Normal Distribution

The normal distribution is expressed mathematically as $f(x)=\dfrac{1}{\sqrt{2\pi}\sigma}e^{-\dfrac{1}{2}\left( \dfrac{x-\mu}{\sigma}\right)^{2}}$ $-\infty$ <$x$,$\mu$ < $+\infty$, $\sigma$ > 0.The function $f(x)$ is called probability density function. By taking $Z = \dfrac{X-\mu}{\sigma}$ in formula of Normal distribution, we have $f(z)=\dfrac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}z^{2}}$ $-\infty$ <$z$ < $+\infty$. The function $f(z)$ is called Standard Normal Distribution. The graph of $f(z)$ is called standard normal curve. Computing probability of an event in case of Normal Probability model is explained in notes provided here. The area under the standard normal curve represent corresponding ,Probability (Click here)(proportion or percentage in frequency approach) of observing random variable lying in interval. (Note: This PDF document contains embedded video demonstrating area properties, read a statistical table. You must have latest version of pdf reader to view these videos in file.) For a normal distribution with mean $\mu$ and standard deviation $\sigma$, approximately

1.  68.27$\%$ of the population values lie within one standard deviation ($\pm1\sigma$) of the mean, 
2.  95.45$\%$ of the population values lie within two standard deviation ($\pm2\sigma$) of the mean, and 
3.  99.73$\%$ of the population values lie within three standard deviation ($\pm3\sigma$) of the mean.

The following Application explains Area properties of Normal distribution. You can select markers on $x-$Axis and move right or left. The change in area is shown numerically. First Select Icon with + sign (see right corner) and move the graph in center region, now select Arrow point Icon (see upper left corenr) and go to $x-$axis. Try this!!! (If you cannot see in Mobile Version send your comments. Thank you)