The Normal distribution is an example of a continuous probability distribution.

It can be used to model many naturally occurring events, such as:
- heights and weights of a population,
- IQs of people,- variations in manufacturing.

Characteristics of the Normal distribution

The Normal distribution:

- has parameters \(\mu\) (the population mean) and \(\sigma^2\) (the population variance),
- is symmetrical about the mean (mean = median = mode),
- has a bell shape,
- has asymptotes at each end,
- has a total area under the curve of 1,
- has points of inflection at \(\mu + \sigma\) and \(\mu - \sigma\).

For a Normally distributed variable:
- approximately 68% of the data lies within 1 standard deviation of the mean,
- approximately 95% of the data lies within 2 standard deviations of the mean, and
- approximately 99.7% of the data lies within 3 standard deviations of the mean.

The notation for Normal distribution is: \(X∼N(\mu, \sigma^2)\)

You can use a calculator to do most problems involving the Normal distribution.

Standard Normal distribution

The standard Normal distribution has a population mean (\(\mu\)) of \(0\) and a population variance (\(\sigma^2\)) of \(1\).

For a normally distributed variable \(X∼N(\mu, \sigma^2)\), \(X\) can be coded using the formula \(Z=\dfrac{X-\mu}{\sigma}\) to form the standard Normal variable \(Z∼N(0, 1)\).

Finding an unknown mean or variance

You can find an unknown mean or variance by using the inverse Normal function to calculate the \(Z\) value.