📚 IGCSE OCR Statistics Formula & Theorem Quick Reference Handbook | IGCSE OCR 统计：公式定理速查手册
This article serves as a compact revision guide for all the essential formulas, definitions and theorems in the IGCSE OCR Statistics course. Each section presents key concepts with worked-style explanations, enabling you to quickly recall the mathematical tools needed for data analysis, probability and inference. All formulas are presented using standard mathematical notation with Unicode symbols, ensuring clarity for quick reference before the exam.
本文为 IGCSE OCR 统计课程中所有必备公式、定义和定理的紧凑复习指南。每个小节以结合实例的方式解释关键概念，帮助你迅速回忆起数据分析、概率和统计推断所需的数学工具。所有公式均采用标准数学符号配合 Unicode 字符呈现，确保考前速查时清晰明了。
1. Types of Data | 数据类型
Data can be classified as qualitative (categorical) or quantitative (numerical). Quantitative data is further divided into discrete (countable values, e.g. number of students) and continuous (measurable values, e.g. height, time).
数据可分为定性（分类）数据和定量（数值）数据。定量数据又分为离散型（可数数值，例如学生人数）和连续型（可测量数值，例如身高、时间）。
Knowing the data type determines which statistical diagrams and measures are appropriate. For example, bar charts suit categorical data, while histograms are used for continuous data.
了解数据类型决定了采用何种统计图表和度量。例如，条形图适用于分类数据，而直方图用于连续数据。
2. Charts and Graphical Representations | 图表与图形表示
Bar chart: Used for qualitative or discrete data, with gaps between bars. Height corresponds to frequency.
条形图：用于定性或离散数据，条形之间有间隔，高度对应频数。
Pie chart: Each sector angle = (frequency / total frequency) × 360°.
饼图：每个扇形的角度 = (频数 ÷ 总频数) × 360°。
Histogram: For continuous data with no gaps; frequency is proportional to the area of the bar. On a histogram with unequal class widths, frequency density = frequency / class width.
直方图：用于连续数据，条形之间无间隔；频数与条形面积成正比。在组距不等的直方图中，频数密度 = 频数 ÷ 组距。
Frequency polygon: Plotting midpoints of each class interval joined by straight lines.
频数多边形：描出每个组区间的中点并用直线连接。
3. Measures of Central Tendency | 集中趋势的度量
Mean (x̄): x̄ = (Σx) / n, where Σx is the sum of all values and n is the number of values. For grouped data: x̄ ≈ (Σfx) / Σf, using midpoints x of each class.
平均数 (x̄)：x̄ = (Σx) / n，其中 Σx 是所有数值之和，n 是数值个数。对于分组数据：x̄ ≈ (Σfx) / Σf，使用每个组的中点值 x。
Median: The middle value when data are ordered. For n values, median position = (n+1)/2 th value. For grouped data, use linear interpolation within the median class: Median = L + ( (n/2 – cf_below) / f_median ) × w, where L is the lower class boundary, cf_below is the cumulative frequency before the median class, f_median is the frequency of the median class, and w is the class width.
中位数：数据排序后居中的值。对于 n 个数值，中位数的位置为第 (n+1)/2 个值。对于分组数据，在中位数组内进行线性插值：中位数 = L + ( (n/2 – cf_below) / f_median ) × w，其中 L 为下组界，cf_below 为低于中位数组的累积频数，f_median 为中位数组的频数，w 为组距。
Mode: The most frequently occurring value. For grouped data, the modal class is the class with the highest frequency density (on a histogram) or simply the class with highest frequency.
众数：出现次数最多的数值。对于分组数据，众数组是频数密度（在直方图上）最高或频数最高的组。
4. Measures of Dispersion | 离散程度的度量
Range: Range = maximum value – minimum value.
极差：极差 = 最大值 – 最小值。
Interquartile range (IQR): IQR = Q₃ – Q₁, where Q₁ is the lower quartile (25th percentile) and Q₃ is the upper quartile (75th percentile). To find quartiles, use position: Q₁ at (n+1)/4 th value, Q₃ at 3(n+1)/4 th value. For grouped data, use interpolation similar to the median.
四分位距 (IQR)：IQR = Q₃ – Q₁，其中 Q₁ 为下四分位数（第25百分位数），Q₃ 为上四分位数（第75百分位数）。求四分位数时采用位置：Q₁ 位于第 (n+1)/4 个值，Q₃ 位于第 3(n+1)/4 个值。对于分组数据，使用与中位数类似的插值法。
Standard deviation (s): For a sample: s = √[ Σ(x – x̄)² / (n-1) ]. For a population: σ = √[ Σ(x – μ)² / N ]. For grouped data, use x as class midpoints weighted by frequency: s = √[ Σf(x – x̄)² / (Σf – 1) ] (if sample).
标准差 (s)：样本标准差：s = √[ Σ(x – x̄)² / (n-1) ]。总体标准差：σ = √[ Σ(x – μ)² / N ]。对于分组数据，以组中点值 x 按频数加权计算：s = √[ Σf(x – x̄)² / (Σf – 1) ]（若为样本）。
5. Cumulative Frequency and Box Plots | 累积频数与箱线图
Cumulative frequency curves plot cumulative frequency against the upper class boundary. From the graph, the median is read at 50% of total frequency, Q₁ at 25%, Q₃ at 75%. The interquartile range can be found directly, and the number of values above or below a certain threshold can be estimated.
累积频数曲线以各组的组上界为横坐标、累积频数为纵坐标作图。从图上可读出总频数50%处对应的中位数，25%处的 Q₁，75%处的 Q₃。四分位距可直接由图上读取，还可估计高于或低于某一界限的数值个数。
A box plot (box-and-whisker diagram) displays the minimum, Q₁, median, Q₃ and maximum. It enables quick visual comparison of spread and skewness across datasets.
箱线图（箱须图）显示最小值、Q₁、中位数、Q₃ 和最大值。可快速横向比较不同数据集的离散程度和偏态。
6. Basic Probability | 基础概率
The probability of an event A is given by: P(A) = Number of favourable outcomes / Total number of equally likely outcomes. 0 ≤ P(A) ≤ 1.
事件 A 的概率为：P(A) = 有利结果数 / 等可能结果总数。0 ≤ P(A) ≤ 1。
The sum of probabilities of all mutually exclusive outcomes equals 1. For complementary events: P(not A) = 1 – P(A).
所有互斥结果的概率之和为 1。对于互补事件：P(非 A) = 1 – P(A)。
The expected frequency of an event in n trials is n × P(A).
在 n 次试验中，事件的期望频数为 n × P(A)。
7. Combined Events, Tree Diagrams and Conditional Probability | 复合事件、树图与条件概率
The addition rule for mutually exclusive events: P(A or B) = P(A) + P(B). If not mutually exclusive: P(A or B) = P(A) + P(B) – P(A and B).
互斥事件的加法法则：P(A 或 B) = P(A) + P(B)。若事件不互斥：P(A 或 B) = P(A) + P(B) – P(A 且 B)。
The multiplication rule for independent events: P(A and B) = P(A) × P(B). For dependent events, use conditional probability: P(A and B) = P(A) × P(B|A), where P(B|A) is the probability of B given that A has occurred.
独立事件的乘法法则：P(A 且 B) = P(A) × P(B)。对于非独立事件，使用条件概率：P(A 且 B) = P(A) × P(B|A)，其中 P(B|A) 表示在 A 已发生的条件下 B 发生的概率。
Tree diagrams systematically represent successive events. Multiply along branches for ‘and’, and add probability products at the ends for multiple outcomes.
树图可系统地表示相继事件。沿分支相乘得到“且”的概率，将各末端的概率积相加得到多个结果的概率。
Conditional probability formula: P(B|A) = P(A and B) / P(A). This is essential when analysing two-way tables and without-replacement experiments.
条件概率公式：P(B|A) = P(A 且 B) / P(A)。在处理双向表和不放回试验时至关重要。
8. Permutations and Combinations | 排列与组合
A permutation is an arrangement of objects where order matters. The number of ways to arrange r objects from n distinct objects is given by: nPr = n! / (n – r)!.
排列是指顺序有要求的物体排列方式。从 n 个不同对象中选取 r 个进行排列的方法数为：nPr = n! / (n – r)!。
The factorial n! = n × (n-1) × … × 1, with 0! = 1.
阶乘 n! = n × (n-1) × … × 1，且 0! = 1。
A combination is a selection where order does not matter. The number of combinations of r objects from n is: nCr = n! / (r! (n – r)!). This is often read as “n choose r”.
组合是指顺序无关的选取方式。从 n 个对象中选取 r 个的组合数为：nCr = n! / (r! (n – r)!)，常称作“n 选 r”。
Permutations and combinations are frequently used to calculate probabilities in equally likely sample spaces.
排列与组合常用于计算等可能样本空间中的概率。
9. Discrete Random Variables and Expectation | 离散随机变量与期望值
A discrete random variable X takes a countable number of values. Its probability distribution lists each possible value with its probability P(X = x). The sum of all these probabilities is 1.
离散随机变量 X 取值可数。其概率分布列出每个可能取值及其概率 P(X = x)。所有概率之和为 1。
The expected value (mean) of X is E(X) = Σ [x · P(X = x)]. The expectation can be used to predict the long-run average outcome.
X 的期望值（均值）为 E(X) = Σ [x · P(X = x)]。期望值可用于预测长期的平均结果。
Variance of X: Var(X) = Σ [x² · P(X = x)] – [E(X)]². Alternatively, Var(X) = Σ [(x – μ)² · P(X = x)]. The standard deviation is √Var(X).
X 的方差：Var(X) = Σ [x² · P(X = x)] – [E(X)]²，或者 Var(X) = Σ [(x – μ)² · P(X = x)]。标准差为 √Var(X)。
10. Scatter Diagrams and Correlation | 散点图与相关
A scatter diagram plots bivariate data to investigate the relationship between two variables. The pattern may suggest positive, negative or no correlation.
散点图用于绘制双变量数据，探究两个变量之间的关系。图形可能呈现正相关、负相关或无相关。
Correlation is not causation. The strength and direction of linear association can be described qualitatively (strong, weak, moderate) or eventually with a correlation coefficient (not required in the core IGCSE OCR specification but useful). A line of best fit may be drawn by eye to make predictions (interpolation within the data range; extrapolation outside is unreliable).
相关不等于因果。线性关联的强度和方向可定性描述（强、弱、中等），或进一步用相关系数表示（IGCSE OCR 核心内容不要求计算，但有助于理解）。通过目测可画出一条最佳拟合线用于预测（在数据范围内进行内插；外推不可靠）。
11. Sampling Methods | 抽样方法
Simple random sampling: every member of the population has an equal chance of being chosen. Methods include using random number generators or lottery.
简单随机抽样：总体中每个成员被抽中的机会均等。方法包括使用随机数生成器或抽签。
Stratified sampling: the population is divided into strata (groups sharing a characteristic), and a random sample proportional to size is taken from each stratum. This ensures representation.
分层抽样：将总体划分为若干层（具有共同特征的组），并按各层规模比例随机抽取样本。这可确保代表性。
Systematic sampling: selecting every kth individual from an ordered list after a random start.
系统抽样：从有序名单中随机确定起点后，每隔固定间隔 k 选取一个个体。
Bias in sampling can lead to unreliable conclusions. Understanding the strengths and limitations of each method is crucial for interpreting survey results.
抽样中的偏差会导致结论不可靠。了解每种方法的优缺点对于解读调查结果至关重要。
12. Quick Reference Formula Sheet | 公式速查表
Below is a compact table of the most frequently used formulas in IGCSE OCR Statistics.
以下是 IGCSE OCR 统计中最常用公式的紧凑表格。
| Concept | Formula |
|---|---|
| Mean (ungrouped) | x̄ = Σx / n |
| Mean (grouped) | x̄ = Σfx / Σf |
| Median position | (n + 1) / 2 |
| Interquartile range | IQR = Q₃ – Q₁ |
| Range | Max – Min |
| Sample standard deviation | s = √[ Σ(x – x̄)² / (n-1) ] |
| Probability of A | P(A) = favourable / total |
| Complement rule | P(A’) = 1 – P(A) |
| Addition rule (non-mutually exclusive) | P(A or B) = P(A) + P(B) – P(A and B) |
| Conditional probability | P(B|A) = P(A and B) / P(A) |
| Permutations | nPr = n! / (n-r)! |
| Combinations | nCr = n! / [r!(n-r)!] |
| Expected value of X | E(X) = Σ [x · P(X=x)] |
| Frequency density | Frequency / class width |
Use this table as a last-minute memory aid. Remember that applying the correct formula depends on recognizing the context and assumptions (e.g. equally likely outcomes, independent events, sample vs. population). Practise past papers to internalize the swift application of these tools.
可将此表格作为考前最后的记忆辅助。记住，选用正确公式取决于识别背景和假设（如等可能性、独立事件、样本与总体）。通过练习历年真题，将这些工具的快速运用内化。
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