The Normal Distribution

The standard normal curve is given by the following formula:

$$y=\frac{e^{-\frac{1}{2}x^2}}{\sqrt{2\pi}}$$

The graph of this curve is shown below:

Note, it has the following properties...

  • The graph of the normal distribution curve is bell-shaped (unimodal, and symmetric) and continuous.
  • The ''x''-axis is a horizontal asymptote for the curve
  • The total area under the curve is 1
  • The domain is the set of all reals
  • The range is the set of all positive reals

When viewed as a continuous probability density function, we can show that the standard normal curve has a mean of 0, and a standard deviation of 1.

Some additional properties of interest...

  • The curve has a maximum height of $1/\sqrt{2\pi} \doteq 0.399$ at $x=0$.
  • Points of inflection occur at $$\left(\pm 1, \frac{e^{-1/2}}{\sqrt{2\pi}}\right) \doteq (\pm 1, 0.242)$$

By altering the formula for a standard normal curve slightly we build the entire family of normal curves...

Consider the following,

$$y = \frac{e^{-\frac{1}{2} \left(\frac{x-\mu}{\sigma}\right)^2}}{\sigma\sqrt{2\pi}}$$

We have replaced $x$ with $\frac{x - \mu}{\sigma}$ which shifts the original curve to the right $\mu$ units, and horizontally stretches it by a factor of $\sigma$.

The presence of another $\sigma$ in the bottom denominator then vertically compresses the resulting graph by a factor of $\sigma$.

The net effect of which preserves the overall "bell-shape" of the original graph, while moving the mean to $\mu$, and changing the standard deviation to $\sigma$, leaving the area under the curve unchanged. (The area is still 1, since the curve was horizontally stretched and vertically compressed by the same factor.)

We associate this curve with a Normal distribution of mean $\mu$ and standard deviation $\sigma$.