📚 IB Mathematics Applications and Interpretation: Complete Revision Guide | IB数学应用与解释：完整复习指南
The IB Mathematics: Applications and Interpretation (AI) course focuses on mathematical modelling, statistics, technology and real-world problem solving. Whether you are preparing for Standard Level or Higher Level, a clear grasp of the syllabus structure, assessment style and key techniques is essential for success. This revision guide draws on the IB Prepared Oxford 2021 approach, combining concise explanations with exam-focused practice.
IB数学：应用与解释（AI）课程侧重于数学建模、统计学、技术应用和现实世界问题解决。无论你准备标准级别还是高级级别，清晰掌握课程大纲结构、评估风格和关键技巧对成功至关重要。本复习指南借鉴IB Prepared Oxford 2021的方法，将简明讲解与考试导向练习相结合。
1. Course Overview and Assessment Objectives | 课程概览与评估目标
The AI course is assessed through written papers and an internal exploration. At SL, Paper 1 and Paper 2 both allow a graphical display calculator, with Paper 1 focusing on short responses and Paper 2 on extended problem solving. At HL, three papers include further statistics, graph theory and calculus content. Each question targets command terms such as 'calculate', 'interpret', 'justify' and 'hence'.
AI课程通过笔试和内部探索进行评估。在标准级别，试卷一和试卷二都允许使用图形显示计算器，试卷一侧重简答题，试卷二侧重扩展性问题解决。在高级级别，三份试卷包含更深入的统计学、图论和微积分内容。每道题目的指令词包括"计算"、"解释"、"证明"和"因此"。
Assessment objectives are weighted across knowledge and understanding, problem solving, communication and use of technology. You must show reasoning clearly, especially when a question asks you to interpret a result in context. Marks are often awarded for method, accuracy and final answer, so systematic working is important even when using a calculator.
评估目标在知识与理解、问题解决、沟通和科技运用之间分配权重。你必须清晰地展示推理过程，尤其是当题目要求结合情境解释结果时。即使使用计算器，也要系统性地书写过程，因为分数通常分配给方法、准确性和最终答案。
2. GDC Skills and Technology | 图形计算器技能与技术
Your graphical display calculator is an integral tool in AI. You need to enter data lists, calculate one-variable and two-variable statistics, sketch functions, find intersections and zeros, and run probability distributions. Practise using the statistics mode, regression options and solver until these operations become automatic.
图形显示计算器是AI课程中不可或缺的工具。你需要输入数据列表、计算单变量和双变量统计量、绘制函数图像、求交点和零点，以及运行概率分布。练习使用统计模式、回归选项和求解器，直到这些操作变得自动化。
Always write down the calculator command or the values you enter, not just the output. For example, if finding a binomial probability, state n, p and the bound. This allows examiners to follow your method and gives you partial credit if the final answer is slightly wrong.
始终写下计算器命令或你输入的数值，而不仅仅是输出结果。例如，如果求二项分布概率，应说明 n、p 和边界。这样考官才能追踪你的方法，并在最终答案略有错误时给予部分分数。
3. Descriptive Statistics and Data Representation | 描述性统计与数据表示
Data can be summarised using measures of centre and spread. The mean is x̄ = Σxᵢ/n and the standard deviation is s = √[Σ(xᵢ - x̄ idpoints and cumulative frequency to estimate the mean, median, quartiles and percentiles.
数据可以用中心趋势和离散程度的度量来概括。均值公式为 x̄ = Σxᵢ/n，标准差公式为 s = √[Σ(xᵢ - x̄)²/(n-1)]。对于分组数据，使用组中值和累积频率来估计均值、中位数、四分位数和百分位数。
n outlier. Box plots, histograms and cumulative frequency graphs help you compare distributions visually and comment on skew, symmetry or bimodality.
异常值可以使用四分位距来识别：任何低于 Q₁ - 1.5 × IQR 或高于 Q₃ + 1.5 × IQR 的值都是异常值。箱线图、直方图和累积频率图帮助你可视化比较分布，并评价偏斜、对称或双峰性。
4. Probability and Distributions | 概率与分布
Basic probability rules include P(A ∪ B) = P(A) + P(B) - P(A ∩ B) and conditional probability P(A|B) = P(A ∩ ce tables are useful for multi-stage or overlapping events. Independent events satisfy P(A ∩ B) = P(A) × P(B).
基本概率法则包括 P(A ∪ B) = P(A) + P(B) - P(A ∩ B) 和条件概率 P(A|B) = P(A ∩ B)/P(B)。树状图、维恩图和样本空间表适用于多阶段或重叠事件。独立事件满足 P(A ∩ B) = P(A) × P(B)。
Key discrete distributions are binomial and Poisson. For a binomial variable X ~ B(n, p), P(X = r) = C(n, r) pʳ (1-p)ⁿ⁻ʳ. The Poisson model X ~ Po(λ) has P(X = r) = λʳ e⁻λ / r!. The normal distribution N(μ, σ²) uses the z-score Z = (X - μ)/σ to find probabilities.
关键的离散分布是二项分布和泊松分布。对于二项变量 X ~ B(n, p)，P(X = r) = C(n, r) pʳ (1-p)ⁿ⁻ʳ。泊松模型 X ~ Po(λ) 的概率为 P(X = r) = λʳ e⁻λ / r!。正态分布 N(μ, σ²) 使用 z 分数 Z = (X - μ)/σ 来求概率。
5. Hypothesis Testing and Confidence Intervals
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