# what do the inflection points on a normal distribution represent? where do they occur?

## Question:

what do the inflection points on a normal distribution represent? where do they occur?

## Normal distribution

The normal distribution is symmetric distribution. It has a bell-shaped curve with top of the bell at mean value. A random variable X is said to follow normal distribution with parameters {eq}{\mu _X}\,and\,\,\,\sigma _X^2 {/eq} if it's probability distribution function is given by

{eq}f\left( {x\,;{\mu _X},\sigma _X^2} \right) = \dfrac{1}{{{\sigma _X}\sqrt {2\pi } }}\exp \left( {\dfrac{{ - 1}}{2}{{\left( {\dfrac{{x - {\mu _X}}}{{{\sigma _x}}}} \right)}^2}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \infty < x < \infty {/eq}

Inflection point of a curve is the point on the curve where it changes concavity i.e. either it changes from concave up to concave down or changes from concave down to concave up.

The inflection points of the normal distribution occur at points {eq}{\rm{x = \mu + \sigma }}\,\,{\rm{and}}\,\,{\rm{x = \mu - \sigma }} {/eq}.

i.e. the inflection points occur at {eq}\pm {/eq} 1 standard deviation of the mean of the normal distribution.

The graph of the normal curve changes its concavity at these points.