Calculations involving Venn diagrams use the formulae: and if the eventsandare independent,where for example, P(A) indicates the probability of A happening. The best way to illustrate the use of these examples is by way of examples. A Venn diagram can be solved completely with three conditions. Example:andare independent. Sketch the Venn diagram. Ifandare independent, – […]

## Space Diagrams

A space diagram is a table which displays all the possible outcomes. They are usually used for discrete distributions and are especially useful when 2 or more trials are carried out simultaneously and the results must be combined in some way. For example, the phase space diagrams could show all the possible outcomes if we […]

## Cumulative Distribution Functions

A probability distribution tells you the probability of something happening,the probability of the random variabletaking the valueA Cumulative distribution tells you the probability of the random variabletaking a value less than or equal to a valueoften written Given a probability distributionwe can find the cumulative distribution: Ifis presented in a table, add the entries as […]

## Venn Diagrams 1

Venn diagrams are a means to display categories of data graphically. They are more flexible than contingency tables, allowing complex reasoning, and have many applications in set theory. Typically each set is illustrated by a bubble, allowing intersection, with one or more of the other sets, and an intersection simultaneously of all the sets. All […]

## Censuses, Random/Systematic/Stratified/Quota/Sampling

If you need to find peoples’ opinions you need to ask them questions. You could ask everybody. In the United Kingdom there are 60 million people. It is a big task to ask everybody. A task this big can only be done when the issues involved are major, so a census is taken every ten […]

## Statistical Models and Modelling

All statistically models assume randomness. Not complete randomness, so that all possible outcomes are equally likely, but a randomness that makes some outcomes more or less likely. We have to model each situation so that the predictions of the model are in line with the observations. We might use intuition and logic to say that […]

## Introduction to Hypothesis Testing

Hypothesis testing is used to test whether the value of a mean or standard deviation has changed over time, whether the mean or standard deviation of two samples are equal, whether dogs prefer Dogslop or Kanineswill etc. You are testing always a null hypothesis, labelledagainst an alternative hypothesisThe null hypothesis assumes as little as possible […]

## Mutually Exclusive, Independent and Exhaustive Events Illustrated on Venn Diagrams

Mutually exclusive events are events that cannot both happen at the same time: You either like bovril or hate it. You either do what we say or you die! (George Bush’s sentiment). We can illustrate mutually exclusive events A and B on a Venn diagram. The sets A and B have no intersection so are […]

## Mean and Standard Deviation of Lists of Numbers – Bias

To find the mean of a list of numbers, we add the numbers up and divide by how many numbers there are. The standard deviation is a measure of spread – the more spread out the numbers are, the higher the standard deviation will be. The graphs above show two normal distributions corresponding to two […]

## The Binomial Distribution

The name “The Binomial Distribution” is derived from the binomial expansion, because if you use the formula for binomial expansion, you can substituteandand then the probability of obtainingsuccesses inattempts is given by theth term There are three conditions necessary for the binomial to be a possible distribution. is a fixed number. There aretrials or attempts. […]