Statistics 101, 201, and 202: Three Shiny Apps for Teaching Probability Distributions, Inferential Statistics, and Simple Linear Regression

This paper introduces three open-source Shiny web applications—Statistics 101, 201, and 202—that facilitate the teaching of probability distributions, inferential statistics, and simple linear regression by providing interactive, code-free interfaces that display numerical results, visualizations, and mathematical derivations side by side.

The Gist

Imagine you are trying to learn how to drive a car. In a traditional statistics class, the teacher might hand you a map, a compass, and a complex manual on engine mechanics, then ask you to calculate the exact speed and fuel consumption of a car you've never seen, all while doing the math by hand. It's slow, confusing, and easy to make a mistake.

This paper introduces three digital "driving simulators" for statistics students. Created by Antoine Soetewey, these are free, interactive web apps called Statistics 101, 201, and 202. They are designed to let students focus on understanding the road (the concepts) rather than getting stuck on how to turn the steering wheel (the coding).

Here is a breakdown of what these tools do, using simple analogies:

The Big Idea: "No Coding Required"

Usually, to do statistics in the modern world, you need to learn a programming language called R. It's like learning to build a car from scratch before you're allowed to drive it. These apps remove that barrier. You don't need to write a single line of code. You just open a website, type in your numbers, and the app does the heavy lifting.

But here is the magic trick: The app doesn't just give you the answer; it shows you the "magic trick" happening right in front of your eyes. It displays the math formulas, the graphs, and the final number all on the same screen, so you can see how they connect.

The Three "Simulators"

1. Statistics 101: The "Probability Playground"

• What it does: This app helps you understand Probability Distributions. Think of these as different "shapes" that data can take (like a bell curve, a spike, or a flat line).
• The Analogy: Imagine you are a baker trying to guess how many cookies will burn in the oven. There are 18 different "recipes" (distributions) for how cookies might burn.
• How it helps: You pick a recipe, type in your numbers (like oven temperature), and the app instantly draws a picture of the cookie distribution. It shades the area where the cookies burn and writes out the math formula for that specific shape. It answers: "If I bake 100 cookies, what are the odds exactly 10 burn?" without you needing to memorize complex tables.

2. Statistics 201: The "Detective's Toolkit"

• What it does: This app handles Confidence Intervals and Hypothesis Tests. This is where you act like a detective trying to prove if a claim is true or false based on evidence.
• The Analogy: Imagine you are a detective trying to figure out if a suspect is guilty. You have a pile of evidence (data).
  • Confidence Interval: This is like drawing a "fence" around your best guess. The app draws the fence and tells you, "We are 95% sure the truth is inside this fence."
  • Hypothesis Test: This is the courtroom trial. The app sets up the "Prosecution" (the null hypothesis) and the "Defense" (the alternative). It calculates the "evidence score" (test statistic) and draws a graph showing if the evidence is strong enough to convict (reject the null) or if the suspect goes free.
• How it helps: Instead of flipping through a textbook to find the right formula for a "t-test" or a "z-test," the app picks the right tool for your specific case, runs the numbers, and shows you the verdict with a clear visual graph.

3. Statistics 202: The "Crystal Ball" (Linear Regression)

• What it does: This app teaches Simple Linear Regression. This is about finding a straight line that connects two things (like studying hours vs. exam scores).
• The Analogy: Imagine you are trying to predict how tall a sunflower will grow based on how much water you give it. You have a scatter of dots (your data).
• How it helps: You drop your data into the app, and it draws the best-fit line through the dots.
  • It doesn't just draw the line; it breaks down the math step-by-step, showing you exactly how it calculated the slope (how much the sunflower grows per drop of water).
  • It gives you a "report card" telling you how good the line is.
  • It even checks the "health" of your data (diagnostics) to make sure the line makes sense, like checking if the sunflower is actually growing straight or if something is weird.

Why This Matters

The author noticed that in his own classes, students were spending 80% of their time struggling with the math mechanics (looking up numbers, typing code) and only 20% of their time actually thinking about what the results meant.

These apps are like a transparent calculator. They do the boring math instantly, but they keep the "curtain" open so you can see the gears turning. This frees up the student's brain to focus on the big picture: "What does this result actually mean for the real world?"

The Best Part?

Everything is free and open source. It's like giving everyone the blueprints to the simulator so teachers can tweak it, students can use it at home, and anyone can learn statistics without needing a degree in computer science.

Technical Summary

Based on the provided paper, here is a detailed technical summary of the Statistics 101, 201, and 202 suite of Shiny applications.

1. Problem Statement

Introductory statistics courses across disciplines (economics, biostatistics, social sciences) require students to perform complex computations, including:

• Calculating probabilities from standard distributions.
• Constructing confidence intervals.
• Conducting hypothesis tests.
• Fitting simple linear regression models.

Current limitations identified by the author include:

• Manual Calculation: Doing these by hand is time-consuming and prone to arithmetic errors.
• Programming Barriers: Using professional tools like R, Python, or SPSS requires programming knowledge that beginners often lack.
• Existing Tools:
  • Commercial textbook calculators are often proprietary.
  • General-purpose environments require code adaptation.
  • Web-based calculators (e.g., Wolfram Alpha) often lack comprehensive coverage (missing specific distributions, hypothesis testing, or regression modules) and fail to display the underlying mathematical logic alongside results.
  • Existing Shiny tools tend to focus on isolated topics rather than a cohesive introductory sequence.

2. Methodology and Technical Architecture

The author developed a suite of three open-source, interactive web applications using the R programming language and the Shiny framework. The design philosophy prioritizes a "transparent" interface where computation and statistical reasoning are presented side-by-side.

Key Technical Components:

• Core Framework: Built with R and Shiny for interactivity.
• Visualization: Plots are rendered using ggplot2 and made interactive via plotly (in the regression app).
• Mathematical Typesetting: Inline mathematical derivations and formulas are typeset using MathJax to match lecture notation.
• Data Handling: Supports both raw data input (comma-separated values) and summary statistics.
• Deployment: Hosted on shinyapps.io for immediate web access; source code is hosted on GitHub.
• Licensing: Released under a CC-BY-4.0 license to encourage adoption and modification.

3. Key Contributions: The Three Applications

A. Statistics 101: Probability Distributions

• Scope: Covers 18 probability distributions (e.g., Beta, Binomial, Normal, Poisson, Student's t, Weibull).
• Functionality:
  • Users select a distribution and input parameters via a dynamic sidebar.
  • Computes probabilities for lower tails, upper tails, and intervals.
• Output:
  1. Solution: Full mathematical notation of the probability statement (e.g., $P(X \le x)$) with numerical results.
  2. Plot: Density/mass function with the relevant probability region shaded.
  3. Details: Displays the PDF/PMF formula, Expectation, Standard Deviation, and Variance evaluated at the user's specific parameters.

B. Statistics 201: Confidence Intervals and Hypothesis Tests

• Scope: Covers seven inference settings:
  • One/Two means (z-test, t-test, paired t-test, Welch's correction).
  • One/Two proportions (z-test).
  • One/Two variances ($\chi^2$ test, F-test).
• Functionality:
  • Accepts raw data or summary statistics.
  • Allows customization of significance levels ($\alpha$) and hypothesis directions.
• Output: A structured four-step pedagogical workflow:
  1. Data: Summary statistics and raw data display.
  2. Confidence Interval: Formula with numerical substitution.
  3. Hypothesis Test: Step-by-step derivation (Hypotheses, Test Statistic, Critical Value, Conclusion).
  4. Interpretation: Plain-language conclusion including p-values.
  • Visualization: A sampling distribution plot (t, N, $\chi^2$, or F) showing the rejection region and the observed test statistic.

C. Statistics 202: Simple Linear Regression

• Scope: Fits the model $y = \beta_0 + \beta_1x + \varepsilon$.
• Functionality:
  • Accepts paired predictor ($x$) and response ($y$) data.
  • Validates data length and variance.
• Output:
  1. Data Table: Interactive table with export options.
  2. Step-by-Step Derivation: Explicit calculation of OLS estimators ($\hat{\beta}_1$ and $\hat{\beta}_0$) using the user's data, bridging the gap between symbolic formulas and numerical results.
  3. R Output: Full summary(lm(...)) output (coefficients, SE, t-stats, p-values, $R^2$).
  4. Interactive Plot: Scatter plot with the fitted line and confidence bands (via plotly).
  5. Interpretation: Plain-language explanation of coefficients based on significance.
  6. Diagnostics: Collapsible panel with six model-checking plots (linearity, homoscedasticity, etc.) generated by performance::check_model().
  7. Report Generation: Generates a downloadable HTML report via R Markdown, with an option to include/hide source code.

4. Results and Pedagogical Impact

• Bridging the Gap: The primary result is the successful integration of symbolic mathematics, numerical computation, and graphical representation in a single interface. This addresses the common student difficulty of connecting abstract formulas to software output.
• Efficiency: By automating the "mechanics" of looking up tables or computing test statistics, the tools free up class time for conceptual discussion and interpretation.
• Usability: The apps require no installation, registration, or programming knowledge. Students can access them immediately via a URL.
• Validation: The suite has been actively used in doctoral research, teaching assistantships, and as an invited lecturer tool at UCLouvain and UNamur for courses in probability, inferential statistics, and linear regression.

5. Significance

• Open Science: The suite is fully open-source, allowing instructors to fork, adapt, and extend the tools for their specific curricula.
• Comprehensive Coverage: Unlike fragmented web calculators, this suite covers the entire typical two-semester introductory statistics sequence (Probability $\to$ Inference $\to$ Regression) within a consistent design language.
• Pedagogical Innovation: It shifts the focus from "how to calculate" to "what the result means," serving as a complement to lectures rather than a replacement for conceptual learning.
• Reproducibility: The inclusion of R Markdown templates and clear GitHub repositories ensures that the statistical workflows are transparent and reproducible.