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Normal Curve Distribution

Normal curve distribution is a symmetrical distribution, which has a bell shape and identical scores for the mean (i.e., the average score), median (i.e., the middle score splitting the bottom 50% from the top 50% in the distribution), and mode (i.e., most frequent value). A bell-shaped curve (see an example in Figure 1) characterizing the normal distribution can be represented by the equation below:

Y = N 2 π σ e − ( X − μ ) 2 2 σ 2

where Y = frequency of a given value of X; X = any score in the distribution; µ = mean of the distribution (the mean or the average score is the ratio of the sum of all scores to the number of scores in the distribution); σ = standard deviation of the distribution (standard deviation is one of the variability measures representing the average of the number of scores about the mean); N = the number of data points (or frequency) in the distribution; π = a constant of 3.1416; and e = a constant of 2.7183.

Figure 1 The Normal Curve

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For users of statistics who are primarily interested in applying their statistical knowledge in social scientific research to analyze data, Equation 1 is of little practical use. But, it illustrates the notion that the normal curve is a theoretical distribution that can be mathematically represented.

In the curve depicted in Figure 1 each side of the mean has an inflection point, situated where the direction of the curvature changes from curving down to curving up. For any normally distributed bell curve, the inflection points are located 1 standard deviation above and below the mean. At each tail of the distribution, the curve gets close to the x axis, although theoretically, it never touches the horizontal axis. In statistics, such a curve is described as being asymptotic to the horizontal axis.

Relationship Between Mean, Standard Deviation and Area Under the Curve

In any normal distribution, the mean and the standard deviation have a special connection to the area under the curve. In a normally distributed set of scores 34.13% of the area below the curve is located between the mean and 1 standard deviation or µ + 1σ. Furthermore,

  • 13.59% of the area is between µ + 1σ and µ + 2σ;
  • 2.15% of the area is between µ + 2σ and µ + 3σ; and
  • 0.13% of the area can be found above µ + 3σ.

Taken together, these percentages comprise 50% of the area. Because the bell curve is symmetrical, the same relationship between the mean, standard deviation, and the area under the curve applies for scores below the mean.

Let’s consider an example of a population of 10,000 intelligence scores, which is normally distributed with µ = 100 and σ = 16. By applying the aforementioned information about the relationship between the mean, standard deviation, and the area below the curve to this example, one can determine that 34.13% of the scores are located between 100 and 116 ( μ + 1σ = 100 + 16 = 116), 13.59% of the scores are between 116 and 132 (μ + 2σ = 100 + 32 = 132), 2.15% are between 132 and 148, and 0.13% are above 148. Likewise, 34.13% of the scores are located between 84 and 100, 13.59% of scores are between 68 and 84, 2.15% are between 52 and 68, and 0.13% are below 52.

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