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In statistics, a sample is a subset of a population. Typically, the population is very large, making a census or a complete enumeration 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 inferences or extrapolations from the sample to the population. This process of collecting information from a sample is referred to as sampling.

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.

A sample that is not random is called a nonrandom sample or a nonprobability sample. Some examples of nonrandom samples are convenience samples, judgment samples, purposive samples, quota samples, snowball samples, and quadrature nodes in Quasi-Monte Carlo methods.

In mathematical terms, given a random variable X with distribution F, a random sample of length n =1,2,3,... is a set of n independent, identically distributed (iid) random variables with distribution F. ^[1]

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 n individuals, X_i 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, X_i is a function representing the measurement at the i-th experiment and x_i = X_i(ω) 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 statistics computed from it, such as the sample mean and covariance.

1. ^ Samuel S. Wilks, Mathematical Statistics, John Wiley, 1962, Section 8.1

• Estimation theory
• Sampling (statistics)
• Sample size

• Statistical Terms Made Simple