S1.6 - The Normal distribution

A Normal distribution is a continuous, bell-shaped model that is symmetric about its mean $\mu$. Write $X\sim N(\mu,\sigma^2)$, where $\sigma^2$ is the variance and $\sigma>0$ is the standard deviation. Symmetry makes the mean, median and mode coincide, and probabilities are areas under the curve.

Here $\Phi(z)=P(Z\le z)$ is read from the cumulative Normal table or calculator. For $X\sim N(210,25^2)$,$$P(190<X<240)=\Phi(1.2)-\Phi(-0.8)=0.8849-0.2119=0.6730.$$Draw the event mentally before using the table: a right tail needs a complement, while an interval needs a difference.

For an unknown boundary or parameter, turn the stated cumulative probability into its $z$-value and use $x=\mu+z\sigma$. If $P(X<152)=0.05$ and $P(X<180)=0.60$, then$$152=\mu-1.6449\sigma,\qquad180=\mu+0.2533\sigma.$$Subtracting gives $\sigma=28/1.8982\approx14.8$, then $\mu\approx176$. The same method finds percentiles, quartiles and symmetric limits.

After finding a single-observation probability $p$, ordinary probability rules still apply. For independent observations, use products or a complement such as $1-(1-p)^n$. For a conditional event, divide the joint probability by the probability of the condition; the Normal table supplies the component areas, not the final conditional ratio.