# Engineering Tables/Normal Distribution

The normal distibution is an extremely important family of continuous probability distributions. It has applications in every engineering discipline. Each member of the family may be defined by two parameters, location and scale: the mean ("average", μ) and variance (standard deviation squared, σ2) respectively.

The probability density function, or pdf, of the distribution is given by:

${\displaystyle \varphi _{\mu ,\sigma ^{2}}(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right).}$

The cumulative distribution function, or cdf, of the normal distribution is:

${\displaystyle \Phi _{\mu ,\sigma ^{2}}(x)=\int _{-\infty }^{x}\varphi _{\mu ,\sigma ^{2}}(u)\,du={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{x}\exp {\Bigl (}-{\frac {(u-\mu )^{2}}{2\sigma ^{2}}}\ {\Bigr )}\,du}$

These functions are often impractical to evaluate quickly, and therefore tables of values are used to allow fast look-up of required data. The family of normal distributions is infinite in size, but all can be "normalised" to the case with mean of 0 and SD of 1:

Given a normal distribution ${\displaystyle X\sim N(\mu ,\sigma ^{2})\,}$, the standardised normal distribution, Z, is:
${\displaystyle Z={\frac {X-\mu }{\sigma }}\!\sim N(0,1)}$

Due to this relationship, all tables refer to the standardised distribution, Z.