stat basic definitions

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Statistical Inference

Experiment

Experimental (or Sampling) Unit

Population

Sample

Parameter

Statistic

Sampling Distribution

Estimate

Estimator

Estimation

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Statistical Inference

Statistical Inference makes use of information from a sample to draw conclusions (inferences)about the population from which the sample was taken.

Experiment

An experiment is any process or study which results in the collection of data, the outcome ofwhich is unknown. In statistics, the term is usually restricted to situations in which theresearcher has control over some of the conditions under which the experiment takes place.

ExampleBefore introducing a new drug treatment to reduce high blood pressure, the manufacturercarries out an experiment to compare the effectiveness of the new drug with that of onecurrently prescribed. Newly diagnosed subjects are recruited from a group of local general

practices. Half of them are chosen at random to receive the new drug, the remainder receivingthe present one. So, the researcher has control over the type of subject recruited and the wayin which they are allocated to treatment.

Experimental (or Sampling) Unit

A unit is a person, animal, plant or thing which is actually studied by a researcher; the basicobjects upon which the study or experiment is carried out. For example, a person; a monkey; asample of soil; a pot of seedlings; a postcode area; a doctor's practice.

Population

A population is any entire collection of people, animals, plants or things from which we maycollect data. It is the entire group we are interested in, which we wish to describe or drawconclusions about.

In order to make any generalisations about a population, a sample, that is meant to berepresentative of the population, is often studied. For each population there are many possiblesamples. A sample statistic gives information about a corresponding population parameter. Forexample, the sample mean for a set of data would give information about the overall population

mean.

It is important that the investigator carefully and completely defines the population beforecollecting the sample, including a description of the members to be included.
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ExampleThe population for a study of infant health might be all children born in the UK in the 1980's.The sample might be all babies born on 7th May in any of the years.

Sample

A sample is a group of units selected from a larger group (the population). By studying thesample it is hoped to draw valid conclusions about the larger group.

A sample is generally selected for study because the population is too large to study in itsentirety. The sample should be representative of the general population. This is often bestachieved by random sampling. Also, before collecting the sample, it is important that theresearcher carefully and completely defines the population, including a description of themembers to be included.

ExampleThe population for a study of infant health might be all children born in the UK in the 1980's.

The sample might be all babies born on 7th May in any of the years.

Parameter

A parameter is a value, usually unknown (and which therefore has to be estimated), used torepresent a certain population characteristic. For example, the population mean is a parameterthat is often used to indicate the average value of a quantity.

Within a population, a parameter is a fixed value which does not vary. Each sample drawn fromthe population has its own value of any statistic that is used to estimate this parameter. For

example, the mean of the data in a sample is used to give information about the overall meanin the population from which that sample was drawn.

Parameters are often assigned Greek letters (e.g. ), whereas statistics are assignedRoman letters (e.g. s).

Statistic

A statistic is a quantity that is calculated from a sample of data. It is used to give informationabout unknown values in the corresponding population. For example, the average of the data in

a sample is used to give information about the overall average in the population from which thatsample was drawn.

It is possible to draw more than one sample from the same population and the value of astatistic will in general vary from sample to sample. For example, the average value in asample is a statistic. The average values in more than one sample, drawn from the samepopulation, will not necessarily be equal.

Statistics are often assigned Roman letters (e.g. m and s), whereas the equivalent unknown

values in the population (parameters ) are assigned Greek letters (e.g. and ).

Sampling Distribution


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The sampling distribution describes probabilities associated with a statistic when a randomsample is drawn from a population.

The sampling distribution is theprobability distributionorprobability density functionof thestatistic.

Derivation of the sampling distribution is the first step in calculating a confidence interval orcarrying out a hypothesis test for a parameter.

ExampleSuppose that x1, ......., xn are a simple random sample from a normally distributed population

with expected value and known variance .

Then the sample mean is a statistic used to give information about the population parameter

; is normally distributed with expected value and variance /n.

Estimate

An estimate is an indication of the value of an unknown quantity based on observed data.

More formally, an estimate is the particular value of an estimator that is obtained from aparticular sample of data and used to indicate the value of a parameter.

ExampleSuppose the manager of a shop wanted to know the mean expenditure of customers in hershop in the last year. She could calculate the average expenditure of the hundreds (or perhapsthousands) of customers who bought goods in her shop, that is, the population mean. Insteadshe could use an estimate of this population mean by calculating the mean of a representativesample of customers. If this value was found to be 25, then 25 would be her estimate.

Estimator

An estimator is any quantity calculated from the sample data which is used to give information

about an unknown quantity in the population. For example, the sample mean is an estimator ofthe population mean.

Estimators of population parameters are sometimes distinguished from the true value by usingthe symbol 'hat'. For example,

= true population standard deviation

= estimated (from a sample) population standard deviation

Example
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The usual estimator of the population mean is

where n is the size of the sample and X1, X2, X3, ......., Xn are the values of the sample.

If the value of the estimator in a particular sample is found to be 5, then 5 is the estimate of thepopulation mean .

Estimation

Estimation is the process by which sample data are used to indicate the value of an unknownquantity in a population.

Results of estimation can be expressed as a single value, known as a point estimate, or arange of values, known as a confidence interval.

By P. B. Moradiya .________________________________________________________________
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