Randomization is a statistical process in which a random mechanism is employed to select a sample from a population or assign subjects to different groups.^[1][2][3] The process is crucial in ensuring the random allocation of experimental units or treatment protocols, thereby minimizing selection bias and enhancing the statistical validity.^[4] It facilitates the objective comparison of treatment effects in experimental design, as it equates groups statistically by balancing both known and unknown factors at the outset of the study. In statistical terms, it underpins the principle of probabilistic equivalence among groups, allowing for the unbiased estimation of treatment effects and the generalizability of conclusions drawn from sample data to the broader population.^[5][6]

Randomization is not haphazard; instead, a random process is a sequence of random variables describing a process whose outcomes do not follow a deterministic pattern but follow an evolution described by probability distributions. For example, a random sample of individuals from a population refers to a sample where every individual has a known probability of being sampled. This would be contrasted with nonprobability sampling, where arbitrary individuals are selected. A runs test can be used to determine whether the occurrence of a set of measured values is random.^[7] Randomization is widely applied in various fields, especially in scientific research, statistical analysis, and resource allocation, to ensure fairness and validity in the outcomes.^[8][9][10]

In various contexts, randomization may involve

• Generating Random Permutations: This is essential in various situations, such as shuffling cards. By randomly rearranging the sequence, it ensures fairness and unpredictability in games and experiments.
• Selecting Random Samples from Populations: In statistical sampling, this method is vital for obtaining representative samples. By randomly choosing a subset of individuals, biases are minimized, ensuring that the sample accurately reflects the larger population.
• Random Allocation in Experimental Design: Random assignment of experimental units to treatment or control conditions is fundamental in scientific studies. This approach ensures that each unit has an equal chance of receiving any treatment, thereby reducing systematic bias and improving the reliability of experimental results.
• Generating Random Numbers: The process of random number generation is central to simulations, cryptographic applications, and statistical analysis. These numbers form the basis for simulations, model testing, and secure data encryption.
• Data Stream Transformation: In telecommunications, randomization is used to transform data streams. Techniques like scramblers randomize the data to prevent predictable patterns, which is crucial for securing communication channels and enhancing transmission reliability."

Randomization has many uses in gambling, political use, statistical analysis, art, cryptography, gaming and other fields.

In the world of gambling, the integrity and fairness of games hinge significantly on effective randomization. This principle serves as a cornerstone in gambling, ensuring that each game outcome is unpredictable and not manipulable. The necessity for advanced randomization methods stems from the potential for skilled gamblers to exploit weaknesses in poorly randomized systems. High-quality randomization thwarts attempts at prediction or manipulation, maintaining the fairness of games. A quintessential example of randomization in gambling is the shuffling of playing cards. This process must be thoroughly random to prevent any predictability in the order of cards.^[11] Casinos often employ automatic shuffling machines, which enhance randomness beyond what manual shuffling can achieve.

With the rise of online casinos, digital random number generators (RNGs) have become crucial. These RNGs use complex algorithms to produce outcomes that are as unpredictable as their real-world counterparts.^[12] The gambling industry invests heavily in research to develop more effective randomization techniques. To ensure that gambling games are fair and random, regulatory bodies rigorously test and certify shuffling and random number generation methods. This oversight is vital in maintaining trust in the gambling industry, ensuring that players have equal chances of winning.

The unpredictability inherent in randomization is also a key factor in the psychological appeal of gambling. The thrill and suspense created by the uncertainty of outcomes contribute significantly to the allure and excitement of gambling games.^[13]

In summary, randomization in gambling is not just a technical necessity; it is a fundamental principle that upholds the fairness, integrity, and thrill of the games. As technology advances, so too do the methods to ensure that this randomization remains effective and beyond reproach

The concept of randomization in political systems, specifically through the method of allotment or sortition, has ancient roots and contemporary relevance, significantly impacting the evolution and practice of democracy.

In the fifth century BC, Athenian democracy was pioneering in its approach to ensuring political equality, or isonomia.^[14][15] Central to this system was the principle of random selection, seen as a cornerstone for fair representation.^[14] The unique structure of Greek democracy, which translates to "rule by the people," was exemplified by administrative roles being rotated among citizens, selected randomly through lot. This method was perceived as more democratic than elections, which the Athenians argued could lead to inequalities. They believed that elections, which often favored candidates based on merit or popularity, contradicted the principle of equal rights for all citizens. Furthermore, the random allotment of positions like magistrates or jury members served as a deterrent to vote-buying and corruption, as it was impossible to predict who would be chosen for these roles.^[15]

In modern times, the concept of allotment, also known as sortition, is primarily seen in the selection of jurors within Anglo-Saxon legal systems, such as those in the UK and the United States. However, its political implications extend further. There have been various proposals to integrate sortition into government structures. The idea is that sortition could introduce a new dimension of representation and fairness in political systems, countering issues associated with electoral politics.^[16] This concept has garnered academic interest, with scholars exploring the potential of random selection in enhancing the democratic process, both in political frameworks and organizational structures.^[17] The ongoing study and debate surrounding the use of sortition reflect its enduring relevance and potential as a tool for political innovation and integrity.

Randomization is a core principle in statistical theory, whose importance was emphasized by Charles S. Peirce in "Illustrations of the Logic of Science" (1877–1878) and "A Theory of Probable Inference" (1883). Its application in statistical methodologies is multifaceted and includes critical processes such as randomized controlled experiments, survey sampling and simulations.

In the realm of scientific research, particularly within clinical study designs, constraints such as limited manpower, material resources, financial backing, and time necessitate a selective approach to participant inclusion.^[2][4] Despite the broad spectrum of potential participants who fulfill the inclusion criteria, it is impractical to incorporate every eligible individual in the target population due to these constraints. Therefore, a representative subset of treatment groups is chosen based on the specific requirements of the research.^[8] A randomized sampling method is employed to ensure the integrity and representativeness of the study. This method ensures that all qualified subjects within the target population have an equal opportunity to be selected. Such a strategy is pivotal in mirroring the overall characteristics of the target population and in mitigating the risk of selection bias.

The selected samples (or continuous non-randomly sampled samples) are grouped using randomization methods so that all research subjects in the sample have an equal chance of entering the experimental group or the control group and receiving corresponding treatment. In particular, the random grouping after the research subjects are stratified can make the known or unknown influencing factors between the groups basically consistent, thereby enhancing the comparability between the groups.^[4]

Survey sampling uses randomization, following the criticisms of previous "representative methods" by Jerzy Neyman in his 1922 report to the International Statistical Institute. It randomly displays the answer options to survey participants, which prevents order bias caused by the tendency of respondents to choose the first option when the same order is presented to different respondents.^[18] To overcome this, researchers can give the answer options in a random order so that the respondents allocate some time to read all the options and choose an honest answer. For example, consider an automobile dealer who wants to conduct a feedback survey and ask the respondents to select their preferred automobile brand. The user can create a study with randomized answers to display the different automobile brands so that the respondents do not see them in the same order.

Some important methods of statistical inference use resampling from the observed data. Multiple alternative versions of the data-set that "might have been observed" are created by randomization of the original data-set, the only one observed. The variation of statistics calculated for these alternative data-sets is a guide to the uncertainty of statistics estimated from the original data.

In many scientific and engineering fields, computer simulations of real phenomena are commonly used. When the real phenomena are affected by unpredictable processes, such as radio noise or day-to-day weather, these processes can be simulated using random or pseudo-random numbers. One of the most prominent uses of randomization in simulations is in Monte Carlo methods. These methods rely on repeated random sampling to obtain numerical results, typically to model probability distributions or to estimate uncertain quantities in a system.

Randomization also allows for the testing of models or algorithms against unexpected inputs or scenarios. This is essential in fields like machine learning and artificial intelligence, where algorithms must be robust against a variety of inputs and conditions.^[19]

Randomization plays a fascinating and often underappreciated role in literature, music, and art, where it introduces elements of unpredictability and spontaneity. Here is how it manifests in each of these creative fields:

Pioneered by surrealists and later popularized by writers like William S. Burroughs, automatic writing and cut-up techniques involve randomly rearranging text to create new literary forms. It disrupts linear narratives, fostering unexpected connections and meanings.^[20]

In aleatoric music, elements of the composition are left to chance or the performer's discretion. Composers like John Cage used randomization to create music where certain elements are unforeseeable, resulting in each performance being uniquely different. Modern musicians sometimes employ computer algorithms that generate music based on random inputs. These compositions can range from electronic music to more classical forms, where randomness plays a key role in creating harmony, melody, or rhythm.

Some artists in abstract expressionism movement, like Jackson Pollock, used random methods (like dripping or splattering paint) to create their artworks. This approach emphasizes the physical act of painting and the role of chance in the artistic process.Also, contemporary artists often use algorithms and computer-generated randomness to create visual art. This can result in intricate patterns and designs that would be difficult or impossible to predict or replicate manually.

Although historically "manual" randomization techniques (such as shuffling cards, drawing pieces of paper from a bag, spinning a roulette wheel) were common, nowadays automated techniques are mostly used. As both selecting random samples and random permutations can be reduced to simply selecting random numbers, random number generation methods are now most commonly used, both hardware random number generators and pseudo-random number generators.

Randomization is used in optimization to alleviate the computational burden associated to robust control techniques: a sample of values of the uncertainty parameters is randomly drawn and robustness is enforced for these values only. This approach has gained popularity by the introduction of rigorous theories that permit one to have control on the probabilistic level of robustness, see scenario optimization.

Common randomization methods including

• Simple randomization (coin flipping, drawing lots and random number method）^[3]
• Stratified randomization (stratified sampling and stratified allocation)^[3]
• Block randomization^[3]
• Systematic randomization
• Cluster randomization
• Multistage sampling
• Quasi-randomization
• Covariate Adaptive Randomization^[3]

• Randomized algorithm
• Bias
• Random number generation

• RQube - Generate quasi-random stimulus sequences for experimental designs
• RandList - Randomization List Generator