Distribution, Normal

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Distribution, Normal

The normal distribution is the single most important distribution in the social sciences. It is described by the bell-shaped curve defined by the probability density function

where exp is the exponential function, μthe mean of the distribution, σ the standard deviation, and σ² the variance. As a matter of convenience, this distribution is often expressed as X ~ N (μ, σ ²). If X ~ N (0, 1) so that μ = 0 and σ ² = 1, the outcome is the standard normal distribution. The resulting curve is shown in Figure 1, where the horizontal axis indicates values of X in terms of positive and negative integer values of the standard deviation. The curve’s shape is typical of normally distributed variables, even when they have different means and variances.

The normal distribution has two significant features. First, the curve is perfectly symmetrical about the mean of the distribution. As a result, the distribution mean is identical to the two alternative measures of central tendency, namely, the mode (the most frequent value of X ) and the median (the middle value of X ). Second, the mathematical function provides the basis for specifying the number of observations that should fall within select portions of the curve. In particular, approximately 68.3 percent of the

observations will likely fall within one standard deviation of the mean. In the case of the standard normal deviation, this would indicate that more than two-thirds of the observations would have a value between –1 and +1. Moreover, about 95.4 percent of the observations would fall within two standard deviations above and below the mean, and about 99.7 percent would fall within three standard deviations below and above the mean. Hence, relatively fewer observations are expected in the upper and lower tails of the distribution; the more extreme the departure from the mean the lower the score’s probability of occurrence.

HISTORY

The normal distribution was first associated with errors of measurement. In the latter half of the seventeenth century Galileo Galilei (1564–1642) noticed that the errors in astronomical observations were not totally random. Instead, not only did small errors outnumber large errors, but also the errors tended to be symmetrically distributed around a central value. In the first decade of the nineteenth century the mathematicians Adrien-Marie Legendre (1752–1833) and Carl Friedrich Gauss (1777–1855) worked out the precise mathematical formula, and Gauss demonstrated that this curve provided a close fit to the empirical distribution of observational errors. Gauss also derived the statistical method of least squares from the assumption that errors were normally distributed.

However, the normal distribution also appeared in other mathematical contexts. In the early eighteenth century Abraham de Moivre (1667–1754) showed that certain binomial distributions could be approximated by the same general curve. In fact, the normal curve is the limiting case for a binomial when events have a fifty-fifty chance of occurring and when the number of trials goes to infinity. A commonplace illustration is the distribution of coin tosses. In the early nineteenth century Pierre-Simon Laplace (1749–1827), when working on the central limit theorem, showed that the distribution of sample means tends to be normally distributed: The larger the number of samples, the closer is the fit to normality—a result that holds regardless of whatever the population distribution might be. Even if the scores in the population are highly skewed, the distribution of sample means will tend toward the normal curve.

Despite the fact that many mathematicians contributed to the emergence of the concept, it is Gauss whose name became most strongly linked with the discovery. As a consequence, the eponymic term Gaussian is often used instead of “normal” or “bell-shaped.”

APPLICATIONS

Although the normal distribution was first applied to the description of measurement errors, scientists later began to realize that it also described variation in human phenomena independent of errors of measurement. In 1835 Adolphe Quetelet (1796–1874) applied the normal distribution to many physical attributes, such as height, and in 1869 Francis Galton (1822–1911) extended the same distribution to cover individual differences in ability. The latter application is seen in those psychometric instruments in which test scores are actually defined according to the normal distribution. For instance, the IQ scores on most intelligence tests are assigned in terms of a person’s position in the distribution. Thus, under the assumption that IQ has a mean of 100 and a standard deviation of 15, a score of 130 would place the individual in the upper 2 percent of the population in intellectual ability.

Indeed, the concept of the normal distribution has become so universal that it now provides the basis of almost all parametric statistical methods. For example, multiple regression analysis and the analysis of variance both assume that the errors of prediction, or residuals, are normally distributed with a mean of zero and a uniform variance. More sophisticated methods such as canonical correlation, discriminant analysis, and multivariate analysis of variance all require a more complex assumption, namely, multivariate normality. This means that the joint distribution of the variables is normally distributed. In the special case of bivariate normality, this assumption signifies that the joint distribution will approximate the shape of a three-dimensional bell. To the extent that the normality assumption is violated, the population inferences associated with these statistical methods will become approximate rather than exact.

Given the prominent place of the normal distribution in the social sciences, it is essential to recognize that not all human attributes or behavioral events are normally distributed. For example, many phenomena display extremely skewed distributions with long upper tails. Examples include the distributions of annual income across households, the box-office performance of feature films, the output of journal articles by scientists, and the number of violent acts committed by male teenagers. Sometimes these departures from normality can be rectified using an appropriate data transformation. For instance, a lognormal distribution becomes normal after a logarithmic transformation. Yet many important variables cannot be normalized in this way. In such cases, researchers may use statistics based on the specific nonnormal distribution or else employ various nonparametric or distribution-free methods. Furthermore, it is likely that the causal processes that generate normal distributions are intrinsically different from those that generate nonnormal distributions. As an example, the former tend to emerge when multiple causal processes are additive, whereas the latter tend to appear when those processes are multiplicative.

SEE ALSO Central Limit Theorem; Central Tendencies, Measures of; Distribution, Poisson; Distribution, Uniform; General Linear Model; Mean, The; Mode, The; Regression; Regression Analysis; Social Science; Standard Deviation; Variables, Random; Variance