## Normal Distribution

**Before you start:**

__Drawing a diagram__for each step helps you see what you are doingIn normal distribution

**mean, median and mode**are the

**same**and the data is

**symmetrical**

No matter what set of numbers there are, if the data is normally distributed it will take the form of the graph above

The data for the distribution is given in relation to the normal 'N' (The central line)

X - Random variable

N - Normal

~ - Distributed

μ - Mean

σ - Standard deviation

N - Normal

~ - Distributed

μ - Mean

σ - Standard deviation

**How is this data used in probability?**

## Variable Z

The variable Z has a set of

These values are that

This means you have to be able to

As the graph's are the same shape the

To move from one to the other:

**probabilities pre-set**for it's values when**Z ~ N(0, 1)**These values are that

**big list of z and p values**at the back of the S1 part of the__formula book__This means you have to be able to

**move the X values to the Z values**As the graph's are the same shape the

**probability is exactly the same**To move from one to the other:

**E.g.**

After referring to the table putting 1 in as a z value we find:

**P(Z < 1) = 0.8413****P(X < 2.3) = 0.8413**## Using Probability with the Graph

Remember that the probability under the whole graph is always 1

So if

Then

So if

**P(X < 2.3) = 0.8413**Then

**P(X > 2.3) = 0.1587**## Using the Symmetry of the Graph

As the graph is

**perfectly symmetrical**then this means you can manipulate the data to suit the Z values in the table**The table only gives P(Z < z) and values on the right hand side of the normal****E.g.****X ~ N(25, 16)****Remember σ = 4****Find P(X < 23)****P(Z < -1/2)**As it's on the wrong side

**flip it round****P(Z < -1/2) = P(Z > 1/2)**This is on the wrong side however.

As we saw above you need to turn this round

In the end we find

meaning:

meaning:

hence:

As we saw above you need to turn this round

**P(Z > 1/2) = 1 - P(Z < 1/2)**In the end we find

**P(Z < 1/2) = 0.6915**meaning:

**P(Z > 1/2) = 0.3085**meaning:

**P(Z < -1/2) = 0.3085**hence:

**P(X < 23) = 0.3085**## Finding the mean or standard deviation

You may be asked to find the mean or standard deviation

In answering this you'll be given the probability, standard deviation and the z value

E.g.

P(X < 2.3) = 0.7123 with standard deviation of 0.4

Hence

In answering this you'll be given the probability, standard deviation and the z value

E.g.

P(X < 2.3) = 0.7123 with standard deviation of 0.4

Hence

Using the table in the formula book

So:

0.4

μ = 2.3 - 0.224

**0.7123 requires a z value of 0.56**So:

__2.3 - μ__= 0.560.4

μ = 2.3 - 0.224

**μ = 2.006**## Percentage Points

There are exact probabilities you should use to get a special

They are represented on the formula booklet for

**4 decimal z value**They are represented on the formula booklet for

**P(Z > z)**(not like the other table of P(Z < z) )If you get these exact values you

You'll know when to use them as a percentage is used in the question generally

Remember though that they are going the wrong way with there probabilities

On this table

**must use this table**You'll know when to use them as a percentage is used in the question generally

Remember though that they are going the wrong way with there probabilities

On this table

**P(Z > z) = 0.0500**means: