The normal distribution is a commonly occuring continuous probability distribution. Considered to be the most prominent probability distribution, the normal distribution is very important in statistics as well as in natural and social sciences.
The graph of the probability density $f\left(x\right)$ is known as the "bell curve", as it exhibits symmetry about the center with approximately 50% of the values less than the mean and 50% greater than the mean.
A random variable X is normal or Gaussian if its probability density function is given by:
$f\left(x\right)equals;\frac{1}{\mathrm{\sigma}\cdot \sqrt{2\cdot \mathrm{\pi}}}{\cdot ExponentialE;}^{\frac{{\left(x\mathrm{\mu}\right)}^{2}}{2\cdot {\mathrm{\sigma}}^{2}}}$
where $\mathrm{\μ}$ is its mean and $\mathrm{\σ}$ its standard deviation.
For any distribution $X$, the mean, denoted $\mathrm{\μ}$, is the expected value of X. The variance , ${\mathrm{\σ}}^{2}$, is the expected value of the square of the difference between the value of the X and its mean. The square root of the variance, $\mathrm{\σ}$, is called the standard deviation.
If the mean $\mathrm{\mu}$ = 0, and standard deviation ${\mathrm{\sigma}}^{}\=1$, the distribution is called the standard normal distribution.
