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In statistics
Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
, a sample is a subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 of a population
Statistical population

In statistics, a statistical population is a Set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population....
. Typically, the population is very large, making a census
Census

A census is the procedure of systematically acquiring and recording information about the members of a given population. It is a regularly occurring and official count of a particular population....
 or a complete enumeration
Enumeration

In mathematics and theoretical computer science, the broadest and most abstract definition of an enumeration of a Set is an exact listing of all of its element s ....
 of all the values in the population impractical or impossible. The sample represents a subset of manageable size. Samples are collected and statistics are calculated from the samples so that one can make inference
Inference

Inference is the act or process of deriving a logical consequence from premises.Inference is studied within several different fields.* Human inference is traditionally studied within the field of cognitive psychology....
s or extrapolation
Extrapolation

In mathematics, extrapolation is the process of constructing new data points outside a discrete set of known data points. It is similar to the process of interpolation, which constructs new points between known points, but the results of extrapolations are often less meaningful, and are subject to greater uncertainty....
s from the sample to the population. This process of collecting information from a sample is referred to as sampling
Sampling (statistics)

Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference....
.

The best way to avoid a biased or unrepresentative sample is to select a random sample
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....
, also known as a probability sample.






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In statistics
Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
, a sample is a subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 of a population
Statistical population

In statistics, a statistical population is a Set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population....
. Typically, the population is very large, making a census
Census

A census is the procedure of systematically acquiring and recording information about the members of a given population. It is a regularly occurring and official count of a particular population....
 or a complete enumeration
Enumeration

In mathematics and theoretical computer science, the broadest and most abstract definition of an enumeration of a Set is an exact listing of all of its element s ....
 of all the values in the population impractical or impossible. The sample represents a subset of manageable size. Samples are collected and statistics are calculated from the samples so that one can make inference
Inference

Inference is the act or process of deriving a logical consequence from premises.Inference is studied within several different fields.* Human inference is traditionally studied within the field of cognitive psychology....
s or extrapolation
Extrapolation

In mathematics, extrapolation is the process of constructing new data points outside a discrete set of known data points. It is similar to the process of interpolation, which constructs new points between known points, but the results of extrapolations are often less meaningful, and are subject to greater uncertainty....
s from the sample to the population. This process of collecting information from a sample is referred to as sampling
Sampling (statistics)

Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference....
.

The best way to avoid a biased or unrepresentative sample is to select a random sample
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....
, also known as a probability sample. A random sample is defined as a sample where the probability
Probability

Probability, or wikt:chance, is a way of expressing knowledge or belief that an Event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about t...
 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 sample
Simple random sample

In statistics, a simple random sample is a subset of individuals chosen from a larger set . Each individual is chosen randomization 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...
s, systematic samples
Systematic sampling

Systematic sampling is a statistics 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 k, the sampling interval , is calculated as:...
, stratified random samples
Stratified sampling

In statistics, stratified sampling is a method of sampling from a population.When sub-populations vary considerably, it is advantageous to sample each subpopulation independently....
, and cluster random samples
Cluster sampling

Cluster sampling is a sampling technique used when "natural" groupings are evident in a statistical population. It is often used in marketing research....
.

A sample that is not random is called a nonrandom sample or a nonprobability sample
Nonprobability sampling

sampling is the use of a subset of the population to represent the whole population. Probability sampling, or random sampling, is a sampling technique in which the probability of getting any particular sample may be calculated....
. Some examples of nonrandom samples are convenience samples, judgment samples, purposive samples, quota samples, snowball samples
Snowball sampling

In social science research, snowball sampling is a technique for developing a research sampling where existing study subjects recruit future subjects from among their acquaintances....
, and quadrature nodes in quasi-Monte Carlo method
Quasi-Monte Carlo method

In numerical analysis, a quasi-Monte Carlo method is a method for the computation of an integral that is based on low-discrepancy sequences. This is in contrast to a regular Monte Carlo method, which is based on sequences of pseudorandom numbers....
s.

Mathematical description of random sample


In mathematical terms, given a random variable
Random variable

In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions....
 X with distribution
Probability distribution

In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval ....
 F, a random sample of length n =1,2,3,... is a set of n independent
Independent variable

The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects....
, identically distributed (iid
Independent and identically-distributed random variables

In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed if each has the same probability distribution as the others and all are mutually statistical independence....
) random variables with distribution F.

A sample concretely represents n experiments in which we measure the same quantity. For example, if X represents the height of an individual and we measure individuals, will be the height of the i-th individual. Note that a sample of random variables (i.e. a set of measurable functions) must not be confused with the realizations of these variables (which are the values that these random variables take). In other words, is a function representing the measurement at the i-th experiment and is the value we actually get when making the measurement.

The concept of a sample thus includes the process of how the data are obtained (that is, the random variables). This is necessary so that mathematical statements can be made about the sample and statistic
Statistic

A statistic is the result of applying a function to a Data set.More formally, statistical theory defines a statistic as a function of a sample where the function itself is independent of the sample's distribution: the term is used both for the function and for the value of the function on a given sample....
s computed from it, such as the sample mean and covariance.

For example: Sample

The sample of the study consisted of 350 people who are university students.

Sampling The sample of the study consisted of 350 people through purposive sampling.

See also

  • Estimation theory
    Estimation theory

    Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data....
  • Sampling (statistics)
    Sampling (statistics)

    Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference....
  • Sample size
    Sample size

    The sample size of a statistical sample is the number of observations that constitute it. It is typically denoted n, a positive integer ....
  • Multiset#Cumulant generating function
    Multiset

    In mathematics, a multiset is a generalization of a Set . A Element of a multiset can have more than one Element , while each member of a set has only one membership....


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