normal distribution

Statistics.

a theoretical frequency distribution represented by a normal curve. Also called Gaussian distribution.

[1895-1900]

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In statistics, a frequency distribution in the shape of the classic bell curve.

It accurately represents most variations in such attributes as height and weight. Any random variable with a normal distribution has a mean (see mean, median, and mode) and a standard deviation that indicates how much the data as a whole deviate from the mean. The standard deviation is smaller for data clustered closely around the mean value and larger for more dispersed data sets.

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▪ statistics

also called  Gaussian distribution 

      the most common distribution function for independent, randomly generated variables. Its familiar bell-shaped curve is ubiquitous in statistical reports, from survey analysis and quality control to resource allocation.

 The graph of the normal distribution is characterized by two parameters: the mean, or average, which is the maximum of the graph and about which the graph is always symmetric; and the standard deviation, which determines the amount of dispersion away from the mean. A small standard deviation (compared to the mean) produces a steep graph, whereas a large standard deviation (again compared to the mean) produces a flat graph. See the figure—>.

      The normal distribution is produced by the normal density function, p(x) = ^{e−(x − μ)2/2σ2}/_σ√2π. In this exponential function e is the constant 2.71828…, is the mean, and σ is the standard deviation. The probability of a random variable falling within any given range of values is equal to the proportion of the area enclosed under the function's graph between the given values and above the x-axis. Because the denominator (σ√2π), known as the normalizing coefficient, causes the total area enclosed by the graph to be exactly equal to unity, probabilities can be obtained directly from the corresponding area—i.e., an area of 0.5 corresponds to a probability of 0.5. Although these areas can be determined with calculus, tables were generated in the 19th century for the special case of  = 0 and σ = 1, known as the standard normal distribution, and these tables can be used for any normal distribution after the variables are suitably rescaled by subtracting their mean and dividing by their standard deviation, (x − μ)/σ. Calculators have now all but eliminated the use of such tables. For further details see probability theory.

      The term “Gaussian distribution” refers to the German mathematician Carl Friedrich Gauss (Gauss, Carl Friedrich), who first developed a two-parameter exponential function in 1809 in connection with studies of astronomical observation errors. This study led Gauss to formulate his law of observational error and to advance the theory of the method of least squares approximation. Another famous early application of the normal distribution was by the British physicist James Clerk Maxwell (Maxwell, James Clerk), who in 1859 formulated his law of distribution of molecular velocities—later generalized as the Maxwell-Boltzmann distribution law.

      The French mathematician Abraham de Moivre (Moivre, Abraham de), in his Doctrine of Chances (1718), first noted that probabilities associated with discretely generated random variables (such as are obtained by flipping a coin or rolling a die) can be approximated by the area under the graph of an exponential function. This result was extended and generalized by the French scientist Pierre-Simon Laplace (Laplace, Pierre-Simon, marquis de), in his Théorie analytique des probabilités (1812; “Analytic Theory of Probability”), into the first central limit theorem, which proved that probabilities for almost all independent and identically distributed random variables converge rapidly (with sample size) to the area under an exponential function—that is, to a normal distribution. The central limit theorem permitted hitherto intractable problems, particularly those involving discrete variables, to be handled with calculus.

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