Normal Distribution

The normal distribution (sometimes referred to as the Gaussian distribution) has been applied to problems in the social sciences many times. Many continuous random variables can be modeled as a normal PROBABILITY distribution whose density function produces a symmetric, bell-shaped curve. The normal density function for a continuous random variable Y is

where −∞ <y< ∞, −∞ <µ< ∞, and σ> 0, and µ and σ are the POPULATION mean and STANDARD DEVIATION, respectively, such that E(Y) = µ, and V(Y) = σ2. Notice that these are the only two PARAMETERS that define the density of the normal distribution.

The normal distribution has a special property that holds across parameter values: 68% of the distribution lies within 1 standard deviation above and below the mean, 95% of the distribution lies within 2 standard deviations above and below the mean, and 99.7% of the distribution lies within 3 standard deviations of the mean. However, because the distribution lies between −∞ and ∞, a normally distributed random variable has events with nonzero probability occurring anywhere on the real-number line. In addition, because the distribution is symmetric and has a unique maximum, its MEAN, MEDIAN, and MODE are the same. Furthermore, the two inflection points are located 1 standard deviation away from the mean. Figures 1 and 2 demonstrate these points. Note that the two distributions are both centered around 3, but the spread of the first is greater than that of the second. However, a very high or very low number is a possible event (even with very low probability) in both distributions.

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A major reason for the centrality of the normal distribution in the social sciences is the CENTRAL LIMIT THEOREM. The theorem links any probability function (as long as its µ and σ are finite) to the normal distribution by proving that regardless of the distribution of the original variable, the mean and the sum of the variable are normally distributed for large enough SAMPLES (or, more strictly, as n approaches infinity). A particular application of the theorem is the normal approximation to the BINOMIAL distribution. The application establishes that a fraction of successes in a series of nBERNOULLI trials with a probability of success in each one, p, is approximately normally distributed with a mean p and variance p(1−p)/n .

In addition, following the ASSUMPTION that the ERROR terms in the frequently used LEAST SQUARES REGRESSION are distributed normally, the least squares estimators are distributed normally, a property that helps with many questions of STATISTICAL INFERENCE.

The Standard Normal Distribution

To compare values of two normally distributed variables, each drawn from a different normal distribution, a standardization is required. The standardization shifts the means of the distributions and adjusts their variances to comparable grounds, such that a comparison of, say, a grade in Advanced Akkadian and a grade in Introduction to Astrophysics is made possible.

The standardization procedure is Zi =Yi−µ/σ, such that each value Zi represents the difference from Yi to its mean in units of its standard deviation (σ) of the original variable Yi. Therefore, the mean value of Zi must be zero, and the standard deviation is equal to 1.

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