# Normal Distributions

The standard normal distribution is the distribution that is normal with μ = 0 and σ = 1. A normal distribution is of a bell shape, bring uni-modal and symmetrical. The mean gives centre of the distribution, while the variance provides the spread of the distribution. The area under the distribution can be regarded as the likeliness that the range of values would be chosen.

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## Standardisation

It is important to standardise a distribution before applying standard normal distribution formulas and using standard distribution probability tables.

## Probability

Probability can be found for a given Z value. The Z is the value on the x-axis of the distribution graph. The probability can not be found for a single Z value, as the area under the graph at one point is 0, and therefore we can only find the probability that a range of values can be found.

This is denoted by:

P(Z < z) = ϕ(z)

Where the probability can be found by looking at the Standard Normal table. Note that the table gives the area to the left of the point z, and that z = 0 is at the centre of the distribution.

**Example:**

The probability that z is less than 1.25.

P(Z < 1.25)

From the table, 89.44%

**Example:**

The probability that z is greater than -2.52.

P(Z > 2.5) = 1 - P(Z < 2.52) = 1 - 0.0059

Therefore, 99.41%

The process can work backwards to find the z value for a given probability. This involves searching for the probability in the Standard Normal tables and finding the corresponding z value.

## Checking the Distribution for Normality

### Histograms

Normality of a data set can be checked using the histogram of the data. The histogram should have the following features:

- Uni-modal, bell-shaped
- Not skewed, symmetrical
- No outliers

### Quantile Plot

Normality of data can be checked using a quantile plot. The data points should follow a diagonal straight line to be considered normal.

## End

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