This article explains the simple linear regression model, its six key assumptions, how to evaluate the fit using ANOVA and the coefficient of determination, and covers hypothesis testing and confidence intervals for regression coefficients with practical examples.
This article walks through a complete F‑test example comparing the monthly return variances of IBM and HP stocks over 36 months, detailing hypothesis formulation, calculation of the test statistic, critical value lookup, and interpretation of results to determine if the variances differ significantly.
This article explains the null and alternative hypotheses for one‑ and two‑tailed variance tests, demonstrates how to compute the F‑statistic and critical value, and walks through a practical example comparing the monthly return standard deviations of IBM and HP stocks, concluding that their variances are not significantly different.
This article explains the paired comparison test for non‑independent samples, outlines the null and alternative hypotheses, presents the t‑test statistic formula, and walks through a detailed example evaluating a Dow‑10 investment strategy versus a buy‑and‑hold approach.
This article explains how to conduct hypothesis tests for equality of means between two independent normal populations, covering both equal‑variance and unequal‑variance cases, detailing test statistics, degrees of freedom, critical values, and decision rules through examples on S&P 500 returns and bond recovery rates.
Since the mid‑20th century, the infusion of mathematical theory and computer‑driven modeling has transformed biology, spawning fields such as mathematical ecology, numerical taxonomy, statistical genetics, and numerous model‑based approaches that quantify ecological dynamics, classification, and evolutionary mechanisms.
This article explains the concept of one‑factor (one‑way) ANOVA, walks through a lamp‑life example with four manufacturing processes, derives the within‑ and between‑group sum‑of‑squares formulas, and shows how to execute the test in Python using statsmodels.
This article explains the fundamentals of statistics with a model‑centric approach, covering data collection, exploratory data analysis, descriptive statistics, visualization, and the three‑step Bayesian modeling process—including hypothesis formulation, model fitting, and evaluation—while emphasizing simplicity and practical programming tools such as Python.
Exploring how probability quantifies uncertainty, the article examines questions like the chance of life on Mars, explains subjective versus objective interpretations, outlines Bayesian reasoning, Cromwell’s Rule, conditional probability, and why all probabilities can be viewed as conditional, linking logic and statistical inference.
This article demystifies Bayes’ theorem by deriving it from probability multiplication rules, explaining its components—prior, likelihood, posterior, and evidence—and illustrating its significance with intuitive examples, while also discussing its role in sequential data analysis and model updating.
This article explains statistical inference focusing on parameter estimation, distinguishing point estimation from interval estimation, and demonstrates how to construct confidence intervals for population means using sample data, including a practical example calculating a confidence interval for the average Sharpe ratio of equity funds.
This article explains the fundamentals of hypothesis testing, covering its definition, significance testing, underlying small‑probability principle, test statistics, rejection and acceptance regions for one‑ and two‑tailed tests, step‑by‑step procedures, and the two types of errors involved.
Discover a curated collection of beginner-friendly Python books covering statistics, data analysis, machine learning, deep learning, and computer vision, each with concise descriptions, key features, and practical examples, plus a chance to win a free copy by participating in the event.
This article introduces Bayes’ theorem, explains its derivation from probability multiplication rules, clarifies the meanings of prior, likelihood, posterior, and evidence, and illustrates why these components differ, providing intuitive examples and highlighting its importance for sequential data analysis.
This article explains the method of moments for estimating population parameters, outlines its step‑by‑step derivation, and demonstrates a Python implementation that estimates a basketball player's shooting odds from binary outcome data using.
This article explains statistical inference focusing on parameter estimation, distinguishing point and interval estimates, describing how confidence levels relate to significance levels, and illustrating the calculation of confidence intervals for population means with a practical example using Sharpe ratios from a sample of 100 observations.
This article explains key measures of statistical dispersion—including range, mean absolute deviation, variance, standard deviation, coefficient of variation, skewness, and kurtosis—illustrating how each quantifies data spread and why they matter in financial risk analysis.
This article explains the four types of averages—arithmetic, geometric, weighted, and harmonic—along with median, mode, and various quantiles, showing how each is calculated and when to use them for reliable data analysis.
This article introduces Bayesian inference through the classic coin‑toss example, explaining how to model bias with a beta prior and binomial likelihood, derive the posterior distribution, and understand convergence of different priors as data accumulates.
Bayes' theorem, a cornerstone of probability theory, relates prior knowledge, likelihood, and evidence to compute posterior probabilities, highlighting why prior and likelihood differ, and explaining concepts such as prior, likelihood, posterior, and evidence with intuitive examples and their relevance to sequential data analysis.
This article explains how statistics—through data collection, exploratory analysis, descriptive metrics, visualization, and model‑centric inference—provides a framework for understanding and predicting phenomena, emphasizing the role of programming (e.g., Python) and Bayesian modeling principles such as simplicity and the Occam razor.
This article surveys intrinsic interpretable machine‑learning models—from classic regression, additive models, and decision trees to modern approaches like Explainable Boosting Machines, GAMINet, RuleFit, and Falling Rule Lists—explaining their principles, parameter estimation, interpretability, advantages, and limitations.
While many claim that machine learning is merely statistics with a flashy veneer, this article explores the nuanced distinctions between the two fields—examining their goals, methodologies, and examples such as linear regression—to clarify why they are related yet fundamentally different.
This article explains the Beta distribution’s definition, its probability density and cumulative distribution functions, key statistical properties such as mode, mean, variance, skewness and kurtosis, and illustrates its practical use in modeling relative humidity in atmospheric studies.
This article explains the Gamma distribution and its inverse, detailing their probability density functions, expectations, variances, special cases, and how the Gamma distribution relates to Poisson and exponential distributions, providing concise mathematical insights and reference sources.
This article explains the chain rule of conditional probability, also known as the multiplication rule, by illustrating how successive events' probabilities combine, providing a clear, intuitive example that demonstrates the step‑by‑step derivation of the formula for calculating joint probabilities.
The Monty Hall problem, a classic probability puzzle from the TV show "Let's Make a Deal," shows that always switching your choice after the host reveals a goat raises the chance of winning the car from one‑third to two‑thirds, a counter‑intuitive result that has fascinated mathematicians for decades.
This article explains the Cramér’s V coefficient, a measure of association between categorical variables, outlines its formula, clarifies key points about the denominator and scaling, provides a step‑by‑step calculation procedure, and demonstrates the method with a practical restaurant survey example.
This article explains how overlooking Moivre’s formula can lead to misleading conclusions about school performance, demonstrates the role of sample size, standard error, confidence intervals, and hypothesis testing using real ENEM data, and clarifies common misconceptions about p‑values and confidence levels.
This article explains how to calculate the number of bins for a frequency distribution histogram using Sturges' formula, illustrating the process with a sample of 50 data points where the maximum value is 980 and the minimum is 500, and shows how to derive the bin width.
This article introduces the fundamentals of statistics, covering its purpose, types, data collection methods, data organization steps, graphical representation, measures of central tendency and dispersion, probability concepts, parameter estimation, hypothesis testing, and the distinction between correlation and regression analysis.
This article explains why randomised experiments are the gold standard for turning correlations into causal claims, illustrates their use in evaluating online versus face‑to‑face learning, and discusses ideal experimental design, assignment mechanisms, and key take‑aways for causal inference.
This article explains how the traditional linear regression assumption can be relaxed by using link functions to transform nonlinear outputs into linear responses, enabling more flexible generalized linear models for probabilities and count data.
This article explains the origins and core concepts of factor analysis, outlines its algorithmic steps, demonstrates how to perform the analysis using Python's factor_analyzer library—including data preparation, adequacy tests, eigenvalue selection, rotation, and visualization—culminating in extracting new latent variables.
This article explains how to build a Python version of the popular Chinese card game ZhaJinHua, covering deck preparation, player enrollment, card dealing, hand‑type detection, scoring rules, winner determination, and statistical analysis of thousands of simulated rounds.
A solid grasp of basic statistical concepts—such as populations, samples, means, variance, probability distributions, the Central Limit Theorem, and hypothesis testing—enables designers of A/B experiments to correctly size samples, interpret p‑values and confidence intervals, and reliably deploy DeWu’s integrated platform for automated experiment allocation, metric monitoring, and one‑click reporting, ultimately driving data‑driven product decisions.
Learn how Pearson, Spearman, and Kendall correlation coefficients measure linear and monotonic relationships between variables, explore their mathematical properties, interpret their value ranges, and see practical Python examples using Pandas to compute each coefficient on generated data.
This article explains the difference between continuous and discrete probability distributions, introduces common examples such as Bernoulli, Binomial, Poisson, Uniform, Exponential, and Normal distributions, and describes how these distributions relate to each other.
This tutorial demonstrates how to generate synthetic datasets with NumPy, organize them into a Pandas DataFrame, and explore their distributions using Seaborn’s histograms, KDE plots, boxplots, violin plots, as well as multivariate visualizations like heatmaps, pair plots, and joint plots.
Principal Component Analysis (PCA) reduces high‑dimensional data to a few uncorrelated components by maximizing variance, enabling noise reduction, visualization, and efficient modeling, with practical steps—including data standardization, covariance matrix computation, eigenvalue extraction, and component selection—illustrated through a clothing‑size measurement case study.
This article walks through a Python tutorial that generates a random list, sorts it, and calculates the minimum, maximum, sum, and average values, providing clear explanations, code snippets, and output screenshots for readers to follow along.
This article outlines thirteen key statistical techniques—including descriptive and inferential methods, hypothesis testing, reliability analysis, contingency tables, regression, clustering, discriminant, factor, and time‑series analysis—explaining their purposes, assumptions, and typical applications for researchers and data analysts.
This article compiles a comprehensive list of links to past posts from the "Model Perspective" public account covering probability and statistics, calculus, linear algebra, and mathematical modeling, while noting upcoming collections on Python programming, academic writing, and competition analysis.
This article explains how to recognize whether a stationary random time series follows a pure AR, pure MA, or mixed ARMA process by examining the patterns of the autocorrelation function (ACF) and the partial autocorrelation function (PACF).
This article explains the concept of differencing for extracting deterministic information, outlines when to use first, second, or seasonal differencing, introduces the ARIMA(p,d,q) model and its special cases, and provides a complete Python example with code and visualizations.
This article explains the three main types of stationary time‑series models—AR, MA, and ARMA—detailing their definitions, back‑shift operator notation, polynomial representations, and the essential stationarity and invertibility conditions required for valid modeling.
This article defines independence for continuous random variables, introduces the weaker concept of mean‑independence, explains how it differs from linear uncorrelatedness, and presents key propositions and proofs showing that mutual independence implies mean‑independence while the converse does not hold.
This article introduces the key numerical characteristics of random variables—including expectation, variance, covariance, correlation, and higher-order moments such as skewness and kurtosis—explains their definitions, properties, and relationships, and extends these concepts to random vectors and matrix representations.
This article explains the fundamentals of probability, conditional probability, independent events, the law of total probability, and both discrete and continuous probability distributions, including joint and marginal densities and conditional distributions for random variables.
Learn a step-by-step approach to predict seasonal time series—such as product sales or climate data—using the seasonal coefficient method, illustrated with a quarterly refrigerator sales case study and a complete Python implementation that computes next year's quarterly forecasts.
This article explains the construction of logistic regression models, the use of the sigmoid function, maximum likelihood estimation, and least‑squares estimation for grouped data, illustrated with a housing‑purchase case study and complete Python code for fitting and predicting probabilities.
This article introduces the mathematical foundations of ordinary least squares, ridge, and LASSO regression, explains why LASSO requires coordinate descent, presents two real-world case studies with data, and provides complete Python code for fitting, visualizing, and interpreting LASSO models.
This article explains the concept of one‑way ANOVA, walks through a real‑world example comparing four manufacturing processes, and demonstrates how to conduct the analysis in Python using statsmodels, interpreting the resulting F‑statistic and p‑value to assess significance.
This guide shows how to calculate common statistical measures such as mean, median, range, variance, standard deviation, covariance, and correlation using NumPy functions, and demonstrates the equivalent operations with Pandas DataFrames, including a table of useful DataFrame methods for statistical analysis.
This article introduces the fundamentals of statistical analysis, covering the distinction between populations and samples, constructing frequency tables and histograms, and explaining essential descriptive measures such as mean, median, variance, standard deviation, range, skewness, kurtosis, covariance, and correlation coefficients.
This article explains how to conduct large‑sample hypothesis tests for a population proportion, covering the formulation of null and alternative hypotheses, constructing 99% confidence intervals using the Central Limit Theorem, calculating standardized test statistics, determining rejection regions, and applying the method to a real‑world soft‑drink preference case study.
This article introduces a collection of lesser‑known NumPy functions—including np.full_like, np.logspace, np.meshgrid, np.triu, np.ravel, np.vstack, np.r_, np.where, np.allclose, np.argsort, np.isneginf, np.polyfit, np.clip, np.count_nonzero, and np.array_split—demonstrating their usage with code examples and visualizations for data‑science and scientific‑computing tasks.
This article explains the fundamentals of hypothesis testing, covering its definition, significance testing, underlying small‑probability principle, test statistics, rejection and acceptance regions, step‑by‑step procedures, and the two types of errors, providing a concise overview for statistical inference.
This article explains the t‑distribution, its derivation from the normal and chi‑square distributions, illustrates how its shape changes with degrees of freedom, and shows how to construct confidence intervals for population means when sample sizes are small and the population variance is unknown.
This article explains how to use Oracle Data Pump's ESTIMATE and ESTIMATE_ONLY parameters to assess the size of a 1 TB production database for migration, compares BLOCKS and STATISTICS estimation methods, and details best practices for gathering accurate optimizer statistics with ANALYZE and DBMS_STATS.
This article introduces the basics of simple (univariate) linear regression and extends to multivariate linear regression, explaining their regression equations, the use of the least‑squares method to estimate parameters, and the practical relevance of multiple predictors in modeling real‑world phenomena.
This article intuitively demonstrates the Central Limit Theorem using uniform and Beta distributions, showing how sample means converge to a normal shape as sample size grows, and provides the formal statistical statement and its significance for inference.
This article explains the Central Limit Theorem, distinguishes standard deviation from standard error, illustrates the 3‑σ rule, and shows how confidence levels, significance levels, and interval estimation combine to derive reliable confidence intervals for large‑sample population mean estimates.
This article introduces fundamental statistical concepts—including populations, samples, measures, parameters, descriptive and inferential statistics—explains the relationship between samples and populations, and details key descriptive metrics such as mean, variance, percentiles, box plots, violin plots, and z‑scores.
The National Bureau of Statistics published 2021 average wage figures showing urban non‑private employees earning 106,837 CNY annually, with the IT‑related sector topping the list, while netizens debated the data’s realism and implications for various professions.
The article explores the concept of causality from philosophical definitions and INUS conditions to statistical approaches like Granger causality, introduces causal network structures (chain, fork, collider), and demonstrates their use in resolving Simpson’s paradox through epidemiological and medical examples.
This article introduces Bayes' theorem, explains its derivation from conditional probability, and demonstrates its counter‑intuitive power through two practical examples—a liar’s dice claim and a medical test scenario—showing how prior probabilities dramatically affect posterior conclusions.
This article explains the concepts of joint probability and conditional probability, illustrates them with coin‑flip and dice examples, and demonstrates how they lead to the law of total probability, providing a foundation for further study of Bayes' theorem.
This article explains the concept of dimensionless processing for evaluation indicators and introduces several common transformation methods—including standard sample, ratio, vector normalization, range, and efficacy coefficient techniques—to convert raw values into comparable evaluation scores.
This article lists the ten Microsoft‑maintained open‑source repositories on GitHub with the highest star counts as of May 9 2022, providing key statistics, primary programming languages, and concise official descriptions for each project.
This article explains the normal (Gaussian) distribution, its probability density function, the 3‑sigma principle, confidence intervals, and extends the discussion to the bivariate normal distribution, illustrating concepts with bell‑shaped curves and contour plots.
This article introduces the Bernoulli, Binomial, Poisson, and Exponential probability distributions, detailing their definitions, key properties, and real‑world examples such as lottery draws, loan‑approval counts, rare market events, and inventory control, while illustrating how the Binomial converges to the Poisson and linking to the Exponential inter‑arrival times.
This article demonstrates how to conduct various hypothesis tests—including Z‑test, one‑sample and two‑sample t‑tests, F‑test, as well as distribution fitting and outlier detection—using Python libraries such as Statsmodels, SciPy, and pandas on both the Iris dataset and a human temperature dataset.
This article presents a series of Python examples using xlwings and pandas to batch‑sort worksheets, filter and aggregate data across multiple workbooks, compute summary statistics, perform correlation, ANOVA, regression, and generate pivot tables and visualizations, illustrating practical Excel automation and data‑analysis techniques.
This article explains why A/B testing is crucial for data‑driven product decisions, outlines ByteDance’s A/B testing system architecture across multiple layers, describes client‑ and server‑side experiment workflows, shares statistical best practices, and presents real‑world case studies illustrating hypothesis generation, evaluation, and future industry trends.
This guide demonstrates how to use a second‑hand housing dataset to create various reports—frequency tables, cross‑tables, and summary statistics—and visualise them with bar and stacked bar charts using Python's pandas library.
This article presents a comprehensive Python-driven analysis of Nobel Prize laureates, examining gender distribution, top awarding countries across each prize category, proportional representation among fields, and average age of winners, accompanied by visualizations generated from the underlying datasets.
This tutorial explains how to perform hypothesis testing, chi‑square, t‑tests, confidence‑interval calculation, and effect‑size measurement in Python, covering data preparation, statistical assumptions, code implementation, and interpretation of results for real‑world datasets.
This article explains the classic Monty Hall problem, walks through the probability reasoning that switching doors raises the win chance from 1/3 to 2/3, and provides a complete Java program that simulates ten‑hundred‑thousand trials to verify the result.
This article presents a comprehensive guide to essential Pandas operations, including creating Series and DataFrames, common methods for data selection, indexing, grouping, reading and writing files, handling missing values, sorting, statistical analysis, and data transformation, with practical code examples for each feature.
This article explains why A/B testing is essential, defines the methodology, details Volcano Engine's multi‑layer A/B testing architecture, outlines client and server experiment flows, shares statistical analysis practices, best‑practice guidelines, future trends, and answers common questions.
This article explains the statistical principles behind A/B testing, covering concepts such as populations, samples, parameters, hypothesis testing, significance levels, t‑tests, metric types, p‑value calculations, and real‑world examples to guide data‑driven product decisions.
This article examines MySQL 8.0.18’s ANALYZE command, explains its role in updating table statistics, demonstrates a series of tests on a CentOS server—including data insertion and case‑sensitive table naming—and reveals a bug where uppercase table names prevent statistics from being refreshed.
This article presents a curated collection of 100 frequently used pandas functions, organized into six categories—statistical aggregation, data cleaning, data selection, plotting and element‑wise operations, time‑series utilities, and miscellaneous helpers—providing concise Chinese explanations for each function’s purpose.
Online experiments, rooted in biomedical randomized controlled trials, have become essential for internet businesses to achieve data‑driven growth by providing causal inference, quantifying value, and managing risk through various designs such as AB, ABn, AA, multivariate and quasi‑experimental tests.
The article explains how to choose appropriate Redis data structures—such as Bitmap, HyperLogLog, Set, List, Hash, and Sorted Set—to efficiently handle massive statistical scenarios like UV counting, ranking, and set‑based aggregation, while providing concrete command examples and performance considerations.
This article provides a comprehensive overview of machine learning, explaining core AI terminology, distinguishing statistics, statistical learning, and machine learning, outlining a three‑part learning roadmap covering mathematical foundations, algorithms, and Python programming practice, and offering curated resources for building a solid knowledge system.
The article introduces causal inference for ride‑hailing businesses, explaining the difference between causality and correlation, common misconceptions, and how randomized experiments and observational techniques like propensity‑score matching can quantify effects of actions such as coupons, driver assignments, and platform growth decisions.
An enthusiastic overview walks through the fundamentals of linear and multivariate regression, explains loss functions and least‑squares optimization, shows Python implementations of fit and predict, and applies the model to the classic Boston housing dataset to illustrate feature impact and prediction.
This article explains how to extend a performance testing framework to record and visualize individual HTTP interface latencies by parsing consolidated logs, grouping timings by URL, and using a Groovy script with StatisticsUtil to generate response‑time charts.
This article explains how NodeSelectorSlot, ClusterBuilderSlot, LogSlot and StatisticSlot work together in a Java backend framework to map resources, create cluster nodes, handle origins, and collect precise QPS and concurrency statistics using sliding windows and LongAdder.
This article explains the fundamentals of linear regression, including regression formulas, loss functions, and error metrics, and provides complete Python code using scikit‑learn to perform both simple and multiple linear regression on the Iris and Boston housing datasets, along with model evaluation and visualization.
This article explains how MySQL calculates table and index statistics, distinguishes between persistent and non‑persistent statistics, details relevant InnoDB parameters, shows when and how statistics are updated, and provides example queries and commands for manual analysis.
The 2020 China University Graduate Employment Report reveals that 2019 graduates earned an average monthly salary of 5,440 yuan, with computer science leading among undergraduates, while inflation‑adjusted starting salaries rose 23.6% for bachelor's and 15.7% for vocational graduates, and highlights green (high‑demand) and red (low‑demand) majors.
This article describes how Meituan's delivery team built a comprehensive A/B testing evaluation platform, covering the motivation for a robust assessment framework, the architecture of the platform with three functional modules, the statistical methods for reliable experiment design, and the practical implementation details that enable data‑driven operational decisions.
Using Python, the author analyzes Shandong’s 2020 Gaokao simulation scores, visualizing distribution patterns, pass rates, subject selection ratios, top‑student combinations, and double‑first‑class university admission thresholds, providing data‑driven insights for students and educators.
This tutorial demonstrates how to use pandas for calculating total and average scores, adding column means, removing duplicate records, and transposing Excel data, providing clear code examples and explanations of key functions such as sum, mean, duplicated, drop_duplicates, and transpose.
This article introduces the fundamentals of binomial and Bernoulli distributions, explains permutations and combinations, provides Python functions to compute them, demonstrates probability calculations and visualizations with matplotlib and plotly, and shows a maximum likelihood estimation example for binomial parameters.
A comprehensive collection of 70 NumPy exercises with detailed problem statements, code snippets, and expected outputs that cover array creation, manipulation, statistical analysis, handling missing data, encoding, sorting, and advanced mathematical operations for Python developers.
This article explains why machine learning relies on big‑data statistical learning, illustrating human learning through induction and deduction, presenting case studies that highlight the limits of anecdotal reasoning, and introducing the law of large numbers and probabilistic trust as foundations for reliable AI models.
