After measurement, we assign an estimated value to a measurand as well as an accompanying uncertainty. The uncertainty is usually expressed as an interval around the estimated value. With any such interval we associate a probability that the actual or true value of the measurand falls within that interval. Measurands are usually continuous quantities such as temperature, voltage and time. However, when discussing probabilities in the context of measurement it is convenient first to consider ‘experiments’ in which the outcomes are discrete, for example tossing a coin, where the outcome is a head or a tail.

Distribution of scores when tossing coins or dice

A fair coin falls heads up with probability 1/2 and tails up also with probability 1/2. A fair coin is an idealised object (since all real coins have a slight bias towards either heads or tails) and presents the simplest case of a ‘uniform’ probability distribution. When a probability distribution is uniform, the possible outcomes of an experiment (tossing a coin in this case) occur with equal probability. We will show how non-uniform probabilities emerge as soon as two or more fair coins are considered. These non-uniformities tend to a characteristic pattern called a Gaussian (or ‘normal’) probability density distribution. For the sake of brevity we shall usually refer to the ‘Gaussian probability distribution’ as simply the ‘Gaussian distribution’. Likewise we shall usually refer to the ‘uniform probability density distribution’ as the ‘uniform distribution’.