Normal distribution: what it is, characteristics and examples in statistics

Normal distribution: what it is

In statistics and probability, the normal distribution, also called the Gaussian distribution (after Carl F. Gauss), Gaussian distribution or Laplace-Gauss distribution, reflects how data are distributed in a population.

This is the most frequent distribution in statistics, and is considered the most important because of the large number of real variables that take its form. Thus, many of the characteristics in the population are distributed according to a normal distribution: intelligence, anthropometric data in humans (for example height, height…), etc.

Let’s look in more detail at what normal distribution is, and several examples of it.

What is normal distribution in statistics?

Normal distribution is a concept that belongs to statistics. Statistics is the science of counting, sorting, and classifying data obtained from observations so that comparisons can be made and conclusions drawn.

A distribution describes how certain characteristics (or data) are distributed in a population . Normal distribution is the most important continuous model in statistics, both because of its direct application (since many variables of general interest can be described by such a model), and because of its properties, which have allowed the development of numerous statistical inference techniques.

The normal distribution is therefore a probability distribution of a continuous variable . Continuous variables are those that can take on any value within a range that is already predetermined. Between two of the values, there can always be another intermediate value, which can be taken as a value by the continuous variable. An example of a continuous variable is weight.

Historically, the name “Normal” comes from the fact that for a time it was believed by doctors and biologists that all natural variables of interest followed this model.


Some of the most representative characteristics of the normal distribution are the following:

Mean and standard deviation

The normal distribution has a mean of zero and a standard deviation of 1 . The standard deviation indicates the gap between any sample value and the mean.

2. Percentages

In a normal distribution, you can determine exactly what percentage of the values will be within any specific range . For example:

About 95% of the observations are within 2 standard deviations of the mean. 95% of the values will be within 1.96 standard deviations from the mean (between -1.96 and +1.96).

About 68% of the observations are within 1 standard deviation from the mean (-1 to +1), and about 99.7% of the observations would be within 3 standard deviations from the mean (-3 to +3).

Examples of Gaussian distribution

Let’s take three examples to illustrate, for practical purposes, what normal distribution is.

1. Height

Let’s think about the height of all Spanish women; that height follows a normal distribution. That is, most women’s height will be close to average. In this case, the average Spanish height is 163 centimetres for women.

On the other hand, a similar number of women will be a little higher and a little lower than 163cm ; only a few will be much higher or much lower.

2. Intelligence

In the case of intelligence, the normal distribution is fulfilled worldwide, for all societies and cultures. This implies that the majority of the population has average intelligence , and that at the extremes (below, people with intellectual disabilities, and above, the gifted), there is a smaller part of the population (the same % below as above, approximately).

3. Maxwell curve

Another example that illustrates the normal distribution is the Maxwell curve. The Maxwell curve, within the field of physics, indicates how many gas particles are moving at a given speed .

This curve rises smoothly from low speeds, reaches a peak in the middle, and then descends smoothly again towards the high speeds. Thus, this distribution shows that most particles are moving at a speed around the average, which is characteristic of the normal distribution (mostly concentrated in the mean).

Bibliographic references:

  • Quintela, A. (2005). Basic Sweetened Statistics. Bookdown.
  • Fontes de Gracia, S. García, C. Quintanilla, L. et al. (2010). Fundamentals of research in psychology. Madrid: UNED. ISBN: 9788436260557.
  • Botella, J. Sueró, M. Ximénez, C. (2012). Data analysis in psychology I. Madrid: Pirámide. ISBN: 9788436815382.

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