A-level Mathematics/OCR/S2/Normal Distribution

Normal distributions of varying means and standard deviations. The green line is the standard normal distribution.

A collection of data can be modelled with the Normal Distribution if the data produces a fairly bell-shaped curve when put into a histogram and has the same mean, median and mode - that is to say most of the data is in the middle of the diagram, with the bins becoming smaller toward the outside. If a sample of data is truly normally distributed, about 68% of the data will fall within one standard deviation, 95% within two standard deviations, and 99.7% within 3 standard deviations of the mean.

The normal distribution's probability density function is: ${\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}$

The standard normal distribution is a special case in which the mean, μ, is 0 and the standard deviation, σ, is 1. Its p.d.f is: ${\frac {1}{\sqrt {2\pi }}}\,e^{-{\frac {x^{2}}{2}}}$

Integrating this function in order to obtain probabilities is a trifle difficult if you're an S2 candidate, so it is usual to use calculator functions or tables of cumulative probabilities (these give probabilities from -∞ to a particular value).