# Random Sample

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A random sample is a statistical sample in which all elements are independent and identically distributed.

**AKA:**Random Experiment Outcome Multiset.**Context:**- It can also be defined as a statistical sample in which every member of the respective population has an equal chance of being selected.
- It can be produced by a Random Sampling Task (or random draws).
- It can be described with a Random Sample Statistic Function.
- It can range from being a Small Sample (such as a random singleton) to being a Large Sample.
- It can range from being a Random Sample with Replacement to being a Random Sample without Replacement.

**Example(s):**- a Multiset of [math]\displaystyle{ n }[/math] Random Experiment Outcomes
- ?? Two-Sample Random Experiment.
- {Hx3, Tx5}, a from a Coin Toss Experiment.

- …

- a Multiset of [math]\displaystyle{ n }[/math] Random Experiment Outcomes
**Counter-Example(s):**- a Nonrandom Sample, such as a biased sample.
- an Sample Space, such as:
`{(Ace,Diamond)x2, (3,Heart)x1, (3,Diadmond)x1, (3,Spade)x1},`

card draw experiment.

**See:**Random Function, Random Trial, Statistical Inference, IID Random Variable Set.

## References

### 2016

- (StatGuide, 2016) ⇒ Statistical Analysis Glossary https://www.quality-control-plan.com/StatGuide/sg_glos.htm
- QUOTE: A random sample of size N is a collection of N objects that are independent and identically distributed. In a random sample, each member of the population has an equal chance of becoming part of the sample.

### 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

- In statistics, quality assurance, and survey methodology,

### 2011

- (Wikipedia - Random Sample, 2009) http://en.wikipedia.org/wiki/Random_sample
- A sample is a subject chosen from a population for investigation. A
**random sample**is one chosen by a method involving an unpredictable component. Random sampling can also refer to taking a number of independent observations from the same probability distribution, without involving any real population. The sample usually is not a representative of the population from which it was drawn— this random variation in the results is termed as sampling error. In the case of random samples, mathematical theory is available to assess the sampling error. Thus, estimates obtained from random samples can be accompanied by measures of the uncertainty associated with the estimate. This can take the form of a standard error, or if the sample is large enough for the central limit theorem to take effect, confidence intervals may be calculated. [Types of random samples include]:- A simple random sample is selected so that all samples of the same size have an equal chance of being selected from the population.
- A self-weighting sample, also known as an EPSEM (Equal Probability of Selection Method) sample, is one in which every individual, or object, in the population of interest has an equal opportunity of being selected for the sample. Simple random samples are self-weighting.
- Stratified sampling involves selecting independent samples from a number of subpopulations, group or strata within the population. Great gains in efficiency are sometimes possible from judicious stratification.
- Cluster sampling involves selecting the sample units in groups. For example, a sample of telephone calls may be collected by first taking a collection of telephone lines and collecting all the calls on the sampled lines. The analysis of cluster samples must take into account the intra-cluster correlation which reflects the fact that units in the same cluster are likely to be more similar than two units picked at random.

- A sample is a subject chosen from a population for investigation. A

### 2009

- (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=random%20sample
- S: (n) random sample (a sample in which every element in the population has an equal chance of being selected)
- S: (n) random sample (a sample grabbed at random)

- http://www.introductorystatistics.com/escout/main/Glossary.htm
- random sample A set of data chosen from a population in such a way that each member of the population has an equal probability of being selected.

- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Simple_random_sample
- In statistics, a simple random sample is a subset of individuals (a sample) chosen from a larger set (a population). Each individual is chosen randomly and entirely by chance, such that each individual has the same probability of being chosen at any stage during the sampling process, and each subset of k individuals has the same probability of being chosen for the sample as any other subset of k individuals (Yates, Daniel S.; David S. Moore, Daren S. Starnes (2008). The Practice of Statistics, 3rd Ed.. Freeman. ISBN 978-0-7167-7309-2.). This process and technique is known as simple random sampling, and should not be confused with Random Sampling.

In small populations and often in large ones, such sampling is typically done "without replacement" ('SRSWOR'), i.e., one deliberately avoids choosing any member of the population more than once. Although simple random sampling can be conducted with replacement instead, this is less common and would normally be described more fully as simple random sampling with replacement ('SRSWR'). Sampling done without replacement is no longer independent, but still satisfies exchangeability, hence many results still hold. Further, for a small sample from a large population, sampling without replacement is approximately the same as sampling with replacement, since the odds of choosing the same sample twice is low.

An unbiased random selection of individuals is important so that in the long run, the sample represents the population. However, this does not guarantee that a particular sample is a perfect representation of the population. Simple random sampling merely allows one to draw externally valid conclusions about the entire population based on the sample.

- In statistics, a simple random sample is a subset of individuals (a sample) chosen from a larger set (a population). Each individual is chosen randomly and entirely by chance, such that each individual has the same probability of being chosen at any stage during the sampling process, and each subset of k individuals has the same probability of being chosen for the sample as any other subset of k individuals (Yates, Daniel S.; David S. Moore, Daren S. Starnes (2008). The Practice of Statistics, 3rd Ed.. Freeman. ISBN 978-0-7167-7309-2.). This process and technique is known as simple random sampling, and should not be confused with Random Sampling.
- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Systematic_sampling
- Systematic sampling is a statistical method involving the selection of elements from an ordered sampling frame. The most common form of systematic sampling is an equal-probability method, in which every kth element in the frame is selected, where [math]\displaystyle{ k }[/math], the sampling interval (sometimes known as the 'skip'), is calculated as:
- sample size (n) = population size (N) /k

- Using this procedure each element in the population has a known and equal probability of selection. This makes systematic sampling functionally similar to simple random sampling. It is however, much more efficient (if variance within systematic sample is more than variance of population).

The researcher must ensure that the chosen sampling interval does not hide a pattern. Any pattern would threaten randomness. A random starting point must also be selected.

- Systematic sampling is a statistical method involving the selection of elements from an ordered sampling frame. The most common form of systematic sampling is an equal-probability method, in which every kth element in the frame is selected, where [math]\displaystyle{ k }[/math], the sampling interval (sometimes known as the 'skip'), is calculated as:
- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Statistical_sample
- … The best way to avoid a biased or unrepresentative sample is to select a Random Sample, also known as a probability sample. A random sample is defined as a sample where the Probability that any individual member from the population being selected as part of the sample is exactly the same as any other individual member of the population. Several types of random samples are Simple Random Samples, systematic samples, stratified random samples, and cluster random samples. ...

### 2000

- (Valpola, 2000) ⇒ Harri Valpola. (2000). “Bayesian Ensemble Learning for Nonlinear Factor Analysis." PhD Dissertation, Helsinki University of Technology.
- Stochastic sampling http://www.cis.hut.fi/harri/thesis/valpola_thesis/node23.html
- QUOTE: In stochastic sampling one generates a set of samples of models, whose distribution approximates the posterior probability of the models [33]. There are several techniques having slightly different properties, but in general the methods yield good approximations of the posterior probability of the models but are computationally demanding. To some extent the trade-off between efficiency and accuracy can be controlled by adjusting the number of generated samples.

For simple problems, the stochastic sampling approach is attractive because it poses the minimal amount of restrictions on the structure of the model and does not require careful design of the learning algorithm. For an accessible presentation of stochastic sampling methods from the point of view of neural networks, see [92].