The bNormal distribution/b is an example of a bcontinuous probability distribution/b.

It can be used to model many naturally occurring events, such as:

• heights and weights of a population,
• IQs of people,- variations in manufacturing.

bCharacteristics of the Normal distribution/b

The Normal distribution:

• has parameters (\mu) (the population mean) and (\sigma²⁾ (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, \sigma²⁾⁾

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

bStandard Normal distribution/b

The standard Normal distribution has a population mean ((\mu)) of (0) and a population variance ((\sigma²⁾⁾ 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)).

bFinding an unknown mean or variance/b

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

[b]uNormal distribution[/u]/b

(X∼N(\mu, \sigma²⁾⁾

[b]uStandard Normal distribution[/u]/b

(Z∼N(0, 1))

(Z=\dfrac{X-\mu}{\sigma})