What Is Normal Distribution?

An illustration of people composing a bell curve, or the normal distribution of data.
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A normal distribution of data is one in which the majority of data points are relatively similar, occurring within a small range of values, while there are fewer outliers on the higher and lower ends of the range of data.

When data are normally distributed, plotting them on a graph results in an image that is bell-shaped and symmetrical. In such a distribution of data, the mean, median, and mode are all the same value and coincide with the peak of the curve. The normal distribution is also often called the bell curve because of its shape.

However, a normal distribution is more of a theoretical ideal than a common reality in social science. The concept and application of it as a lens through which to examine data is through a useful tool for identifying and visualizing norms and trends within a data set.

Properties of the Normal Distribution

One of the most noticeable characteristics of the normal distribution is its shape and perfect symmetry. Notice that if you fold a picture of the normal distribution exactly in the middle, you have two equal halves, each a mirror image of the other. This also means that one-half of the observations in the data fall on each side of the middle of the distribution.

The midpoint of the normal distribution is the point that has the maximum frequency. That is, it is the number or response category with the most observations for that variable. The midpoint of the normal distribution is also the point at which three measures fall: the mean, median, and mode. In a perfectly normal distribution, these three measures are all the same number.

In all normal or nearly normal distributions, there is a constant proportion of the area under the curve lying between the mean and any given distance from the mean when measured in standard deviation units. For instance, in all normal curves, 99.73 percent of all cases will fall within three standard deviations from the mean, 95.45 percent of all cases will fall within two standard deviations from the mean, and 68.27 percent of cases will fall within one standard deviation from the mean.

Normal distributions are often represented in standard scores or Z scores. Z scores are numbers that tell us the distance between an actual score and the mean in terms of standard deviations. The standard normal distribution has a mean of 0.0 and a standard deviation of 1.0.

Examples and Use in Social Science

Even though the normal distribution is theoretical, there are several variables that researchers study that closely resemble a normal curve. For example, standardized test scores such as the SAT, ACT, and GRE typically resemble a normal distribution. Height, athletic ability, and numerous social and political attitudes of a given population also typically resemble a bell curve.

The ideal of a normal distribution is also useful as a point of comparison when data are not normally distributed. For example, most people assume that the distribution of household income in the U.S. would be a normal distribution and resemble the bell curve when plotted on a graph. This would mean that most people earn in the mid-range of income, or in other words, there is a healthy middle class. Meanwhile, the numbers of those in the lower classes would be small, as would the numbers of those in the upper classes. However, the real distribution of household income in the U.S. does not resemble a bell curve. The majority of households fall into the low to lower-middle range, which means that we have more people who are poor and struggling to survive than we have those who are comfortably middle class. In this case, the ideal of the normal distribution is useful for illustrating income inequality.​