Normal distribution

For the normal distribution, the values less than one standard deviation away from the mean account for 68.27% of the set; while two standard deviations from the mean account for 95.45%; and three standard deviations account for 99.73%.

The normal distribution is a probability distribution. It is also called Gaussian distribution because it was first discovered by Carl Friedrich Gauss.

The normal distribution is very important in many fields because many things take this form.^[1] It is often called the bell curve, because the graph of its probability density looks like a bell.

The mean ("average") of the distribution defines its location.
The standard deviation ("variability") defines the scale.

These two parameters are represented by the symbols [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \sigma }[/math], respectively.^[2]

The standard normal distribution (also known as the Z distribution) is the normal distribution with a mean of zero and a standard deviation of one (the green curves in the plots to the right).^[2]

Many values follow a normal distribution.^[3]^[4] Some examples include:^[5]

Height
Test scores
Measurement errors
Light intensity as in laser light)
Intelligence is normally distributed.
Insurance companies use normal distributions to model certain average cases.

Origin

Its origin goes back to the 18th century to De Moivre and in the early 19th century with Gauss.

Use

It is the most widely used piece of statistics by far. It was famously used in World War I by the United States Army to decide when men were so poor mentally that they could not be used by the Army in any job.

Normal Distribution Media

As the number of discrete events increases, the function begins to resemble a normal distribution
Comparison of probability density functions, p(k) for the sum of n fair 6-sided dice to show their convergence to a normal distribution with increasing na, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).
The ground state of a quantum harmonic oscillator has the Gaussian distribution.
Histogram of sepal widths for Iris versicolor from Fisher's Iris flower data set, with superimposed best-fitting normal distribution.
Fitted cumulative normal distribution to October rainfalls, see distribution fitting
Carl Friedrich Gauss discovered the normal distribution in 1809 as a way to rationalize the method of least squares.
Pierre-Simon Laplace proved the central limit theorem in 1810, consolidating the importance of the normal distribution in statistics.