In statistics, a sample is part of a population. The sample is carefully chosen. It should represent the whole population fairly, without bias.
When treated as a data set, a sample is often represented by capital letters such as and , with its elements being represented in lowercase (e.g., ), and the sample size being represented by the letter .
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 with sampling. Sampling is about choosing which data to take for later analysis. As an example, suppose the pollution of a lake should be analysed for a study. Depending on where the samples of water were taken, the studies can have different results.
As a general rule, samples need to be random. This means the chance or probability of selecting one individual is the same as the chance of selecting any other individual.
In practice, random samples are always taken by means of a well-defined procedure. A procedure is a set of rules, 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 an electionpoll. 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 completely neutral sample is never possible. In such cases a statistician will think about how to measure the amount of bias, and there are ways to estimate this.
A similar situation occurs when scientists measure a physical property, say the weight of a piece of metal, or the speed of light. If we weigh an object with sensitive equipment we will get minutely different results. No system of measurement is ever perfect. We get a series of estimates, each one being a measurement. These are samples, with a certain degree of error. Statistics is designed to describe error, and carry out analysis on this kind of data.
There are different kinds of samples:
A complete sample includes all the elements that have a given property.
An unbiased or representative sample is 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 called stratified sampling. Stratified sampling is also known as stratified random sample. Stratified sampling is often represented as proportion, such as percent (%).
Suppose an experiment set out to sample the incomes of adults. Obviously, the incomes of college graduates might 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 with variation. Here larger samples are taken from the more variable sub-populations so that the summary statistics such as the means and standard deviations, are more reliable.
Sampling bias
It happens when the way a sample is chosen makes the sample different from the population it is meant to represent. This can make the results of a study misleading.
A population is the full group that researchers want to learn about. A sample is a smaller group taken from that population. Researchers often use samples because studying every member of a population may take too much time or money.
Sampling bias can happen when some parts of the population have little or no chance of being included. It can also happen when people choose for themselves whether to take part. A sample can be large and still be biased.
Example
Suppose a town wants to know how much exercise its people get each week. Researchers ask 500 people in a large park.
This is a large sample. However, it may stil be biased. People who visit a park may exercise more than other people on the town. People who rarely leave their homes may have little chance of being asked.
The results may therefore make the town seem more active than it really is.
Undercoverage
Undercoverage happens when some members of the population have little or no chance of being selected.
For example, an online survey of all adults may miss people who do not use the Internet. A telephone survey may miss people who cannot easily be reached by telephone.
The United States Census Bureau describes undercoverage as a situation where people or housing units do not have a chance of being selected in a sample.
Convenience sampling
Sometimes researchers choose people because they are easy to reach. This is called a convenience sample.
For example, a researcher may ask the first 100 people who enter a shopping center. These people may not represent everyone in the city.
Convenience sampling is often easier and cheaper than probability-based sampling. However, it can produce sampling bias.
Self-selection
Sampling bias can also happen when people decide for themselves whether to join a study.
For example, a website may ask visitors to vote in an online poll. People with strong opinions may be more likely to answer. People who do not care much about the question may be less likely to answer.
Because of this, the people who answer may be different from the full population.
Non-response
A related problem is non-response bias. It happens when people who answer a survey are different from people who do not answer it.
For example, a survey may select people correctly at first. However, many selected people may refuse to answer. If the people who answer are different from those who do not answer, the final results may be biased.
Sampling bias and non-response bias are closely related, but they are not always treated as the same problem.
Sampling bias and sampling error
Sampling bias is differant from random sampling error.
Sampling error happens because a sample contains just part of a population. Two properly chosen samples may give slightly different results by chance.
A larger sample usually reduces this kind of random error. However, making a biased sample larger does not necessarily remove the bias.
For example, asking 10,000 people in a park instead of 500 people does not solve the problem in the example above. The sample may still include too many people who exercise often.
Reducing sampling bias
Researchers can reduce sampling bias by planning carefully how a sample will be chosen.
One useful way is a random sample. In a simple random sample, each member of the population has the same chance of being selected.
Researchers can also try to make sure that important parts of the population are included. In some studies, the population is divided to groups and people are sampled from aech group. This is called stratified sampling.
Some surveys also use statistical weights.
A person from a group that is less likely to be selected may be given more weight in the final calculation. The Census Bureau uses weighting to adjust survey estimates for differences in selection and non-response.
Good sampling does not guarantee that a study will have no errors. However, it can make the sample more representative of the population.
