Normal distribution 1. Learn about the properties of a normal distribution 2. Solve problems using tables of the normal distribution or your calculator.

The Normal Distribution (Bell Curve) Average contents 50 Mean = μ = 50 Standard deviation = σ = 5

The normal distribution is a theoretical probability * the area under the curve adds up to one*

A standard normal distribution is a theoretical model of the whole population. It is perfectly symmetrical about the central value; the mean μ represented by zero.

As well as the mean the standard deviation (σ) must also be known. The X axis is divided up into deviations from the mean. Below the shaded area is one deviation from the mean.

Two standard deviations from the mean

Three standard deviations from the mean

A handy estimate – known as the Imperial Rule for a set of normal data: 68% of data will fall within 1σ of the μ P(<z<1)=0.683=68.3%

95% of data fits within 2σ of the μ P(-2<z<2)=0.954=95.4%

99.7% of data fits within 3σ of the μ P(-3<z<3)=0.997=99.7%

Using the standard normal distribution There is a standard normal distribution, such that the mean = 0 and the standard deviation = 1. 0 We can convert any other distribution to the standard one. For now, we´ll practise using the standard normal distribution, which we call ´Z´

Using the standard normal distribution Formula for P(Z < x) Formula for P(Z < x) Again, life is too short. We look up values in a table or, even better, use your graphics calculator.

Using the standard normal distribution

Use the standard normal distribution – Draw a diagram 1) Find P(0 < Z < 0.5) 2) Find P(-0.6 < Z < 1.5) 1) Find P(0 < Z < 0.5) 2) Find P(-0.6 < Z < 1.5) 0 0

Use the standard normal distribution – Draw a diagram 3) Find P(Z > 0.563) 4) find P(Z 0.563) 4) find P(Z < -1.54) 0 0

Using your graphics calculator To put your calculator in the correct mode press To put your calculator in the correct mode press Menu  2 (Stat)  F5 (Dist)  F1 (Norm)  F2 (Ncd) If you have a blue calculator press F2 (Var).

Using your calculator Use the arrows keys to insert your Lower and Upper bounds. Use the arrows keys to insert your Lower and Upper bounds. Make sure the s.d (σ) = 1 and mean ( μ) = 0 Make sure the s.d (σ) = 1 and mean ( μ) = 0

Using your calculator If you do not have a lower bound put in a large negative number (-1000) like example 4 If you do not have a lower bound put in a large negative number (-1000) like example 4 If you don not have an upper bound put in a large positive number (1000) like example 3 If you don not have an upper bound put in a large positive number (1000) like example 3