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List of statistics articles
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Statistics

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List of statistics articles

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The list of statistics articles serves as a comprehensive index of key concepts, methods, and subfields within , the branch of mathematics dedicated to collecting, analyzing, interpreting, and drawing conclusions from in the presence of and variability. This compilation encompasses foundational topics such as —including measures of like the , , and mode, as well as dispersion metrics such as range and standard deviation—probability theory with discrete and continuous distributions, and inferential techniques like testing, intervals, and sampling distributions. It also extends to , including linear and multiple regression models, , and analysis of variance (ANOVA), which are essential for modeling relationships in . Beyond introductory elements, the list highlights advanced and specialized areas that address complex real-world applications, such as for updating probabilities with new evidence, time series analysis for modeling temporal data, and nonparametric methods for distributions without strong assumptions. These topics reflect the interdisciplinary nature of modern statistics, integrating with fields like for health research, environmental statistics for ecological modeling, spatial statistics for geographic patterns, and for high-dimensional data analysis. Overall, such a list provides an organized reference for scholars, practitioners, and students navigating the evolution of statistical methodologies from classical foundations to contemporary computational approaches.

Foundational Concepts

Core Definitions

Statistics is the mathematical science concerned with the collection, , interpretation, presentation, and organization of to make informed decisions or inferences about real-world phenomena. This discipline encompasses two primary branches: , which involves summarizing and organizing from a sample or to describe its main features, such as through measures of and variability; and inferential statistics, which uses sample to draw conclusions or make predictions about a larger , often incorporating probability to account for . Fundamental terms in statistics include , referring to the entire set of entities or observations under study; sample, a subset of the population selected for analysis; parameter, a numerical characteristic describing a population, such as its mean or variance; and statistic, a numerical characteristic computed from a sample to estimate a parameter. These concepts were formalized in the late 19th and early 20th centuries, with Karl Pearson playing a pivotal role in establishing modern statistical terminology and methodology through his work in biometrics and probability, including the introduction of systematic distinctions between populations and samples in his 1895 contributions to curve fitting and data analysis. Ronald A. Fisher later refined the term statistic in the 1920s to specifically denote sample-based estimates of population parameters. Data in statistics is classified by type and scale to determine appropriate analytical methods. Qualitative data, also known as categorical data, describes qualities or categories without numerical meaning, such as or color, while quantitative data involves numerical values that can be measured or counted, such as height or income. Quantitative data is further divided into discrete types, which take on countable integer values (e.g., number of children), and continuous types, which can assume any value within an interval (e.g., ). Scales of measurement, as defined by psychologist Stanley Smith Stevens in , provide a framework for these classifications: nominal scale for unordered categories (e.g., ); ordinal scale for ordered categories without equal intervals (e.g., Likert scales); interval scale for ordered numerical data with equal intervals but no true zero (e.g., temperature); and ratio scale for ordered numerical data with equal intervals and a true zero (e.g., weight). Bias in statistics refers to a systematic error that skews results away from the , often arising from flawed or non-representative sampling, leading to consistently inaccurate estimates. In hypothesis testing, errors are categorized as Type I error, the incorrect rejection of a true (false positive), and Type II error, the failure to reject a false (false negative), with their probabilities denoted as α and β, respectively. Precision describes the consistency or of measurements, reflecting low variability among repeated observations, whereas accuracy measures how close those measurements are to the ; high precision does not guarantee accuracy if is present, and vice versa. Probability serves as a foundational tool for quantifying in these concepts, with deeper exploration in subsequent sections on probability basics.

Probability Basics

Probability theory begins with the concept of a random experiment, which is any process or phenomenon whose outcome cannot be predicted with certainty, such as tossing a or drawing a card from a deck. The , denoted as Ω\Omega, is the set of all possible outcomes of such an experiment, while are subsets of the sample space representing specific outcomes or collections of outcomes. For instance, in a coin toss, the sample space is {heads,tails}\{\text{heads}, \text{tails}\}, and the event "heads" is the subset containing only that outcome. These foundational elements allow for the modeling of in a structured manner. The axioms of probability, formalized by in 1933, provide the rigorous mathematical foundation for assigning probabilities to events. The first states that the probability of any event is a non-negative : P(E)≥0P(E) \geq 0 for any event EE. The second requires that the probability of the entire is 1: P(Ω)=1P(\Omega) = 1. The third , known as countable additivity, asserts that for a countable collection of mutually exclusive events E1,E2,…E_1, E_2, \dots, the probability of their union is the sum of their individual probabilities: P(⋃i=1∞Ei)=∑i=1∞P(Ei)P\left(\bigcup_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty P(E_i). These s ensure that probability measures are consistent and applicable to infinite sample spaces. From these axioms, basic rules of probability follow directly. The addition rule for two events AA and BB states that P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B), accounting for overlap to avoid double-counting. The multiplication rule for independent events, where the occurrence of one does not affect the other, simplifies to P(A∩B)=P(A)⋅P(B)P(A \cap B) = P(A) \cdot P(B). More generally, for any events, P(A∩B)=P(A)⋅P(B∣A)P(A \cap B) = P(A) \cdot P(B \mid A), introducing P(B∣A)P(B \mid A), which is the probability of BB given that AA has occurred, defined as P(B∣A)=P(A∩B)P(A)P(B \mid A) = \frac{P(A \cap B)}{P(A)} when P(A)>0P(A) > 0. Events AA and BB are independent if P(B∣A)=P(B)P(B \mid A) = P(B). Bayes' theorem, derived from the multiplication rule, reverses conditional probabilities: P(A∣B)=P(B∣A)⋅P(A)P(B)P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}. This theorem is crucial for updating beliefs based on new evidence. A classic example in medical testing involves a affecting 1% of the , with a test that is 99% accurate (true positive and true negative rates both 99%). If a person tests positive, the probability they have the is approximately 50%, not 99%, because the low disease prevalence () outweighs the test's accuracy, leading to many false positives. This illustrates base rate neglect and the theorem's practical importance in diagnostics. Expected value, variance, and covariance extend these principles to quantify averages and spreads under uncertainty. The expected value (or mean) of an event or quantity is the probability-weighted average, representing the long-run average outcome over many repetitions. For a discrete random variable XX taking values xix_i with probabilities pip_i, it is E[X]=∑xipiE[X] = \sum x_i p_i. Variance measures the expected squared deviation from the mean, Var⁡(X)=E[(X−E[X])2]=E[X2]−(E[X])2\operatorname{Var}(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2, capturing dispersion. Covariance between two variables XX and YY, Cov⁡(X,Y)=E[(X−E[X])(Y−E[Y])]=E[XY]−E[X]E[Y]\operatorname{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]E[Y], assesses their joint variability; positive values indicate tendency to move together. These measures are probability-weighted summaries essential for risk assessment and modeling dependencies. Random variables formalize outcomes as functions on the sample space, building on event probabilities for numerical analysis.

Random Variables and Processes

Random variables provide a formal framework for modeling uncertainty in statistical analysis by associating numerical outcomes with events in a probability space. A random variable XX is defined as a measurable function from the sample space Ω\Omega to the real numbers R\mathbb{R}, where the probability measure on Ω\Omega induces a distribution on XX. This concept extends basic probability by quantifying outcomes through expected values and variability, serving as a foundation for more advanced inferential techniques. Discrete random variables take on of possible values, such as integers, and are characterized by their (PMF), denoted pX(x)=P(X=x)p_X(x) = P(X = x), which satisfies ∑xpX(x)=1\sum_x p_X(x) = 1 and pX(x)≥0p_X(x) \geq 0 for all xx. For example, the number of heads in coin flips is a discrete random variable with a binomial PMF. In contrast, continuous random variables assume values in an uncountable interval of the reals and are described by a probability density function (PDF), fX(x)f_X(x), where the probability over an interval is given by the integral P(a≤X≤b)=∫abfX(x) dxP(a \leq X \leq b) = \int_a^b f_X(x) \, dx, with ∫−∞∞fX(x) dx=1\int_{-\infty}^{\infty} f_X(x) \, dx = 1 and fX(x)≥0f_X(x) \geq 0. Unlike PMFs, PDFs can exceed 1 but represent densities rather than direct probabilities, as in the uniform distribution over [0,1]. For multiple random variables, such as XX and , the joint distribution captures their combined behavior. The joint PMF for discrete variables is pX,Y(x,y)=P(X=x,Y=y)p_{X,Y}(x,y) = P(X=x, Y=y), while the joint PDF for continuous variables is fX,Y(x,y)f_{X,Y}(x,y), satisfying ∬fX,Y(x,y) dx dy=1\iint f_{X,Y}(x,y) \, dx \, dy = 1. Marginal distributions are obtained by summing or integrating out the other variable: the marginal PMF of XX is pX(x)=∑ypX,Y(x,y)p_X(x) = \sum_y p_{X,Y}(x,y), and similarly for the PDF fX(x)=∫fX,Y(x,y) dyf_X(x) = \int f_{X,Y}(x,y) \, dy. Conditional distributions describe the distribution of one variable given the other, with the conditional PMF pY∣X(y∣x)=pX,Y(x,y)pX(x)p_{Y|X}(y|x) = \frac{p_{X,Y}(x,y)}{p_X(x)} for pX(x)>0p_X(x) > 0, and analogously for PDFs, enabling analysis of dependencies. Stochastic processes model sequences of random variables indexed by time or another parameter, representing evolving systems under uncertainty. Markov chains are discrete-time stochastic processes where the future state depends only on the current state, not the past, defined by transition probabilities P(Xn+1=j∣Xn=i)=pijP(X_{n+1} = j | X_n = i) = p_{ij}, forming a transition matrix for finite states. They are foundational for modeling systems like queueing or . Poisson processes, as continuous-time counterparts, count events occurring randomly at a constant average rate λ\lambda, with interarrival times exponentially distributed; the number of events in interval [0,t][0,t] follows a with parameter λt\lambda t, and increments are independent. Moment-generating functions (MGFs) and characteristic functions facilitate the analysis of s by encoding their moments and distributions. The MGF of a XX is MX(t)=E[etX]M_X(t) = E[e^{tX}], defined for tt in some neighborhood of 0 where the expectation exists, and the nn-th moment is obtained via dndtnMX(0)=E[Xn]\frac{d^n}{dt^n} M_X(0) = E[X^n]. MGFs uniquely determine distributions when they exist and simplify s for sums of independent variables, as MX+Y(t)=MX(t)MY(t)M_{X+Y}(t) = M_X(t) M_Y(t). Characteristic functions, always defined as ϕX(t)=E[eitX]\phi_X(t) = E[e^{itX}] where i=−1i = \sqrt{-1}
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