The correlation coefficient measurements the goodness of fit of a set of pointsto a straight line but two quantities may in fact observe a perfect correlation without all the points lying on a straight line. The quantities x and y have a perfect though non linear relationship as shown on the graph below. The relationship […]

## Confidence Intervals for the Normal Distribution

Sometimes it happens that we have have a list of data. We can calculate the mean easily, but the mean is specific to each sample. If we take another sample and calculate a new mean, the new mean and the old mean may be different. We might want to know how reliable our estimate of […]

## The Goodness of Fit or Chi – Squared Distribution

Thetest can only be used the goodness of fit of a data set to a hypothesised probability distribution. The observed data is sorted into frequency classes, and for each frequency class, the expected number of observations that would fall into that frequency class is calculated. The difference between each observed frequency O-i and expected frequency […]

## Sampling Distribution of the Mean of a Sample – The Central Limit Theorem

Supposeis a random variable from a population with meanand standard deviation Supposeis a random sample of size n with meanthen the expected mean of the population of sample means is The expected variance of the population of sample means is If very many samples were taken and the mean of each sample calculated then the […]

## Sampling Distribution for a Selection from a Discrete Distribution

In the UK coins come as 1p, 2p, 5p, 10p, 20p, 50p, £1 or £2 coins. Suppose you have 5 1p coins, 3 2p coins and 2 5p coins. You select three coins and are only interested in the total value of the selection. To find the sampling distribution X for the value of the […]

## Estimating Population Mean and Variance From a Sample

In practice most of statisitics involves taking and analysing samples. The most important statistical measures for any sample are the meanand variance(or the standard deviation). We usually do not know the mean and standard deviation of the population, and use the sample to obtain estimators for these. To estimate these values we take a sampleof […]

## Degrees of Freedom

The number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. A simple example is shown in calculating the mean of a sample. Supposeis a sample from a population. Let Theare estimates of the residualswhereis the true mean of the population. The sum […]

## The Different Treatments of a Transformation of a Random Variable and a Combination of Random Variables

It is often desirable to recalibrate the unites in which a quantity is measured – for example from inches to cm. There are about 2.58 cm in each inch, so a length in cm with be 2.58 times the same length in inches. Ifis the length in cm andis the length in inches, then If […]

## Quota Sampling

Quota sampling is a non – random method of sampling a population when a population separates into several disjoint categories. The Procedure for taking a sample is: 1. Decide on the categories into which a population can be divided 2. Decide on the number to be sampled in each catgory 3. Collect the data from […]

## Simple Random Sampling

Sampling involves observing or testing part of the population. The accuracy of of any results increases with the sample size, so a set level of accuracy requires a certain sample size. For a simple random sample, every member of the population must have an equal chance of being sampled. Random sampling can be carried out […]