Every probability distribution has a probability density function, in terms of which it is usually defined. The probability density function for the normal distribution isfor the uniform distribution it isfor values betweenandand zero outside this interval. Both of these examples are continuous distributions, where any value in an interval may occur, though there are many […]

## The Uniform Distribution

When we throw a fair dice the scores 1, 2, 3, 4, 5, 6 are all equally likely. No other scores are possible. In a situation where each outcome is equally likely we are said to have a uniform distribution- all the possible outcomes are equally likely. It is important to know that the outcome […]

## Regression Lines

Many variables are related by a straight line, or more strictly, over a certain range for both variables the relationship between them can be expressed as a straight line. Often we have to find the equation of the line from a table of raw data. This may be a a table of temperatures at certain […]

## Skewness

The normal distribution is symmetrical – small values are as likely as large values. The probability distribution for the normal distribution is symmetrical about the mean. This is not the case for most distributions however. Most probability distributions are unsymmetrical and some are extremely so. The two graphs below are both positively skewed – the […]

## Discussion of the Correlation Coefficient

If two quantities are related, there will probably be a correlation between them. An increase in one may cause an increase in the other, so that the two quantities are positively correlated, or an increase in one may cause a decrease in the other, so that the two quantities are negatively correlated. However, two quantities […]

## The Correlation Coefficient

Regression coefficients tell us how good a fit our data is to a straight line. The regression coefficient is a number between -1 and 1. -1 means a perfect relationship. All the points are on a straight line, and the graph slopes down. 1 means a perfect relationship. All the points are on a straight […]

## Arrangements of Objects in Circles

If we have three objects, labelled A, B and C, there are six distinct ways thes objects can be arranged in a line: ABC, ACB, BAC, BCA, CAB and CBA If these objects are arranged in a circle, we may be only interested in their relative seating positions, which is next to/to the left of/to […]

## Boxplots

Boxplots are used to visually represent a data set consisting of a list of numbers. It displays the lower and upper quartiles, the median, and any outliers or extreme values. Suppose we have an ordered list of numbers representing the lengths, in cm, of some worms: 2, 3, 7, 7, 12, 14, 16, 17, 17, […]

## The Probability Distribution for the Score on Two Dice

If two fair dice are thrown, we can draw up the table below to represent the possible outcomes. The score shown in the body of the table is obtained by adding the scores on the two dice. Dice 2 Dice 1 + 1 2 3 4 5 6 1 2 3 4 5 6 7 […]

## Hypothesis Testing With The Binomial Distribution – One Tailed Tests

If a manufacturer claims superiority for any of their products or a great deal rests the proportion of components that exceed a certain lifetime, then that claim or proportion probably needs to be tested for legitimacy or accuracy. We can test a wide range of statistical parameters or models by subjecting it to a hypothesis […]