, and as scores drift from the center, their frequency decreases.

Normal distributions belong to the family of continuous probability distributions or probability density functions. A probability density function is a function meant to communicate the likelihood of a random variable to assume a given value. This function is graphed by plotting the variable, x, by the probability of that variable occurring, y. These normal probability distributions are characterized by the aforementioned symmetric bell shape but can have any real mean, labeled µ, and any positive real standard deviation, labeled σ. Specifically, the normal distribution is characterized by continuous data, meaning the data can occupy any range of values. In special cases, the normal distribution can be standardized in which the mean becomes 0 and the standard deviation becomes 1. All normal distributions can be transformed or standardized to the standard normal distribution.

The normal distribution is commonly named the “Gaussian distribution,” after Carl Friedrich Gauss, a German mathematician who made significant advancements of statistical concepts. Less frequently, the normal distribution may be called the “Laplace distribution,” after Pierre-Simon Laplace. The remainder of this entry reviews the history of normal distribution, explains the defining function of normal distribution, explores its properties, highlights the differences between normal distribution and standard normal distribution, and reviews assumptions and tests of normality.