A central limit theorem dynamic parameter visualization system and platform special for teaching

Abstract

The application discloses a teaching special central limit theorem dynamic parameter visual system and platform, belongs to the cross field of education technology and data visualization, and aims to solve the pain points of the obscure teaching and the abstract convergence process of the theorem. The system covers six core modules of distribution generation, sample simulation, real-time visualization, parameter interaction, animation demonstration and convergence measurement. The distribution module covers 11 teaching core distributions, the sample simulation relies on a matrix architecture, is presented in a superimposed manner by means of an experience histogram and a normal curve, supports progressive animation demonstration of sample capacity of 1 to 500, can independently adjust distribution parameters, sample capacity and simulation times, and quantitatively displays indexes such as mean value and standard deviation. The system dynamically displays the convergence process through the closed loop process of distribution generation, sampling calculation and visual rendering, makes up for the defects of the prior art such as staticization and incomplete distribution coverage, can help students to intuitively understand the theorem and improve the teaching efficiency, and has teaching practicability, expandability and college application prospect.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of educational technology, computer data visualization, and probability and statistics simulation, specifically referring to the teaching scenario of probability and statistics courses. Specifically, it involves an interactive visualization system and platform designed to overcome the teaching difficulties of the central limit theorem. By dynamically displaying the process of sample mean approaching the normal distribution under different population distributions, it supports teachers and students to independently set population distributions and parameters, realizing intuitive teaching and exploratory learning of abstract theorems. It is suitable for probability and statistics courses in universities and vocational colleges, as well as extended mathematics teaching in middle schools. Background Technology

[0002] The Central Limit Theorem (CLT) is a core theory in probability and statistics. It reveals that the distribution of the sample mean of a large number of independent and identically distributed random variables converges to a normal distribution. It is a key link between probability theory and mathematical statistics, and its applications span multiple fields such as statistical inference, data analysis, and risk assessment. It is a key focus and core content of probability and statistics courses.

[0003] In probability and statistics teaching practice, the Central Limit Theorem, due to its abstract mathematical logic and complex proof process, has become a common teaching pain point for both teachers and students: students struggle to understand the core mechanism of "sample mean gradually converges to a normal distribution" through pure formula derivation and static charts, resorting to rote memorization of the conclusion without truly grasping the theorem's meaning and application scenarios; teachers lack efficient visualization teaching tools, failing to transform abstract concepts into observable and operable teaching processes, relying solely on verbal explanations and static blackboard writing, making it difficult to break down teaching difficulties, resulting in poor teaching effectiveness and low student interest. Existing related technologies are mainly geared towards scientific research simulation or general data visualization scenarios, exhibiting the following shortcomings that are seriously out of sync with teaching needs:

[0004] 1. Static presentations cannot break down teaching difficulties: Current technology can only present the final distribution results under a specific sample size, and cannot dynamically demonstrate the entire process of the sample mean distribution converging to the normal distribution as the sample size gradually increases. When teachers explain the core concept of "convergence," there is a lack of intuitive visual support, making it difficult to guide students to understand "how sample size affects the convergence effect." Students' understanding of the theorem remains superficial and cannot form a deep understanding.

[0005] 2. Insufficient support for population distributions and limited adaptability to teaching cases: Existing visualization tools mostly support only a few simple population distributions such as normal and uniform distributions, failing to cover the core distribution types in the probability and statistics curriculum syllabus, such as skewed exponential distribution, discrete Poisson distribution, chi-square distribution, t-distribution, and F-distribution. This prevents teachers from designing diverse teaching cases, makes it difficult to verify the universality of theorems, and limits students' comprehensive understanding of convergence rules under different distributions, thus failing to meet the syllabus's requirements for the breadth and depth of knowledge points.

[0006] 3. Lack of interactivity and inconsistency with teaching principles: Existing tools are mostly pre-set demonstration modes, which prevent students from independently adjusting key variables such as sample size and distribution parameters. They can only passively accept pre-set demonstration results, lacking the exploratory learning process of "hands-on operation - observation of phenomena - summarizing rules". This makes it difficult to deepen the understanding of theorems and also fails to cultivate experimental thinking and data analysis skills. Teachers also cannot adjust demonstration parameters in real time according to the teaching progress and students' acceptance level, making it impossible to achieve personalized teaching guidance and adapt to the learning needs of students at different levels.

[0007] 4. Poor adaptability to teaching and difficulty in meeting classroom needs: Traditional tools do not consider the special characteristics of teaching scenarios, have complex operation procedures, and parameter settings that do not conform to teaching logic. Students need to spend a lot of time learning how to operate the tools, which affects learning efficiency. At the same time, problems such as calculation delays and rendering lag are prone to occur in real-time classroom demonstrations, which cannot adapt to the fast pace of classroom teaching and have low teaching practicality. In addition, some tools have problems such as garbled Chinese characters and poor cross-platform compatibility, which affect the user experience of teachers and students and cannot run stably on different teaching devices.

[0008] To address the teaching pain points of the central limit theorem being "obscure and difficult to prove," and the numerous shortcomings of existing technologies in teaching scenarios, urgent

[0009] There is a need for a dedicated tool designed specifically for teaching scenarios, which is highly interactive, visually intuitive, easy to operate, and adaptable to the pace of the class. This tool would help teachers efficiently overcome teaching difficulties and enable students to understand the theorem's meaning by setting parameters independently and observing dynamic processes, thereby improving the teaching quality and learning outcomes of probability and statistics courses. Summary of the Invention

[0010] (a) Purpose of the invention

[0011] The core objective of this invention is to provide a dedicated visualization system and platform for the Central Limit Theorem, specifically designed to address the pain points of abstract and difficult-to-understand concepts and the lack of intuitive demonstration tools in teaching the Central Limit Theorem. Through the deep integration of dynamic parameter adjustment and visualization, the abstract theorem is transformed into an observable, operable, and explorable teaching process. This helps teachers quickly break down teaching difficulties such as "convergence" and "universality," design personalized teaching cases, and adapt to different teaching scenarios and student levels. Students can independently set the population distribution and its parameters, intuitively observe the convergence process, and deeply understand the core logic of the theorem through exploratory learning, achieving the teaching goal of "efficient teaching by teachers and easy learning by students," while simultaneously improving teaching efficiency and learning outcomes.

[0012] (II) Technical Solution

[0013] To achieve the aforementioned teaching-oriented objectives, the present invention adopts the following technical solution:

[0014] 1. System Architecture

[0015] A dynamic parameter visualization system for the central limit theorem, specifically designed for teaching, includes a distribution generation module, a sample simulation module, a real-time visualization module, a parameter interaction module, and a convergence metric module. These modules work collaboratively to meet teaching needs, enabling a dynamic, pedagogically-oriented demonstration of the theorem's convergence process, as detailed below:

[0016] (1) Distribution Generation Module: Designed specifically for teaching scenarios, its core function is to generate more than 10 core population distributions from the probability and statistics course syllabus, comprehensively covering the key points of the course, including but not limited to 0-1 distribution, binomial distribution, geometric distribution, uniform distribution, exponential distribution, normal distribution, chi-square distribution, t distribution, F distribution, and Poisson distribution. This module can dynamically adjust the shape of the population distribution according to the distribution parameters input by teachers and students. The distribution parameters include but are not limited to the success probability of 0-1 distribution, the number of trials and success probability of binomial distribution, the success probability of geometric distribution, the interval range of uniform distribution, the rate parameter of exponential distribution, the mean and standard deviation of normal distribution, the degrees of freedom of chi-square distribution, the degrees of freedom of t distribution, the numerator and denominator degrees of freedom of F distribution, and the rate parameter of Poisson distribution. It accurately matches the needs of teaching cases, ensures coverage of the core teaching content of the course, and supports teachers in designing diverse teaching scenarios to verify the universality of theorems.

[0017] (2) Sample Simulation Module: Based on the sample size n and simulation times N set by teachers and students, samples are randomly drawn from the population distribution generated by the distribution generation module to construct an N x n sample data matrix (each row represents n samples in one sampling). The mean of each sampling is efficiently calculated, and finally a set of means containing the means of N samples is obtained, providing data support for subsequent visualization. This module adopts an optimized calculation algorithm to ensure fast response and no delay or stuttering in scenarios with large sample sizes (n≤5000) and high simulation times (N≤10000) commonly used in teaching, adapting to the fast pace of classroom teaching and ensuring the smoothness of real-time demonstration.

[0018] (3) Real-time visualization module: Designed for teaching demonstration needs, its core function is to visualize the set of sample means in an intuitive and easy-to-understand way. Specifically, this includes drawing an empirical distribution histogram of the sample means and calculating the corresponding theoretical normal distribution parameters based on the central limit theorem (mean is the population mean, standard deviation is the population standard deviation / ...). The system can plot the theoretical normal distribution curve and the empirical distribution histogram for comparison, making it easy for teachers and students to visually observe the fitting effect of the two. At the same time, it supports progressive animation rendering as the sample size gradually increases. Through smooth animation transition, it clearly shows the complete process of the sample mean distribution converging from non-normal to normal, helping teachers to break down the teaching focus of "the influence of sample size on convergence", allowing students to intuitively perceive the dynamic formation mechanism of "convergence" and transforming abstract concepts into concrete visual experiences.

[0019] (4) Parameter interaction module: Based on teaching and operation habits, a simple and easy-to-understand graphical user interface (GUI) is designed. The core function is to provide full parameters.

[0020] The system features real-time interactive functionality, allowing teachers and students to independently adjust teaching demonstration parameters. Interactive controls include drop-down menus and sliders. The drop-down menus categorize population distribution types according to teaching logic, such as "discrete distribution," "continuous distribution," and "skewed distribution," facilitating quick location of the corresponding distribution and adaptation to the teaching process. The sliders indicate commonly used parameter ranges and key critical values ​​(e.g., the sample size slider indicates 30, 100, and other critical values ​​commonly used for the Central Limit Theorem), allowing teachers and students to select the population distribution type, adjust population distribution parameters, and set the sample size n and the number of simulations N in real time. This module can capture parameter adjustments made by teachers and students in real time and quickly transmit the adjusted parameters to the distribution generation module and sample simulation module, triggering system recalculation and visualization updates. This meets the dual needs of real-time classroom demonstrations and student self-exploration, offering convenient operation without the need for complex training.

[0021] (5) Animation Demonstration Module: Adapts to the real-time demonstration needs of classroom teaching, and realizes a smooth animation display of the convergence process under increasing sample size, specifically including:

[0022] S1: Animation status control: Preset running / termination status flags to avoid repeated triggering and program errors;

[0023] S2: Interactive controls: Set "Start" and "Stop" demo buttons. The start button is automatically disabled when the animation is running, and the stop button can stop the animation and reset the state.

[0024] S3: Animation gradient: Set the animation gradient with a sample size range of 1 to 500 and an increment step of 10 to balance the demonstration effect and system resources;

[0025] S4: Visualization and rendering: Render the mean distribution histogram and fitted normal curve of the corresponding sample size frame by frame, adapt the chart size to the teaching display, and label the distribution type and the current sample size;

[0026] S5: Real-time indicator display: Synchronously updates sample mean, expected value, standard deviation, skewness (color-coded to indicate convergence), kurtosis, and normality assessment results;

[0027] S6: Resource optimization: Release chart resources after each frame is rendered, and set a 0.5-second inter-frame delay to ensure smooth animation;

[0028] (6) Convergence Measurement Module: A teaching enhancement module whose core function is to quantify the degree of fit between the sample mean distribution and the theoretical normal distribution, providing quantitative reference for teaching explanation and student self-verification. This module focuses on the quantitative auxiliary module for teaching the Central Limit Theorem. Its core function is to comprehensively present the characteristic data of the sample mean distribution, transforming the process of "distribution converging towards normality" into observable and verifiable indicators to support teaching explanation and student self-exploration. The module will simultaneously display four types of core information: First, basic statistics, presenting the expected value of the sample mean and the standard deviation of the sample mean, which intuitively correspond to the conclusion in the Central Limit Theorem that "the expected value of the sample mean is equal to the population expected value, and the standard deviation is related to the sample size"; Second, distribution shape indicators, calculating and displaying skewness and kurtosis, where skewness is used to describe the degree of asymmetry of the distribution, and kurtosis is used to describe the steepness of the distribution. The closer the skewness is to 0 and the closer the kurtosis is to 3, the closer the sample mean distribution is to the normal distribution; Finally, based on these indicators, a normality judgment is given ("close or not?"). This conclusion simultaneously reflects the degree of fit between the sample mean distribution and the theoretical normal distribution. The entire module not only matches the teaching logic of explaining the characteristics of the normal distribution, but also allows teachers to quantify and decompose the convergence process through the combination of "basic statistics + morphological indicators + conclusions". At the same time, it helps students establish a cognitive model of "intuitive observation of graphs + verification by multi-dimensional indicators", transforming abstract theorems into perceptible and verifiable concrete data, and deepening their understanding of the central limit theorem.

[0029] 2. Core Methodology and Flow

[0030] A visualization system and platform for dynamic parameters of the central limit theorem specifically for teaching, implemented using the aforementioned system, includes the following steps:

[0031] S1: System initialization. The parameter interaction module loads default parameters that meet the needs of introductory teaching, including the default population distribution type (such as uniform distribution, which is simple and easy for students to understand the basic logic of the central limit theorem), default population distribution parameters (such as the interval of uniform distribution being [0,1]), default sample size n=30 (which meets the critical value commonly used in teaching the central limit theorem and is convenient for teachers to explain the concept of "large sample"), and default number of simulations N=2000 (balancing visualization effects and calculation speed to ensure a fast initial presentation).

[0032] S2: The distribution generation module generates an initial parent distribution based on default parameters, preparing for a quick start demonstration in classroom teaching. Teachers can directly explain basic concepts without additional configuration.

[0033] S3: The sample simulation module generates an N-row, n-column sample data matrix based on the initial population distribution, the default sample size n, and the number of simulations N, and calculates the sample mean by row to obtain a set of means containing N sample means.

[0034] S4: The real-time visualization module draws an empirical distribution histogram of the sample mean and overlays it with the theoretical normal distribution curve to complete the initial visualization. Teachers can use this visualization to explain basic concepts such as "the relationship between the sample mean distribution and the normal distribution" to students.

[0035] S5: Teachers and students can adjust parameters through the parameter interaction module, including but not limited to switching the population distribution type (e.g., switching from a uniform distribution to a skewed exponential distribution to compare the convergence differences between different distributions), modifying population distribution parameters (e.g., adjusting the success probability of a binomial distribution to observe the impact of parameter changes on convergence), adjusting the sample size n (e.g., gradually adjusting from n=1 to n=10, 30, 50, 100 to visually demonstrate the convergence process), or the number of simulations N (e.g., adjusting N=1000, 5000, 10000 to observe the impact of the number of simulations on the visualization effect).

[0036] S6: The parameter interaction module captures the parameter adjustment operations of teachers and students in real time and quickly transmits the adjusted parameters to the distribution generation module and the sample simulation module.

[0037] S7: The distribution generation module regenerates the population distribution based on the adjusted parameters, and the sample simulation module regenerates the sample data matrix based on the new population distribution and parameters, and calculates the new set of sample means row by row to ensure that the response speed meets the pace of classroom teaching. The data update is completed within 1-2 seconds after the parameter adjustment.

[0038] S8: The real-time visualization module redraws the empirical distribution histogram and the theoretical normal distribution curve based on the new sample mean set, achieving real-time updates; if the adjusted parameter is the sample size n, a progressive animation rendering method is used, with a smooth transition animation of 0.3-0.5 seconds, gradually showing the dynamic process of the sample mean distribution converging to the normal distribution as n increases, helping students intuitively understand the core conclusion that "the larger the sample size, the more obvious the convergence effect";

[0039] S9: Optionally, the convergence metric module calculates the convergence index (such as skewness and kurtosis) of the new sample mean set and displays it clearly through the GUI interface. The index is labeled with simple and easy-to-understand explanations (such as "skewness ≈ 0: the distribution is close to normal"). Teachers can combine the index to quantitatively explain the convergence effect to students, and students can also use the index to independently verify the convergence law under different parameters.

[0040] S10: Repeat steps 5-9 until the teaching demonstration is completed or the students' independent exploration ends. Teachers are encouraged to guide students to observe key phenomena step by step according to the teaching process, and students are also encouraged to conduct free experiments and summarize the rules independently.

[0041] 3. Key Technology Implementation Details

[0042] (1) Cross-platform font and display optimization technology for teaching adaptation: In the real-time visualization module, in order to solve the problem of Chinese character encoding when using different devices in the teaching scenario, cross-platform font adaptation technology is adopted. By detecting the operating system type (Windows / MacOS), the corresponding Chinese font is automatically set - the SimHei font is used for Windows system and the Arial Unicode MS font is used for MacOS system. At the same time, it ensures that special characters such as minus signs and mathematical symbols are displayed normally, ensuring the readability of visualization charts. In addition, the chart labels adopt concise and easy-to-understand teaching expressions, avoiding the accumulation of professional terms. The coordinate axis labels, titles and other contents are clear and easy to understand, making it easy for students to quickly understand the meaning of the charts and adapt to the teaching display needs.

[0043] (2) Efficient calculation algorithm adapted to classroom pace: The sample simulation module adopts a matrix-based data processing method, storing sample data through an N x n matrix. Vector operations are used to replace the traditional loop calculation method to efficiently calculate the sample mean, improving the calculation efficiency by more than 50% compared to the traditional method. This algorithm ensures that the system can still respond quickly and without delay in scenarios with large sample sizes (n≤5000) and high simulation times (N≤10000) commonly used in teaching. It adapts to the fast pace of classroom teaching, ensures the smoothness of the demonstration when teachers adjust parameters in real time, and avoids affecting the teaching progress due to calculation delays.

[0044] (3) Real-time rendering technology for teaching visualization optimization: The real-time visualization module adopts an incremental data update strategy. When teachers and students adjust the sample size n, only the mean data and visualization content corresponding to the newly added sample size are updated. There is no need to recalculate and render the entire data, which greatly reduces the rendering time. At the same time, the plotting parameters are optimized by setting the number of histogram bins to 50. This value can ensure the clarity of the chart.

[0045] It accurately reflects the shape of the sample mean distribution, while avoiding the slow rendering problem caused by too many bins, balancing visualization effect and rendering speed; the smooth transition animation is used in dynamic rendering, making the changes in the convergence process easier for students to observe and capture, helping students to establish an intuitive understanding of "the sample size gradually increases → the convergence effect gradually improves".

[0046] 4. Technological Innovations

[0047] The core innovation of this invention is deeply designed around teaching needs, and it has significant non-obviousness compared with existing technologies, as detailed below:

[0048] (1) Teaching-oriented multi-distribution dynamic adaptation technology: It specifically covers more than 10 core population distributions in probability and statistics courses, which are highly consistent with the teaching syllabus and solve the problem that existing technologies support only one distribution and cannot meet the diverse teaching cases. By realizing the real-time coupling of population distribution parameters with sample simulation and visualization modules, the system can respond immediately after teachers and students adjust the parameters, regenerate the distribution and update the visualization results. This provides technical support for teachers to design personalized teaching scenarios (such as comparing the convergence differences between discrete and continuous distributions, and the convergence speed of skewed and symmetric distributions), and helps students fully understand the universality of theorems.

[0049] (2) Progressive convergence visualization technology for breaking down teaching difficulties: This innovative technology uses progressive animation rendering to dynamically display the complete process of the sample mean distribution converging from non-normal to normal, transforming the abstract concept of "convergence" into an intuitive and visible dynamic process. Existing technologies can only display the final result, while this technology can break down the entire process of "increasing sample size → changing distribution shape → gradually approaching normality," helping teachers to accurately break down teaching difficulties and allowing students to quickly understand the core logic of the theorem by observing the animation. This overcomes the limitations of traditional static displays and conforms to the cognitive law of "from shallow to deep, gradually guiding" in teaching.

[0050] (3) Interactive teaching design for two-way interaction between teachers and students: Construct a full-parameter interactive system covering "distribution type - distribution parameters - sample size - number of simulations". The interactive controls are optimized around teaching needs - the drop-down menu is classified according to teaching logic, and the slider marks the commonly used teaching range and critical values. The operation is simple and easy to understand, and no complicated training is required to get started. This design supports teachers to adjust demonstration parameters in real time and guide students to observe key phenomena to achieve personalized teaching guidance; it also supports students to set parameters independently and carry out exploratory experiments to achieve a two-way teaching model of "teacher guidance - student autonomy". It is in line with modern educational concepts and solves the problems of lack of interactivity and non-compliance with teaching laws in existing technologies.

[0051] (4) Highly efficient technical solutions for deep adaptation to classroom scenarios: Integrating matrix-based data calculation, incremental updates, and cross-platform adaptation optimization technologies, the technical solutions are deeply integrated with the teaching scenarios. Highly efficient calculation algorithms ensure rapid response under large sample sizes and high simulation times, adapting to the pace of classroom teaching; incremental rendering technology ensures real-time updates after parameter adjustments, avoiding lag; cross-platform font adaptation solves the problem of using different devices, and the ease of operation reduces the workload of teachers in preparing for teaching, comprehensively solving the defects of traditional tools such as low efficiency, complex operation, and incompatibility with teaching scenarios.

[0052] (III) Beneficial Effects

[0053] Compared with existing technologies, this invention is specifically designed for teaching scenarios and has the following significant advantages:

[0054] 1. Addressing teaching pain points and significantly improving teaching quality: Transforming the obscure and difficult-to-prove central limit theorem into an intuitive and dynamic visualization process helps teachers quickly break down core teaching difficulties such as "convergence" and "universality," enabling students to understand the theorem's meaning without relying on complex formula derivations. After trial use in multiple universities, teachers' teaching efficiency has increased by over 30%, and preparation time has decreased by 40%; students' correct understanding of the theorem has increased from 52% to 93%, and their learning interest has significantly improved. It is suitable for various scenarios, including probability and statistics courses in universities and vocational schools, as well as extended mathematics teaching in secondary schools.

[0055] 2. Supporting Exploratory Learning and Cultivating Core Competencies: Students can independently set the population distribution and its parameters, adjust the sample size and number of simulations, and through an exploratory learning process of "hands-on operation - observation of phenomena - analysis of indicators - summarization of patterns," they transform from passively receiving knowledge to actively exploring patterns. This not only enables students to master the central limit theorem but also cultivates core competencies such as experimental thinking, data analysis skills, and logical reasoning abilities, aligning with the requirements of quality education and enhancing students' overall competitiveness.

[0056] 3. Highly adaptable to teaching, convenient and efficient to use: The interactive controls are designed to fit the operating habits of teachers and students, and the parameter range is adapted to commonly used values ​​in teaching cases.

[0057] It can be used without complicated training; cross-platform adaptation technology ensures smooth operation on classroom computers and student personal devices (Windows / MacOS systems) without additional configuration; efficient computing and rendering technology ensures real-time classroom demonstrations without delay, and visualization updates can be completed within 1-2 seconds after parameter adjustments, greatly reducing the workload of teachers in teaching preparation and improving teaching efficiency and user experience.

[0058] 4. Excellent scalability, adaptable to teaching upgrade needs: Adopting a modular architecture design, each module functions independently yet collaboratively, facilitating future functional expansion based on teaching requirements. For example, it can add commonly used custom distribution types to meet the needs of specific teaching cases; it can add convergence metrics (such as goodness-of-fit tests) tailored to teaching, enriching quantitative teaching methods; it can expand online classroom collaboration functions, supporting shared experimental scenarios between teachers and students, adapting to online teaching needs; and it can be adapted to mobile terminals for students' after-class review, achieving a closed loop of "classroom teaching + after-class consolidation," possessing broad prospects for teaching applications. Attached Figure Description

[0059] Figure 1 System architecture diagram (module composition and data flow) in this embodiment of the invention;

[0060] Figure 2: A schematic diagram of the user interface in an embodiment of the present invention;

[0061] Figure 3 : Static simulation flowchart (manually adjustable process) in the embodiments of this invention;

[0062] Figure 4 In this embodiment of the invention, the flowchart (progressive convergence process) is not animated.

[0063] The present invention includes four accompanying drawings to clearly illustrate the system architecture, interactive interface, and core workflow of the interactive simulation teaching platform for the central limit theorem. Specific descriptions of each drawing are as follows:

[0064] Figure 1 System Overall Architecture Diagram (Module Composition and Data Flow)

[0065] in:

[0066] 1 represents the initialization module, which is used to complete the basic configuration when the platform starts up;

[0067] 1.1 represents the font configuration submodule, which is used to implement local loading or system compatibility configuration of Chinese fonts, and solve the problem of Chinese display in visual charts;

[0068] 1.2 Represents the page configuration submodule, used to set the platform's page title, layout style, and initial sidebar state;

[0069] 2 represents the parameter configuration module, deployed in the platform sidebar, providing users with an interactive configuration entry point;

[0070] 2.1 The distribution type selection submodule provides a drop-down selection function for 11 parent distribution types;

[0071] 2.2 Represents the parent parameter adjustment submodule, which provides corresponding parameter adjustment sliders based on the selected distribution type;

[0072] 2.3 represents the CLT sampling parameter adjustment submodule, which provides adjustment functions for sample size n and number of simulations N;

[0073] 3 represents the data generation module, which is used to generate a sample dataset of a specified size according to the configuration parameters and calculate the sample mean array;

[0074] 4 represents the statistical analysis module, which is used to perform statistical calculations on the sample mean array and output indicators such as expected value, standard deviation, skewness, and kurtosis.

[0075] 5 represents the visualization module, used to display data in charts and graphs;

[0076] 5.1 represents the static chart generation submodule, used to draw the empirical distribution histogram of the sample mean and the fitted normal curve after manual parameter adjustment;

[0077] 5.2 represents the animation chart generation submodule, used to draw dynamic convergence charts during the asymptotic process of sample size n;

[0078] 6 represents the interactive control module, which is used to respond to the trigger command of the animation demonstration button, control the dynamic update of the chart placeholders and the animation playback delay.

[0079] The arrows in the diagram represent the data flow between modules. The overall process is as follows: Initialization module → Parameter configuration module → Data generation module → Statistical analysis module → Visualization module → Front-end display layer.

[0080] Figure 2. Schematic diagram of the interactive interface. It intuitively shows the functional area division of the platform.

[0081] in:

[0082] 1 represents the title area, used to display the platform name "Central Limit Theorem (CLT) Interactive Simulation Platform" and functional description text;

[0083] 2 represents the sidebar parameter configuration area, which integrates distribution type selection and parameter adjustment functions;

[0084] 2.1 The drop-down selection box for distribution type is used to display a list of available parent distribution types;

[0085] 2.2 The parent parameter adjustment slider group provides dedicated parameter adjustment controls for the selected distribution;

[0086] 2.3 This represents the sampling parameter adjustment slider group, which provides control for adjusting the sample size n and the number of simulations N;

[0087] 3 represents the main interface function area, used to display core interactive functions and simulation results;

[0088] 3.1 represents the animation demonstration button, used to trigger the animation demonstration process of asymptotic convergence of sample size n;

[0089] 3.2 The chart display area is used to display the empirical distribution histogram of the sample mean and the fitted normal curve;

[0090] 3.3 Represents the statistical indicator display area, which is used to display the expected value, standard deviation, skewness, kurtosis and normality judgment results of the sample mean in the form of cards.

[0091] Figure 3. Static simulation flowchart. This illustrates the core execution steps when manually adjusting parameters.

[0092] in:

[0093] 1. Select the parent distribution type for the user and adjust the corresponding parameters;

[0094] 2. Set the sample size n and the number of simulations N for the user;

[0095] 3. Generate N datasets, each with n samples, for the data generation module;

[0096] 4. To calculate the mean of each sample group, we obtain a sample mean array;

[0097] 5. Calculate the expected value, standard deviation, skewness, kurtosis, and other indicators of the sample mean array for the statistical analysis module;

[0098] 6. The static chart generation submodule generates a histogram of the empirical distribution of the sample mean and a fitted normal curve.

[0099] 7. Output statistical result cards for the statistical indicator display area.

[0100] Figure 4. Animated flowchart demonstrating the core execution steps of the asymptotic convergence process with sample size n.

[0101] in:

[0102] 1. When the user clicks the "Animation Demo" button, the animation process is triggered;

[0103] 2. To initialize the sample size n=1, set the step size to 10;

[0104] 3. Generate an array of sample mean values ​​corresponding to the current n for the data generation module;

[0105] 4. The animation chart generation submodule draws a sample mean distribution chart for the current n, while the statistical analysis module calculates the corresponding indicators.

[0106] 5. Perform a 0.1-second delay operation on the interactive control module to control the animation playback speed;

[0107] 6. To determine if the current n is ≤ 500, if the result is "yes", then execute step 7, increase n by 10 and return to step 3 to repeat the process; if the result is "no", then end the animation process. Detailed Implementation

[0108] The following section, using actual teaching scenarios from university probability and statistics courses, details how students can use this system to intuitively learn and explore the central limit theorem, enabling those skilled in the art to clearly understand the system's teaching application logic and operational procedures, while highlighting the core design goal of "student-led operation and intuitive perception."

[0109] (a) Preparations before using the system

[0110] 1. Equipment and Environmental Requirements

[0111] Student-end devices: Computers, tablets, or mobile phones that support Windows 7 and above, or macOS 10.15 and above (no specific device required).

[0112] High-performance configuration, ordinary learning devices are sufficient);

[0113] Software environment: The system has been pre-deployed on the school's teaching server or an offline installation package has been provided, so students do not need to manually configure the Python environment and dependency libraries;

[0114] Access method: Enter the teaching server address (https: / / clt-interactive-simulation-teaching-platform-vubtf9f6awaszmw9b.streamlit.app / ) in your browser to directly access the system interface without installing any software;

[0115] 2. System initial interface boot

[0116] After students launch the system, they will see a simple and intuitive teaching interface, with the core area divided into two parts:

[0117] The left-hand “Configuration Parameter Area” is logically laid out as “Distribution Selection → Parameter Settings → Sampling Configuration”, with each control accompanied by Chinese prompts (such as “Sample size n: Drag to observe convergence changes”), which can be understood without professional knowledge;

[0118] The "Visualization Area" on the right side displays the empirical distribution histogram of the sample mean and the theoretical normal distribution curve of the "0-1 distribution (p=0.5, n=30, N=2000)" by default, with the title "Interactive Simulation Platform for Central Limit Theorem (CLT)". Key parameters are marked in the charts to help students gain an initial understanding.

[0119] (II) Student-led learning and exploration process

[0120] This system is designed with a three-stage learning path: "basic cognition → hands-on experimentation → pattern summarization". Students can operate independently by following the steps, or they can explore in a targeted manner according to the experimental tasks assigned by the teacher.

[0121] 1. First level: Basic cognition – Intuitively experiencing the core phenomena of the theorem

[0122] (1) Operating steps:

[0123] After starting the system, there is no need to adjust any parameters. Just observe the visualization area on the right: the blue histogram is the "uniformly distributed sample mean distribution", and the red dashed line is the "theoretical normal distribution". Students can intuitively see that "even if the population is uniformly distributed, the sample mean distribution is close to normal".

[0124] Look at the "Key Indicators" section at the bottom of the interface, and pay special attention to "Sample Mean Standard Deviation" and "Theoretical Standard Deviation": Since their values ​​are close (e.g., 0.182 and 0.177 respectively), students can begin to understand that "Sample Mean Standard Deviation = Population Standard Deviation / (Standard Deviation of Sample Mean)". The core formula of "".

[0125] Click the "Select Population Distribution Type" button on the left side of the interface to quickly switch to "Exponential Distribution (Shady Distribution)". Observe that even if the population distribution is asymmetrical, the sample mean distribution still exhibits the basic characteristics of a normal distribution, and get a preliminary sense of the "universality" of the Central Limit Theorem.

[0126] (2) Learning objectives:

[0127] Without complicated operations, the default demonstration and quick switching allow students to intuitively establish a preliminary understanding of "the distribution of sample mean approaches normality", breaking down their fear of abstract theorems.

[0128] 2. Second-order: Hands-on experiment – ​​Adjusting parameters to explore convergence rules

[0129] In this stage, students independently adjust the core parameters and observe the impact of parameter changes on the convergence process, achieving in-depth learning through "hands-on operation → observation of phenomena → correlation theory".

[0130] Experiment 1: Adjust the sample size n and observe the changes in convergence.

[0131] Operating steps:

[0132] (1) In the “CLT Sampling Parameters” area on the left, find the “Sample Size n” slider (marked with teaching critical values: 1, 30, 100, 500). Start from n=1 and gradually drag it to n=10, 30, 50, 100, 500. The maximum value can be dragged to 5000.

[0133] Each time n is adjusted, observe the changes in the visualization area on the right: whether the histogram shape becomes more and more symmetrical, and whether it gets closer and closer to the red normal curve;

[0134] (2) Record the "skewness" value of the "key indicator area" (mark "skewness ≈ 0 indicates normal distribution"), such as skewness = 1.8 when n=1, skewness = 0.32 when n=30, and skewness = 0.11 when n=100;

[0135] (3) Click the “Animation Demonstration” button and the system will automatically play the gradual convergence process of n from 1 to 500, intuitively presenting the dynamic trend that “the larger the sample size, the more obvious the convergence effect”.

[0136] Reflection and Summary: Through observation, students can independently conclude that "the larger the sample size n is, the closer the sample mean distribution is to the normal distribution, and 30 is the critical value for which the central limit theorem applies," and connect this to the concept of "large sample condition" in the textbook.

[0137] Experiment 2: Switching the population distribution type to verify the universality of the theorem.

[0138] Operating steps:

[0139] (1) In the “Select Population Distribution Type” area on the left, select the 0-1 distribution, binomial distribution (p=0.1), Poisson distribution (λ=2), and chi-square distribution (df=5) in the “Discrete distribution / Continuous distribution / Shaysteric distribution” category menu.

[0140] (2) With a fixed sample size of n=30, observe the changes in histogram shape and skewness value after each distribution change: for example, skewness of 0-1 distribution = 0.21, skewness of binomial distribution (p=0.1) = 0.65, and skewness of chi-square distribution (df=5) = 0.73;

[0141] (3) Compare the convergence effects of different distributions and record the differences in convergence speed between "discrete distribution vs. continuous distribution" and "symmetric distribution vs. skewed distribution" (e.g., discrete symmetric distribution converges faster).

[0142] Reflection and Summary: Students independently verify that "regardless of whether the population is discrete or continuous, symmetric or skewed, as long as the sample size is large enough, the sample mean distribution will approach a normal distribution," deepening their understanding of the universality of the theorem.

[0143] Experiment 3: Adjusting the population distribution parameters and observing the effect on convergence speed

[0144] Operating steps:

[0145] (1) Select "binomial distribution", fix the sample size n=30, and adjust the "success probability p" slider from p=0.1 to p=0.5 and p=0.9;

[0146] (2) Observe the changes in skewness values: when p=0.1, skewness = 0.65; when p=0.5, skewness = 0.21; when p=0.9, skewness = 0.63.

[0147] (3) Select “Poisson distribution”, fix n=30, adjust “rate parameter λ” from 2 to 5 and 10, observe that the histogram shape becomes more and more symmetrical, and the skewness value decreases from 0.89 to 0.35.

[0148] Reflection and Summary: Students discovered that "the more symmetrical the population distribution, the faster the convergence speed," and understood the core logic that "the population distribution parameters do not affect the applicability of the theorem, but they do affect the convergence speed."

[0149] 3. Third Stage: Group Collaboration – Completing Exploratory Experimental Tasks

[0150] After the teacher assigns the experimental tasks, students can work in groups to use the system to conduct collaborative exploration and cultivate experimental thinking and data analysis skills.

[0151] Example Task: "Comparison of the Influence of Different Factors on the Convergence Effect of the Central Limit Theorem"

[0152] Task division: Each member of the group is responsible for a sub-experiment on "Influence of Sample Size", "Influence of Distribution Type", and "Influence of Parameter Size" respectively;

[0153] Operating procedures:

[0154] (1) Each group sets a uniform number of simulations N=5000 (to ensure data reliability), and completes parameter adjustment and data recording according to their respective tasks;

[0155] (2) Using an Excel spreadsheet or notebook, record the "skewness value, standard deviation of sample mean, and theoretical standard deviation" for different parameters;

[0156] Summary of Results:

[0157] (1) Compare and analyze the data within the group and draw a "parameter-skewness" relationship diagram (the system supports one-click generation of analysis charts);

[0158] (2) Discuss and summarize the influence of "sample size, distribution type, and population parameters" on the convergence effect, and form an experimental report;

[0159] Classroom presentation: Through the system's "screen sharing" function, group representatives present the operation process, visual results, and experimental conclusions to the whole class, and the teacher provides comments and corrections.

[0160] (III) Application Process of Classroom Teaching Under Teacher Guidance

[0161] In classroom teaching, teachers can guide students to explore collectively and efficiently break down teaching difficulties through the following process:

[0162] 1. Introduction (5 minutes): The teacher starts the system, displays the visualization results of the "exponential distribution (n=1)" (obvious skewness), and asks, "What will this distribution look like when the sample size increases?" to stimulate students' interest;

[0163] 2. Group activity (10 minutes): The teacher guides all students to simultaneously drag the "sample size n" slider, gradually increasing it from n=1 to n=100, and observe the change process of "skewness decreasing from 1.8 to 0.1" together, explaining the core concept of "convergence";

[0164] 3. Group comparison (15 minutes): Divide students into "discrete distribution group", "continuous distribution group" and "skewed distribution group", with each group responsible for exploring one type of distribution and recording convergence data;

[0165] 4. Summary and Conclusion (10 minutes): Each group reports its results. The teacher uses the system to quickly switch between different distribution visualization comparison charts to guide students to summarize the core conclusions of the Central Limit Theorem and connect the textbook formulas with the theory.

[0166] (iv) Verification of teaching application effectiveness

[0167] This system was piloted in a class of over 120 students at East China University of Technology, and the results fully reflected the design goal of "student-led operation and intuitive understanding":

[0168] Learning experience feedback: 92% of students said that "by dragging the parameters and observing the changes, they can quickly understand the concept of convergence", and 88% of students believed that "hands-on experiments are easier to grasp than pure formula explanations".

[0169] Knowledge mastery effect: The accuracy rate of students' understanding of the core logic (convergence and universality) of the central limit theorem has increased from 52% in traditional teaching to 93%, and they can independently complete the entire process of "parameter adjustment - phenomenon observation - conclusion summary";

[0170] Effectiveness of skills development: Teachers reported that "students' experimental thinking and data analysis abilities have significantly improved," and they are able to independently design exploration plans and analyze experimental data in group collaborative experiments.

[0171] Ease of use: The average time for students to start the system and complete the first parameter adjustment is ≤3 minutes. They can operate it independently without professional guidance, which is suitable for the pace of classroom teaching.

[0172] The above verification results show that this system, through its design of "intuitive visualization + autonomous operation + collaborative exploration," effectively lowers the learning threshold of the central limit theorem, enabling students to shift from "passively listening" to "actively doing," fully demonstrating the practicality and advancement of teaching tools, and significantly improving teaching effectiveness and learning efficiency.

Claims

1. A visualization system and platform for dynamic parameters of the central limit theorem specifically for teaching, characterized in that, Designed specifically for probability and statistics courses, this module includes modules for distribution selection, sample simulation, real-time visualization, parameter interaction, and an optional convergence metric. These modules work together to provide a dynamic, pedagogical demonstration of the central limit theorem's convergence process, as detailed below: (1) Distribution selection module: Supports more than 10 core population distributions in the probability and statistics course syllabus, including but not limited to 0-1 distribution, binomial distribution, geometric distribution, uniform distribution, exponential distribution, normal distribution, chi-square distribution, t distribution, F distribution, and Poisson distribution. The population distribution shape can be dynamically adjusted according to the distribution parameters input by teachers and students. The distribution parameters include the success probability of 0-1 distribution, the number of trials and success probability of binomial distribution, and the degrees of freedom of chi-square distribution. (2) Sample simulation module: Based on the sample size n and simulation times N set by teachers and students, samples are randomly drawn from the population distribution generated by the distribution generation module to construct an N rows and n columns of sample data matrix. Each row represents n samples in one sampling. The sample mean of each sampling is calculated to obtain a set of mean values ​​containing N sample mean values. (3) Real-time visualization module: Draws an empirical distribution histogram of the sample mean, and calculates the theoretical normal distribution parameters based on the central limit theorem (mean is the population mean, standard deviation is the population standard deviation / ...). It can plot the theoretical normal distribution curve and the empirical distribution histogram for comparison, and supports progressive animation rendering as the sample size gradually increases, dynamically displaying the convergence process; (4) Parameter interaction module: Provides graphical user interface interactive controls, including drop-down menus and sliders. The drop-down menus display the population distribution types according to the teaching logic. The sliders mark the range and critical values ​​of commonly used teaching parameters, allowing teachers and students to select the population distribution type, adjust the population distribution parameters, set the sample size n and the number of simulations N in real time. The parameter adjustment operation is captured in real time and transmitted to the distribution generation module and the sample simulation module, triggering the system to recalculate and update the visualization. (5) Animation Demonstration Module: Adapts to the real-time demonstration needs of classroom teaching, and realizes a smooth animation display of the convergence process under increasing sample size, specifically including: S1: Animation status control: Preset running / termination status flags to avoid repeated triggering and program errors; S2: Interactive controls: Set "Start" and "Stop" demo buttons. The start button is automatically disabled when the animation is running, and the stop button can stop the animation and reset the state. S3: Animation gradient: Set the animation gradient with a sample size range of 1 to 500 and an increment step of 10 to balance the demonstration effect and system resources; S4: Visualization and rendering: Render the mean distribution histogram and fitted normal curve of the corresponding sample size frame by frame, adapt the chart size to the teaching display, and label the distribution type and the current sample size; S5: Real-time indicator display: Synchronously updates sample mean, expected value, standard deviation, skewness (color-coded to indicate convergence), kurtosis, and normality assessment results; S6: Resource optimization: Release chart resources after each frame is rendered, and set a 0.5-second inter-frame delay to ensure smooth animation; (6) Convergence metric module: Calculates and displays convergence indices such as distribution skewness and kurtosis that are easy to explain in teaching, and quantifies the degree of fit between the sample mean distribution and the theoretical normal distribution.

2. The teaching-specific central limit theorem dynamic parameter visualization system according to claim 1, characterized in that, The real-time visualization module has cross-platform font adaptation capabilities. By detecting the operating system type (Windows / MacOS), it automatically sets the corresponding Chinese font (SimHei font for Windows systems and Arial Unicode MS font for MacOS systems) to ensure that minus signs and mathematical symbols are displayed correctly.

3. The teaching-specific dynamic parameter visualization system for the central limit theorem according to claim 1, characterized in that, The sample simulation module adopts a matrix-based data processing method, which calculates the sample mean through vector operations, thereby improving the calculation efficiency under large sample size and high simulation frequency, and adapting to the pace of classroom teaching.

4. The teaching-specific dynamic parameter visualization system for the central limit theorem according to claim 1, characterized in that, The real-time visualization module adopts an incremental data update strategy. When adjusting the sample size n, it only updates the mean data and visualization content corresponding to the newly added sample size. It also optimizes the plotting parameters, setting the number of histogram bins to 50, thus balancing visualization clarity and rendering speed.

5. A method for visualizing the dynamic parameters of the central limit theorem specifically for teaching, implemented using the system described in any one of claims 1-4, characterized in that, Includes the following steps: S1: System initialization, the parameter interaction module loads default parameters that fit the teaching and introductory needs, including default population distribution type, default population distribution parameters, default sample size n=30, and default number of simulations N=2000; S2: The distribution generation module generates the initial parent distribution based on the default parameters; S3: The sample simulation module generates a sample data matrix and calculates a set of sample means based on the initial population distribution, the default sample size n, and the number of simulations N. S4: The real-time visualization module draws an empirical distribution histogram of the sample mean and overlays it with the theoretical normal distribution curve to complete the initial visualization display; S5: Teachers and students can adjust parameters through the parameter interaction module, including switching the population distribution type, modifying the population distribution parameters, and adjusting the sample size n or the number of simulations N; S6: The parameter interaction module transmits the adjusted parameters to the distribution generation module and the sample simulation module; S7: The distribution generation module regenerates the population distribution based on the adjusted parameters, and the sample simulation module regenerates the sample data matrix and calculates the sample mean set based on the new population distribution and parameters. S8: The real-time visualization module redraws the empirical distribution histogram and the theoretical normal distribution curve based on the new sample mean set, and realizes real-time visualization updates; if the adjusted parameter is the sample size n, the convergence process during the increase of n is displayed using a progressive animation rendering method. 6.S9: Optionally, the convergence metric module calculates the convergence index of the new sample mean set and displays it through the GUI interface to assist teachers and students in analyzing the convergence effect; S10: Repeat steps 5-9 until the teaching demonstration is completed or the students' independent exploration ends.

7. The method for visualizing the dynamic parameters of the central limit theorem for teaching purposes according to claim 5, characterized in that, The drop-down menu of the parameter interaction module displays the parent distribution type according to the teaching logic of "discrete distribution", "continuous distribution" and "skewed distribution". The slider marks the range of commonly used teaching parameters and teaching critical values ​​such as 30 and 100, which are adapted to the operating habits of teachers and students.

8. The method for visualizing the dynamic parameters of the central limit theorem for teaching purposes according to claim 5, characterized in that, The convergence metric module displays the kurtosis index using color coding to indicate the degree of convergence. A skewness absolute value <0.5 is displayed in green, 0.5-1 in yellow, and >1 in red, making it easy for teachers and students to intuitively judge the convergence effect.

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