Sample

A sample is a smaller, manageable version of a larger group. It is a subset containing the characteristics of a larger population. Samples are used in statistical testing when population sizes are too large for the test to include all possible members or observations. A sample should represent the whole population and not reflect bias toward a specific attribute.

In basic terms, a population is the total number of individuals, animals, items, observation, data, etc. of any given subject. For example, as of 2017, the population of the world was 7.5 billion of which 49.6% were female and 50.4% were male. The total number of people in any given country can also be a population size. The total number of students in a city can be taken as a population, and the total number of dogs in a city is also a population size. Scientists, researchers, marketers, academicians, and any related or interested party trying to draw data from a group will find that a population size may be too large to monitor. Consider a team of academic researchers that want to, say, know the number of students that studied for less than 40 hours for the CFA exam in 2016 and still passed. Since more than 200,000 people globally take the exam each year, reaching out to each and every exam participant might be extremely tedious and time consuming. In fact, by the time the data from the population has been collected and analyzed, a couple of years would have passed, making the analysis worthless since a new population would have emerged.

A sample is an unbiased number of observations that is taken from a population. In other words, it is a portion, part, or fraction of a whole group. Following our CFA exam example, the researchers could take a sample of 1,000 CFA participants from the total 200,000 test sitters and run the required data on this number. The mean of this sample would be taken to estimate the average of CFA exam takers that passed even though they only studied for less than 40 hours.

The sample group taken should not be biased. This means that if the sample mean of the 1,000 CFA exam participants is 50, the population mean of the 200,000 test takers should also be approximately 50. In order to achieve an unbiased sample, the selection has to be random so that everyone from the population has an equal and likely chance of being added to the sample group. This is similar to a lottery draw and is the basis for simple random sampling.

Simple random sampling is ideal if every entity in the population is identical. If the researchers don’t care whether their sample subjects are all male or all female or a combination of both sexes in some form, the simple random sampling may be a good selection technique. However, in a case where it will be insightful to know the ratio of men to women that passed the test after studying for less than 40 hours, using a stratified random sample would be preferable to a simple random sample.

From a population of 200,000 test takers that sat for the exam in 2016, 40% were women and 60% were men. The random sample drawn from the population should therefore, have 400 women and 600 men for a total of 1,000 test takers. With a stratified random sample, these fractions are a mini-representation of the population and simulates the population’s characteristics better than a simple random sample.

What if age was an important factor that the researchers would like to include in their data? Using the stratified random sampling technique, they could create layers or strata for each age group. The selection from each strata would have to be random so that everyone in the bracket has a likely chance of being included in the sample. For example, two participants, Alex and David, are 22 and 24 years old, respectively. The sample selection cannot pick one over the other based on some preferential mechanism. They both should have an equal chance of being selected from their age group. The strata could look something like:

From the table, the population has been divided into age groups. For example, 30,000 people within the age range of 20 to 24 years old took the CFA exam in 2016. Using this same proportion, the sample group will have (30,000 ÷ 200,000) x 1,000 = 150 test takers that fall within this group. Alex, or David, or both, or neither may be included among the 150 random sample of exam participants.

There are many more stratas that could be compiled when deciding on a sample size. Some researchers might populate the job functions, countries, marital status, etc. of the test takers when deciding how to create sample.