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Applied Statistics

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Fundamentals of applied statistics for research. Source: Wikipedia (CC BY-SA 4.0).

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Definition, areas, and applications of statistics

  • Definition: study of variability and random processes through probability laws
  • Use in factual sciences: tool for data collection, organization, analysis, and interpretation
  • Methodological objective: drawing conclusions from samples and making inferences about populations
  • Descriptive statistics: description, 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
  • Statistics: term for the summary of descriptive results in specific areas

Historical evolution and origin of statistics

  • Origin of the term: Statistik (1749) by Gottfried Achenwall as the 'science of the State'.
  • Modern concept: Collection and classification of data introduced by Sir John Sinclair (19th century).
  • Bibliographic relationship: Located next to demography in the Dewey classification.
  • Babylon (3000 B.C.): Use of clay tablets for agricultural production and trade.
  • Egypt (3050 B.C.): Censuses since the 1st Dynasty for pyramid planning and state administration.
  • China (2238 B.C.): General census of agricultural, industrial, and commercial activity.
  • Greece (594 B.C.): Censuses for taxation, military service, and voting rights.
  • Rome: Records of births, deaths, and citizen wealth.
  • Mexico (1116): Population census ordered by King Xólotl.
  • Middle Ages: Data collection by Isidore of Seville in Originum sive Etymologiarum.
  • Spain (Early Modern Period): Censuses of Pecheros (1528), Obispos (1587), Millones (1591), and conde de Aranda (1768).
  • England (1500s): Data multiplication due to the plague (births and deaths).

Historical evolution of statistics and probability theory

  • Origin: Pascal-Fermat correspondence (1654) and Huygens' scientific treatment (1657).
  • Bernoulli (1713) and de Moivre (1718) establish probability as a mathematical branch.
  • Kolmogorov: Pillar of the fundamental model of Probability Theory.
  • Simpson (1756): First use of the discussion of observational errors.
  • Error axiom: Positive and negative errors are equally probable and have assignable limits.
  • Laplace (1774): First rule for the combination of observations and formula for the mean of three observations.
  • Laplace (1871): Derives the formula for the law of error ease (Lagrange term, 1744).
  • Bernoulli (1778): Introduces the principle of the maximum product of probabilities.
  • Method of least squares: Published by Legendre (1805), Adrain (1808), and Gauss (1809).
  • Historical use: Gauss predicted the location of Ceres in 1801 using least squares.
  • Quetelet (1796-1874): Introduced the notion of the "average man" for social phenomena.
  • Galton and Pearson (20th Century): Transformed statistics into a rigorous mathematical discipline.
  • Galton's contributions: Standard deviation, correlation, regression analysis, and the study of variation.
  • Pearson's contributions: Pearson correlation coefficient, method of moments, and Pearson distributions.
  • Biometrika: Journal of mathematical statistics and biostatistics founded by Galton and Pearson.
  • Ronald Fisher (1918): First use of the statistical term 'variance'.
  • Ronald Fisher (1935): Development of rigorous models for experimental design.
  • Fisher's concepts: Sufficiency, Fisher information, Fisher's Principle (sex ratio), and Fisherian runaway.

Foundations, applications, and methods of statistics

  • Statistics is a formal science applied to the factual sciences.
  • Purposes of statistical analysis: description and inference.
  • Sample: a selected subset of the population for practical study.
  • Correlation: joint variation of two variables; does not necessarily imply causality.
  • Confounding variable: a third variable that can alter the interpretation of causality.
  • Statistical theory: the branch that uses probabilities and mathematical analysis for theoretical foundations.
  • Validity requirement: the population must meet the method's mathematical assumptions.
  • Statistical culture: the skills and skepticism required to interpret data correctly.
  • Independent variables: predictors that generate an effect.
  • Dependent variables: the observed response or behavior.
  • 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; the zero is arbitrary.
  • Ratio Level: Highest level; zero indicates a total absence of the magnitude.
  • Statistical tests: Include ANOVA, Pearson, Spearman, χ², Fisher, 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: Contains essential characteristics to generalize results.
  • Parameter: Numerical measure of the population (symbols μ or σ).
  • Applications: Key in business for process control (SPC) and decision-making.

Evolution of statistics, misuse of data, and inference models

  • Non-linear models: neural networks, decision trees, generalized linear models, and multilevel models
  • Resampling methods: permutation tests, bootstrap, and Gibbs sampling
  • Selection bias: Manipulation of samples or removal of outliers
  • Professional requirement: Bachelor's degree in Mathematics (specialization in statistics) or a Bachelor's degree in Statistics
  • Type I error: The probability of accepting a spurious association as real
  • p-hacking: Relating variables without a prior hypothesis until significance is found
  • Classical inference: Contrasting data against a null hypothesis
  • Interpretation risk: Exaggerating small differences in large samples without practical relevance

Critiques of hypothesis testing and methodology in statistics education

  • Critique of the p-value: Summarizes the evidence of the effect but not its size.
  • Confidence intervals: Indicate effect size, uncertainty, and statistical significance.
  • Bayesian methods: An alternative to hypothesis testing, also under criticism.
  • Type I and II errors: Tensions in drug approval regarding safety and efficacy.
  • Social Science Teaching: Greater learning with concrete examples and relevant problems.
  • Mixed Approach: Linking quantitative and qualitative methods to quantify descriptive details.
  • Triangulation: Reduces data fallibility by combining different research methods.
  • Curricular Integration: Quantitative reasoning must be embedded into the core courses of the social sciences.
  • Use in Education: The didactic and applied use of statistics predominates.
  • Wikipedia Analysis: Use of visit metrics to identify curriculum content.
  • Pedagogical Objective: Focus on test selection, interpretation, and the use of free software.

Statistical Law and Statistical Criminology

  • Statistical Law: a legal discipline that studies mass phenomena in a quantitative and qualitative manner
  • Objective: to regulate society so that it is fairer and less manipulable
  • Main functions: to guarantee the reliability and validity of data
  • Use in justice: a tool to determine a client's innocence or grievance
  • Risk: misuse of data in judicial proceedings
  • Context: scarce rigorous statistical approaches in Latin America
  • Statistical Criminology: a discipline that provides research principles and techniques for more accurate judgments

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 (through graphs, tables, or measures).

  3. 03

    Inferential statistics is dedicated to the generation of models, inferences, and [____] associated with phenomena.

    predictions

  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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