### Introduction

Mass-produced items should conform to a specification. Usually, a mean is aimed for but due to random errors in the production process a tolerance is set on deviations from the mean. For example if we produce piston rings which have a target mean internal diameter of 45 mm then realistically we expect the diameter to deviate only slightly from this value. The deviations from the mean value are often modelled very well by the
**
normal distribution
**
. Suppose we decide that diameters in the range 44.95 mm to 45.05 mm are acceptable, then what proportion of the output is satisfactory? In this Section we shall see how to use the normal distribution to answer questions like this.

#### Prerequisites

- be familiar with the basic properties of probability
- be familiar with continuous random variables

#### Learning Outcomes

- recognise the shape of the frequency curve for the normal distribution and the standard normal distribution
- calculate probabilities using the standard normal distribution
- recognise key areas under the frequency curve

#### Contents

1 The normal distribution1.1 The central limit theorem

2 The standard normal distribution

3 Probabilities and the standard normal distribution

4 Calculating other probabilities

4.1 Case 1

4.2 Case 2

4.3 Case 3

4.4 Case 4

4.5 Case 5

5 The cumulative distribution function

6 Applications of the normal distribution

7 Probability intervals - standard normal distribution

8 Probability intervals - general normal distribution