Statistical Population Sample

A Statistical Population Sample is a proper subset of a statistical population from which a dataset is collected or selected for statistical analysis.

• AKA: Statistical Sample, Data Sample.
• Context:
  • It is generally denoted as a sequence of [math]\displaystyle{ n }[/math] observed values or measurements [math]\displaystyle{ \{x_1,x_2,...,x_n\} }[/math] of the random variable [math]\displaystyle{ X }[/math]. It can also be denoted as sequence of [math]\displaystyle{ n }[/math] random variables [math]\displaystyle{ \{X_1,X_2, \cdots, X_n\} }[/math] arising from a random sample, the integer number [math]\displaystyle{ n }[/math] is the sample size.
  • It can be produced by a Sampling Task.
  • It can range from being a Non-Random Sample to being a Random Sample.
  • It can range from being an Unbiased Sample to being a Biased Sample.
  • It can range from being an Univariate Sample to being a Multivariate Sample.
  • It can range from being an Independent Sample to being a Paired Sample.
  • It is characterized by a statistic value.
• Example(s):
  • A set of scores measured for each statistical population or experiment treatment.
  • A set of 500 coin flips recorded for a probability study.
  • the planetary systems detected by the space observatory Kepler.
  • Simple Random Sample.
  • Complete Sample.
• Counter-Example(s):
  • Statistical Population.
  • Sample Space.
  • Statistic Function.
  • Sample Size.
  • Experimental Effect.
  • Experiment Treatment.
• See: Sample Space, Statistical Population, Sample Mean, Statistical Hypothesis Testing Task, Sampling Algorithm, Multiset Function, Quality Assurance, Statistical Survey.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Sample_(statistics)
  • In statistics and quantitative research methodology, a data sample is a set of data collected and/or selected from a statistical population by a defined procedure. The elements of a sample are known as sample points, sampling units or observations.

Typically, the population is very large, making a census or a complete enumeration of all the values in the population is either impractical or impossible. The sample usually represents a subset of manageable size. Samples are collected and statistics are calculated from the samples so that one can make inferences or extrapolations from the sample to the population. The data sample may be drawn from a population without replacement, in which case it is a subset of a population; or with replacement, in which case it is a multisubset.

• (Leard Statistics, 2016) ⇒ "Hypothesis Testing - Structure and the Research, Null and Alternative Hypothesis" Laerd Statistics, © 2013 Lund Research Ltd, n.d. Web. Retrieved October 11, 2016, from http://statistics.laerd.com/statistical-guides/hypothesis-testing.php
  • QUOTE: (...) In statistics terminology, the students in the study are the sample and the larger group they represent (i.e., all statistics students on a graduate management degree) is called the population. Given that the sample of statistics students in the study are representative of a larger population of statistics students, you can use hypothesis testing to understand whether any differences or effects discovered in the study exist in the population. In layman's terms, hypothesis testing is used to establish whether a research hypothesis extends beyond those individuals examined in a single study.

Another example could be taking a sample of 200 breast cancer sufferers in order to test a new drug that is designed to eradicate this type of cancer. As much as you are interested in helping these specific 200 cancer sufferers, your real goal is to establish that the drug works in the population (i.e., all breast cancer sufferers).

As such, by taking a hypothesis testing approach, Sarah and Mike want to generalize their results to a population rather than just the students in their sample. However, in order to use hypothesis testing, you need to re-state your research hypothesis as a null and alternative hypothesis

• (Stat Trek, 2016) ⇒ "Populations and Samples", © 2011, Encyclopedia of Mathematics] Retrieved October 11, 2016, from http://stattrek.com/sampling/populations-and-samples.aspx
  • QUOTE: The main difference between a population and sample has to do with how observations are assigned to the data set. A population includes all of the elements from a set of data. A sample consists of one or more observations from the population.

Depending on the sampling method, a sample can have fewer observations than the population, the same number of observations, or more observations. More than one sample can be derived from the same population.

Other differences have to do with nomenclature, notation, and computations. For example, a measurable characteristic of a population, such as a mean or standard deviation, is called a parameter; but a measurable characteristic of a sample is called a statistic.

• (Minitab, 2016) ⇒ http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/introductory-concepts/basic-concepts/sample-and-population/
  • QUOTE: A sample is a subset of people, items, or events from a larger population that you collect and analyze to make inferences. To represent the population well, a sample should be randomly collected and adequately large.

To understand the basic foundation for hypothesis testing and other types of inferential statistics, it’s important to understand how a sample and a population differ.

A population is a collection of people, items, or events about which you want to make inferences. It is not always convenient or possible to examine every member of an entire population. For example, it is not practical to count the bruises on all apples picked at an orchard. It is possible, however, to count the bruises on a set of apples taken from that population. This subset of the population is called a sample.

If the sample is random and large enough, you can use the information collected from the sample to make inferences about the population. For example, you could count the number of apples with bruises in a random sample and then use a hypothesis test to estimate the percentage of all the apples that have bruises.

• (StatGuide,2016) ⇒ Statistical Analysis Glossary: http://www.quality-control-plan.com/StatGuide/sg_glos.htm
  • QUOTE: The population is the universe of all the objects from which a sample could be drawn for an experiment. If a representative random sample is chosen, the results of the experiment should be generalizable to the population from which the sample was drawn, but not necessarily to a larger population. For example, the results of medical studies on males may not be generalizable for females.

• (Encycplopedia of Mathematics, 2016) ⇒ Sample Method. © 2011, Encyclopedia of Mathematics Retrieved October 11, 2016, from https://www.encyclopediaofmath.org/index.php/Sample_method
  • QUOTE: A statistical method for the study of the general properties of a certain population of objects by studying the properties of only a sample (a part) of these objects. The mathematical theory of sample methods is based on two important sections of mathematical statistics — the theory of sampling from a finite population and the theory of sampling from an infinite population. The fundamental difference between the sampling theory for finite and infinite populations consists in the fact that in the former case the theory is usually applied to objects of a non-random, determined nature (for example, the number of defective articles in a given industrial batch of products is not a random variable: it is an unknown constant which must be estimated from the sampling data). In the latter case the theory is usually employed to study the properties of random objects (for example, to study the properties of continuously-distributed random experimental errors, each one of which may be interpreted, in principle, as the realization of one out of an infinite set of possible results).

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/sampling_(statistics) Retrieved:2015-7-10.
  • In statistics, quality assurance, and survey methodology, sampling is concerned with the selection of a subset of individuals from within a statistical population to estimate characteristics of the whole population. Each observation measures one or more properties (such as weight, location, color) of observable bodies distinguished as independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design, particularly stratified sampling. Results from probability theory and statistical theory are employed to guide practice. In business and medical research, sampling is widely used for gathering information about a population.

    The sampling process comprises several stages:

    • Defining the population of concern
    • Specifying a sampling frame, a set of items or events possible to measure
    • Specifying a sampling method for selecting items or events from the frame
    • Determining the sample size
    • Implementing the sampling plan
    • Sampling and data collecting
    • Data which can be selected

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