1. Statistics S2 Year 13 Mathematics

2. 17/04/2015 Unit 1 – The Normal Distribution The normal distribution is one of the most important distributions in statistics. Many measured quantities in the natural sciences follow a normal distribution. It can be a useful approximation to the binomial distribution and the Poisson distribution.

3. 17/04/2015 The Normal Variable X The normal variable X is continuous. Its probability density function f(x) depends on its mean  and standard deviation , where, -  < x <  This is very complicated! Fortunately, you don’t need to know it!

4. 17/04/2015 The Normal Distribution To describe this distribution we write: X  N( ,  2 ) The Normal distribution curve has the following features: –It is bell-shaped –It is symmetrical about  –It extends from -  to +  –The total area under the curve is 1.

5. 17/04/2015 Normal Distribution Curve Approximately 95% of the distribution lies within two standard deviations of the mean. Approximately 99.9% (very nearly all) of the distribution lies between three standard deviations of the mean. The spread of the distribution depends on .

6. 17/04/2015 Finding Probabilities The probability that X lies between a and b is written P(a < X < b). To find this probability, you need to find the area under the normal curve between a and b. One way of finding areas is to integrate, but the normal function is very complicated so tables are used instead.

7. 17/04/2015 The Standard Normal Variable, Z The variable X is standardised so that the mean is always 0 and standard deviation 1. This is so that the same tables can be used for all possible values of  and  2. This standardised normal variable is called Z and Z  N(0, 1). Z is found by

8. 17/04/2015 Using Standard Normal Tables Depending on the tables you have they either give the area under the curve as far as a particular value z or proceeding z. The area as far as is often written  (z). (phi) This area gives the probability that Z is less than a particular value z, so P(Z < z) =  (z). The tables we use give the area proceeding z. It is helpful to draw a bell curve diagram with certain questions.

9. 17/04/2015 Examples 1. Find P(Z > 0.16) 2. Find P(Z > 0.34) 3. Find P(Z < 0.85) 4. Find P(Z < -1.37) 5. Find P(Z > -0.65)