Normal Distribution: Examples in Statistics

The normal or Gaussian distribution is a concept widely used in statistics and psychometrics.


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

It is the most frequent distribution in statistics, and is considered the most important because of the large number of real variables that adopt 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 see in more detail what the normal distribution is, and several examples of this.

What is the normal distribution in statistics?

The normal distribution is a concept belonging to statistics. Statistics is the science that deals with the counting, ordering and classification of the data obtained by observations, in order to make comparisons and draw conclusions.

A distribution describes how certain characteristics (or data) are distributed in a population. The normal distribution is the most important continuous model in statistics, both for its direct application (since many variables of general interest can be described by this model), as well as for 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 any value within the framework of an interval 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 the natural variables of interest followed this model.


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

1. Mean and standard deviation

The normal distribution corresponds to a zero mean and standard or standard deviation of 1. The standard or standard deviation indicates the separation that exists between any value of the sample 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 located within 1.96 standard deviations with respect to the average (between -1.96 and +1.96).

Approximately 68% of the observations are within 1 standard deviation of 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 us give three examples to illustrate, for practical purposes, what is the normal distribution.

1. Height

Think of the stature of all Spanish women; said height follows a normal distribution. That is, the stature of most women will be close to the average height. In this case, the average Spanish height is 163 centimeters in women.

On the other hand, a similar number of women will be a little taller 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 most of the population has an average intelligence, and that at the extremes (below, people with intellectual disability, and above, gifted), there is a smaller part of the population (the same% below that by above, approximately).

3. Maxwell’s 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 move at a certain speed.

This curve rises smoothly from low speeds, reaches the peak at the average, and then gently descends towards high speeds. Thus, this distribution shows that most of the particles move at a speed around the average, characteristic of the normal distribution (concentrating most of the cases in the average).