Normal distribution

Normal

Probability density function The green line is the standard normal distribution
Cumulative distribution function Colors match the image above
Parameters	$\mu$ location (real) $\sigma ^{2}>0$ squared scale (real)
Support	$x\in \mathbb {R} \!$
Probability density function (pdf)	${\frac {1}{\sigma {\sqrt {2\pi }}}}\;\exp \left(-{\frac {\left(x-\mu \right)^{2}}{2\sigma ^{2}}}\right)\!$
Cumulative distribution function (cdf)	${\frac {1}{2}}\left(1+\mathrm {erf} \,{\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\!$
Mean	$\mu$
Median	$\mu$
Mode	$\mu$
Variance	$\sigma ^{2}$
Skewness	0
Excess kurtosis	0
Entropy	$\ln \left(\sigma {\sqrt {2\,\pi \,e}}\right)\!$
Moment-generating function (mgf)	$M_{X}(t)=\exp \left(\mu \,t+{\frac {\sigma ^{2}t^{2}}{2}}\right)$
Characteristic function	$\chi _{X}(t)=\exp \left(\mu \,i\,t-{\frac {\sigma ^{2}t^{2}}{2}}\right)$

The normal distribution is a probability distribution. It is also called Gaussian distribution because it was first discovered by Carl Friedrich Gauss.https://onlinestatbook.com/2/normal_distribution/history_normal.html The normal distribution is a continuous probability distribution that is very important in many fields of science.

Normal distributions are a family of distributions of the same general form. These distributions differ in their location and scale parameters:

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

These two parameters are represented by the symbols $\mu$ and $\sigma$, respectively.

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). It is often called the bell curve, because the graph of its probability density looks like a bell.

Many values follow a normal distribution. This is because of the central limit theorem, which says that if an event is the sum of identical but random events, it will be normally distributed. Some examples include:

• Height
• Test scores
• Measurement errors
• Light intensity (so-called Gaussian beams, as in laser light)
• Intelligence is probably normally distributed. There is a problem with accurately defining or measuring it, though.
• Insurance companies use normal distributions to model certain average cases.

Related pages

• Frequency distribution
• Least squares
• Student's t-distribution

Other websites

• Cumulative Area Under the Standard Normal Curve Calculator from Daniel Soper's Free Statistics Calculators website. Computes the cumulative area under the normal curve (i.e., the cumulative probability), given a z-score.
• Interactive Standard Normal Distribution
• GNU Scientific Library – Reference Manual – The Gaussian Distribution
• Normal Distribution Table
• Download free two-way normal distribution calculator
• Download free normal distribution fitting software