Properties of the normal curve

Bell-shaped and symmetrical about \(x=\mu\).

Maximum value of the curve occurs when \(x=\mu\), therefore mode \(=\) median \(=\) mean \(=\mu\).

Points of inflexion (non-stationary) are at \(x=\mu-\sigma\) and at \(x=\mu+\sigma\).

Probability that the normal random variable \(X\) is within \(1\) standard deviation of the mean, i.e.
\[P(|X-\mu|<\sigma)=0.683\]
or
\[P(\mu-\sigma<X<\mu+\sigma)=0.683\].

Also, we have \(P(|X-\mu|<2\sigma)=0.954\) and \(P(|X-\mu|<3\sigma)=0.998\).

The range of the distribution is therefore approximately \(6\) standard deviations.

The area under the curve from \(x=a\) to \(x=b\) represents the probability \(P(a \leq x \leq b)\).