Sampling (statistics)

Border police looking forillegal drugswith a specially trained dog: If they check every tenth car, they are taking an unbiased sample.

For other uses, seeSampling.

Instatistics,asampleis part of apopulation.The sample is carefully chosen. It should represent the whole population fairly, withoutbias.

When treated as a data set, a sample is often represented bycapital letterssuch as$X$and$Y$,with its elements being represented in lowercase (e.g.,$x_{3}$), and thesample sizebeing represented by the letter$n$.

The reason samples are needed is that populations may be so large that counting all the individuals may not be possible or practical. Therefore, solving a problem in statistics usually starts withsampling.Sampling is about choosing whichdatato take for lateranalysis.As an example, suppose thepollutionof alakeshould be analysed for a study. Depending on where the samples ofwaterwere taken, the studies can have different results.

As a general rule, samples need to berandom.This means the chance orprobabilityof selecting one individual is the same as the chance of selecting any other individual.

In practice,random samplesare always taken by means of a well-definedprocedure.A procedure is a set ofrules,a sequence of steps written down and exactly followed. Even so, some bias may remain in the sample. Consider the problem of designing a sample to predict the result of anelectionpoll.All known methods have their problems, and the results of an election are often different from predictions based on a sample. If you collect opinions by using telephones, or by meeting people in the street, you won't ask people who don't answer phone calls or who don't walk on the street. Therefore, in cases like this a completelyneutralsample is never possible.In such cases astatisticianwill think about how to measure the amount of bias, and there are ways toestimatethis.

A similar situation occurs when scientists measure aphysical property,say the weight of a piece of metal, or thespeed of light.If we weigh an object with sensitiveequipmentwe will get minutely different results. No system ofmeasurementis ever perfect. We get a series of estimates, each one being a measurement. These are samples, with a certain degree oferror.Statistics is designed to describe error, and carry out analysis on this kind of data.

There are different kinds of samples:

• Acomplete sampleincludes all the elements that have a given property.
• Anunbiasedorrepresentative sampleis produced by taking a complete sample and selecting elements from it, in a process that does not depend on the properties of the elements.

The way the sampling is obtained, along with the sample size, will have an impact on how the data is viewed.

Stratified sampling

If a population has obvious sub-populations, then each of the sub-populations needs to be sampled. This is calledstratified sampling.Stratified sampling is also known asstratified random sample.Stratified sampling is often represented asproportion,such as percent (%).

Suppose anexperimentset out to sample theincomesof adults. Obviously, theincomesof collegegraduatesmight differ from that of non-graduates. Now suppose the number of male graduates was 30% of the total male adults (imaginary figures). Then you would arrange for 30% of the total sample to be male graduates picked at random, and 70% of the total to be male non-graduates. Repeat the process for females, because the percentage of female graduates is different from males. That gives a sample of the adult population stratified by sex and college education. The next step would be to divide each of your sub-populations by age groups, because (for example) graduates might gain more income relative to non-graduates in middle age.

Another type of stratified sample deals withvariation.Here larger samples are taken from the more variable sub-populations so that the summary statistics such as themeansandstandard deviations,are more reliable.

Related pages

• Mean (statistics)
• Standard deviation