# Normal Distribution

This is a continuous probability distribution.

Also known as a **Gaussian Distribution**, bell-shaped curve or bell curve.

A normal distribution is symmetrical at the mean and the two tails never touch the horizontal axis.

A normal distribution can be defined by two numbers: the mean and the standard deviation.

The mean determines its height

The standard deviation determines the width

### Standard Deviation Cut-Offs

Standard Deviations cut off fixed propertions of the normal distribution from the mean to infinity in both directions.

approx 68 of the observation lie within 1 standard deviations of the mean

approx 95 of the observation lie within 2 standard deviations of the mean

approx 99.5% of the distribution lies within 3 standard deviations of the mean.

### Different Shapes

Normal distributions can be very tall or very flat.

The exact shape and size will depend on the mean and the standard deviation.

Analysis is often done with normal distributions to determine probabilities

http://www.statsoft.com/textbook/sttable.html

### Cumulative Distribution Graph

SS

### Excel Functions

__NORM.DIST__ - Returns the probability of getting 'less than' or 'equal to' a particular random variable.__NORM.INV__ - Returns the random variable where the probability of getting 'less than' it is equal to P.__Z.TEST__ -

**NORM.DIST(x, mean, standard_dev, cumulative)**

If you want to know the probability that a variable with a normal distribution falls in the range of 'a' to 'b' use:

NORMDIST(b,mean,stdev,true) - NORMDIST(a,mean,stdev,true)

**NORM.INV(probability, mean, standard_dev)**

sometimes referred to as the Inverse Gaussian Density ??

An individual value from a non standard normal distribution is referred to as x

An individual value from a standard normal distribution is referred to as z.

Example - what score will be cut off for 15%

NORMINV(0.85,0,1)

Tails of a normal distribution are asymptotic, indefinitely decreasing but never touching the axis.

The normal distribution is important in inferential statistics because certain theoretical distributions are very close to being a normal distribution.

### Checking for Normality

There are a number of ways you can check for normality

1) Histogram and compare the shape

2) Normal Probability Plot

3) Chi-Squared goodness of Fit

4) Anova, Skew and Kutosis test

4) Kolmogorov Smirnov test

5) Anderson Darling test

6) Shapiro Wilk test

### Important

Is symmetrical around the mean, median and mode

Not all distributions are normal

It is possible to check that you have a normal distribution by using a version of the Chi-Square test to check.

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