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Statistics

University

Introduction to the Degree in Statistics. Source: Wikipedia (CC BY-SA 4.0).

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Definition, applications, and branches of statistics

  • Definition: Study of variability and random processes under probability laws
  • Use in factual sciences: Collection, organization, analysis, and interpretation of data
  • Methodological objective: Extracting conclusions from samples and inferences about populations
  • Descriptive statistics: Summary and visualization of data (e.g., mean, standard deviation, histograms)
  • Inferential statistics: Generation of models, predictions, and inferences about populations
  • Inferential techniques: Hypothesis testing, estimation, forecasting, correlation, and regression
  • Subdivisions of inferential statistics: Parametric and non-parametric
  • Mathematical statistics: Establishes the theoretical foundations of the techniques
  • Differentiation: "Statistics" refers to the summary of descriptive results

Historical evolution and origin of statistics

  • Origin: Gottfried Achenwall (1749), "science of the State".
  • Current meaning: Introduced by Sir John Sinclair in the 19th century.
  • Bibliographic relationship: Located next to demography (Melvil Dewey).
  • Babylon (3000 BC): Clay tablets for agricultural production and trade.
  • Egypt (3050 BC): Censuses for pyramid planning and resource management.
  • China (2238 BC): Yao's general census (agricultural, industrial, and commercial).
  • Greece (594 BC): Censuses for taxes, military service, and voting rights.
  • Rome: Records of births, deaths, and wealth.
  • Mexico (1116): Census ordered by King Xólotl.
  • Middle Ages: Work of Isidore of Seville (Originum sive Etymologiarum).
  • Spain (Early Modern Period): Censuses of Pecheros (1528), Obispos (1587), Millones (1591), and Aranda (1768).
  • England (1500s): Data multiplication due to the plague.

Historical evolution of probability theory and statistics

  • Origin: Pascal-Fermat correspondence (1654)
  • First scientific treatment: Huygens (1657)
  • Mathematical consolidation: Bernoulli (1713) and de Moivre (1718)
  • Modern fundamental model: Kolmogorov
  • Theory of errors: Simpson (1755) introduced the discussion of errors and the axiom of equal probability
  • Combination rules: Laplace (1774) deduced formulas for the mean and the law of error ease
  • Maximum product principle: Bernoulli (1778)
  • Least squares: Published by Legendre (1805), Adrain (1808), and Gauss (1809)
  • Practical application: Gauss (1801) predicted the location of Ceres
  • Social statistics: Quetelet introduced the concept of the "average man"
  • Rigorous discipline: Galton and Pearson in the 20th century
  • Galton's contributions: Standard deviation, correlation, and regression
  • Pearson's contributions: Correlation coefficient, method of moments, and Pearson distributions
  • Institutions: Biometrika and the statistics department at University College London
  • Variance: First use of the term by Fisher (1918)
  • Experimental design: Fisher (1935) developed rigorous models and the concept of sufficiency
  • Biology: Fisher's Principle (sex ratio) and Fisherian runaway (1930)

Foundations, applications, and methods of statistics

  • Definition: Statistics is a formal science and a subfield of mathematics applied to factual sciences.
  • Unit of study: Can be an entire population, a process, or a time series.
  • Sample: A selected subset of the population to facilitate data collection.
  • Purposes of analysis: Description and inference.
  • Correlation: Joint variation of two variables; does not imply causation.
  • Confounding variable: A third variable not taken into account that may explain an apparent correlation.
  • Representativeness: A necessary condition for extending sample conclusions to the population.
  • Statistical theory: The branch that uses probability theory and mathematical analysis.
  • Mathematical assumptions: A mandatory requirement for the validity of any statistical method.
  • Statistical significance: Measure of the extent to which a trend may be caused by random variations.
  • Causality: The study of the effect of independent variables on dependent variables.
  • Types of causality studies: Experimental and observational.

Levels of measurement, statistical techniques, and population concepts

  • Nominal Level: Grouping into classes without interpretable order.
  • Ordinal Level: Possesses an interpretable order but imprecise differences.
  • Interval Level: Constant unit and measurable distance; arbitrary zero.
  • Ratio Level: Highest level; possesses an absolute zero (absence of magnitude).
  • Statistical tests: Include ANOVA, Pearson, Spearman, χ², Student's t, and Mann-Whitney U.
  • Population: The total set of values; its complete analysis is a census.
  • Finite population: A limited number of elements.
  • Infinite population: An unlimited number of observations.
  • Sample: A subset of the population; its selection is sampling.
  • Representative sample: Must contain the essential characteristics of the population.
  • Parameter: Numerical measure of the population (μ or σ).

Evolution of statistics, misuse of data, and inference methods

  • Computational impact: Transition from linear models to neural networks, decision trees, and multilevel models.
  • Resampling methods: Growth of permutation tests, bootstrap, and Gibbs sampling.
  • Chaos theory: Established in the U.S. with 350 books in 1997 and applied in physics and social sciences.
  • Data bias: Introduced by non-random selection or selective removal of outliers.
  • Professional requirement: It is proposed that statistics be a regulated profession for graduates in Mathematics or Statistics.
  • Type I error: A common consequence of technical ignorance in data interpretation.
  • P-hacking: The practice of searching for significant associations without a prior hypothesis, affecting reproducibility.
  • Classical inference: Risk of overestimating the importance of small differences in large samples.
  • Scientific inconsistency: Caused by the use of small samples and poorly applied statistical techniques.

Critiques of hypothesis testing and methodology in statistics education

  • P-value criticism: summarizes evidence of the effect but not its size.
  • Confidence intervals: indicate effect size and uncertainty.
  • Bayesian methods: an alternative to hypothesis testing, also criticized.
  • Type I and Type II errors: create tensions in drug approval.
  • Effective teaching: requires concrete examples and relevant problems.
  • Qualitative-quantitative linkage: allows using descriptive details to quantify and generalize.
  • Triangulation of results: decreases the fallibility of data.
  • Curricular integration: quantitative reasoning must be included in core courses.
  • Wikipedia usage: visit analysis helps identify curriculum topics.
  • Practical approach: prioritizes test selection, interpretation, and free software.

Statistical Law and Statistical Criminology

  • Statistical Law: a legal discipline that studies trends and frequencies of mass phenomena.
  • Objective: to regulate social life to make it fairer and less manipulable.
  • Main function: to provide data for social regulation with reliability and validity.
  • Use in justice: a fundamental tool where data can determine innocence or prejudice.
  • Risk: misuse of statistical data in judicial proceedings.
  • Statistical Criminology: a discipline that provides research principles and techniques for more accurate trials.
  • Context: scarce rigorous statistical approaches in Latin American countries.

Flashcard preview (6 of 40)

  1. 01

    Statistics studies the variability of phenomena and the [____] processes that generate them.

    random

  2. 02

    In factual sciences, what is the objective of descriptive statistics?

    To organize and describe the characteristics of a dataset.

  3. 03

    Inferential statistics is used to draw inferences about the [____] under study.

    population

  4. 04

    Inferential statistics is divided into two types: [____] and non-parametric.

    parametric

  5. 05

    Which discipline establishes the theoretical foundations of statistical techniques?

    Mathematical Statistics

  6. 06

    In 1749, who introduced the German term Statistik to refer to the analysis of State data?

    Gottfried Achenwall

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