📚 Year 11 Cambridge Statistics: Core Knowledge Essentials | 剑桥 Year 11 统计：核心知识点梳理
Statistics is an essential subject that equips students with the skills to collect, analyse, interpret, and present data. In the Cambridge IGCSE Statistics course (Year 11), learners explore a wide range of topics from fundamental data types and graphical representation to probability distributions and regression analysis. This article provides a systematic overview of the core knowledge points that are crucial for exam success, helping students consolidate their understanding and apply statistical reasoning confidently in problem-solving contexts.
统计学是一门重要学科，培养学生收集、分析、解释和展示数据的技能。在剑桥 IGCSE 统计课程（Year 11）中，学生将探索从基本数据类型和图表表示到概率分布和回归分析的广泛主题。本文系统梳理了考试成功所需的核心知识点，帮助学生巩固理解并在实际解题中自信地运用统计推理。
1. Types of Data and Collection Methods | 数据类型与收集方法
In statistics, data can be classified as qualitative (categorical) or quantitative (numerical). Quantitative data is further divided into discrete and continuous types. Understanding the data type is crucial for choosing appropriate graphical representations and statistical measures.
在统计学中，数据可分为定性（分类）数据和定量（数值）数据。定量数据进一步分为离散型和连续型。了解数据类型对于选择合适的图形表示和统计度量至关重要。
Common data collection methods include surveys, experiments, and observational studies. A census collects data from every member of the population, while a sample collects data from a subset. Sampling methods such as random, stratified, and systematic sampling help ensure representative data.
常见的数据收集方法包括调查、实验和观察研究。普查是从总体的每个成员收集数据，而抽样则是从子集中收集数据。随机抽样、分层抽样和系统抽样等抽样方法有助于确保数据具有代表性。
| Qualitative (Categorical) | Colour, gender, favourite food |
| Quantitative Discrete | Number of siblings, goals scored |
| Quantitative Continuous | Height, weight, temperature |
定性（分类）数据指如颜色、性别、最喜爱的食物等属性。定量离散数据是可数的整数，例如兄弟姐妹数量；定量连续数据是测量得到的值，如身高、体重。
2. Presenting Data: Charts & Diagrams | 数据展示：统计图表
Choosing the right diagram is vital for visualising data effectively. Bar charts and pie charts are used for categorical data, while histograms, frequency polygons, and cumulative frequency curves are used for quantitative data. A histogram shows frequencies as areas of bars; the vertical axis represents frequency density for unequal class widths.
选择合适的图表对有效呈现数据至关重要。条形图和饼图用于分类数据，而直方图、频率多边形和累积频率曲线用于定量数据。直方图以条形面积表示频数；若组距不等，纵轴代表频率密度。
Stem-and-leaf diagrams and box-and-whisker plots provide deeper insight into data distribution. The box plot displays the minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃), and maximum, helping identify skewness and outliers.
茎叶图和箱形图能更深入地显示数据分布。箱形图展示最小值、下四分位数（Q₁）、中位数（Q₂）、上四分位数（Q₃）和最大值，有助于识别偏态和异常值。
3. Measures of Central Tendency | 集中趋势测量
The mean, median, and mode are used to describe the centre of a data set. The mean is the arithmetic average, calculated as Σx / n. The median is the middle value when data are ordered, and the mode is the most frequent value. For grouped data, the mean is estimated using midpoints of intervals.
均值、中位数和众数用于描述数据集的中心。均值是算术平均数，计算公式为 Σx / n。中位数是数据排序后的中间值，众数是出现频率最高的值。对于分组数据，均值使用组中值来估算。
Estimated Mean = Σ(f × midpoint) / Σf
估算均值 = Σ（频数 × 组中值）/ Σ 频数
The median is often preferred when data contain extreme values, as it is not affected by outliers. Understanding which average to use depends on the data type and distribution shape.
当数据包含极端值时，中位数通常更适用，因为它不受异常值影响。使用哪种平均数取决于数据类型和分布形态。
4. Measures of Dispersion | 离散程度测量
Dispersion indicates how spread out the data are. The range (maximum – minimum) is the simplest measure but is sensitive to outliers. The interquartile range (IQR = Q₃ – Q₁) measures the spread of the middle 50% of the data and is more robust.
离散度表示数据的离散程度。极差（最大值 – 最小值）是最简单的度量，但易受异常值影响。四分位距（IQR = Q₃ – Q₁）衡量中间 50% 数据的分散程度，更具稳健性。
Variance and standard deviation quantify the average squared deviation from the mean. For a population, standard deviation σ is given by σ = √[Σ(x – μ)² / n]. For a sample, we divide by (n – 1) to obtain the sample standard deviation s. These measures are central to many statistical analyses.
方差和标准差量化了各数据与均值之差的平方的平均水平。对于总体，标准差 σ 的计算公式为 σ = √[Σ(x – μ)² / n]。对于样本，我们用 (n – 1) 除以求得样本标准差 s。这些度量是许多统计分析的核心。
variance = Σ(x – x̄)² / n or Σ(x – x̄)² / (n – 1)
方差 = Σ(x – x̄)² / n 或 Σ(x – x̄)² / (n – 1)
5. Basic Probability Concepts | 基本概率概念
Probability measures the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain). The probability of an event A is P(A) = number of favourable outcomes / total number of outcomes, assuming all outcomes are equally likely.
概率衡量事件发生的可能性，范围从 0（不可能）到 1（必然）。事件 A 的概率 P(A) = 有利结果数 / 总结果数，前提是所有结果等可能。
Events can be combined using the addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B). The multiplication rule applies to independent events: P(A ∩ B) = P(A) × P(B). Tree diagrams and Venn diagrams are essential tools for solving multi-stage probability problems.
事件可以利用加法法则组合：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。如果 A 和 B 互斥，则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。乘法法则用于独立事件：P(A ∩ B) = P(A) × P(B)。树状图和维恩图是解决多阶段概率问题的必备工具。
6. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of values, each with an associated probability. The probability distribution of X lists all possible values and their probabilities, where Σ P(X = x) = 1. The expected value E(X) (or mean μ) is the long-run average outcome given by E(X) = Σ x · P(X = x).
离散随机变量 X 取可数个值，每个值都有对应的概率。X 的概率分布列出了所有可能的取值及其概率，且满足 Σ P(X = x) = 1。期望值 E(X)（或均值 μ）是长期平均结果，计算公式为 E(X) = Σ x · P(X = x)。
The variance of X, Var(X), measures the spread of the distribution: Var(X) = E(X²) – [E(X)]². Understanding these concepts is fundamental before moving on to specific distributions like the binomial distribution.
X 的方差 Var(X) 衡量分布的离散程度：Var(X) = E(X²) – [E(X)]²。在深入学习二项分布等特定分布之前，理解这些概念是基础。
Var(X) = Σ (x – μ)² P(X = x) = Σ x² P(X = x) – μ²
Var(X) = Σ (x – μ)² P(X = x) = Σ x² P(X = x) – μ²
7. Binomial Distribution | 二项分布
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success, p, remains constant. If X ~ B(n, p), then the probability of obtaining exactly r successes is P(X = r) = ⁿCᵣ × pʳ × (1 – p)ⁿ⁻ʳ.
二项分布用于建模在固定次数的独立试验中成功的次数，每次试验只有两种可能结果（成功或失败），且成功概率 p 保持恒定。如果 X ~ B(n, p)，则获得恰好 r 次成功的概率为 P(X = r) = ⁿCᵣ × pʳ × (1 – p)ⁿ⁻ʳ。
The mean (expected value) of a binomial distribution is μ = n p, and the variance is σ² = n p (1 – p). These formulas allow quick calculation of key distribution parameters without summing full probability tables. The binomial distribution is symmetric only when p = 0.5.
二项分布的均值（期望值）为 μ = n p，方差为 σ² = n p (1 – p)。这些公式可以快速计算关键参数，无需汇总完整的概率表。只有当 p = 0.5 时，二项分布才对称。
P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ, μ = np, σ² = np(1 – p)
二项分布公式：P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ, 均值 μ = np, 方差 σ² = np(1 – p)
8. The Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution characterised by its bell-shaped curve. It is symmetric about the mean μ, and its spread is determined by the standard deviation σ. The total area under the curve equals 1, and about 68% of values lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.
正态分布是一种连续概率分布，特点是钟形曲线。它关于均值 μ 对称，离散程度由标准差 σ 决定。曲线下总面积等于 1，约 68% 的值落在 μ ± σ 之内，95% 落在 μ ± 2σ 之内，99.7% 落在 μ ± 3σ 之内。
To find probabilities for any normal distribution, we standardise the variable to the standard normal distribution Z ~ N(0, 1) using the z-score formula: z = (x – μ) / σ. Probability tables or technology can then be used to find the required area under the curve.
为求任意正态分布的概率，我们使用 z 分数公式将变量标准化为标准正态分布 Z ~ N(0, 1)：z = (x – μ) / σ。然后可以使用概率表或技术工具求得曲线下所需的面积。
z = (x – μ) / σ
z 分数：z = (x – μ) / σ
9. Correlation and Scatter Plots | 相关性与散点图
A scatter diagram plots bivariate data points to reveal the relationship between two variables. Correlation describes the strength and direction of the linear association. Positive correlation means that as one variable increases, the other tends to increase; negative correlation means one increases while the other decreases. No correlation suggests no apparent linear pattern.
散点图绘制双变量数据点，以展示两个变量之间的关系。相关性描述了线性关联的强度和方向。正相关表示一个变量增加时另一个也倾向于增加；负相关表示一个增加时另一个减少；无相关表示没有明显的线性模式。
The Pearson product-moment correlation coefficient r quantifies this linear relationship, taking values from –1 (perfect negative) to +1 (perfect positive). The closer r is to –1 or +1, the stronger the linear correlation. Remember that correlation does not imply causation.
皮尔逊积矩相关系数 r 量化了这种线性关系，取值范围从 –1（完全负相关）到 +1（完全正相关）。r 越接近 –1 或 +1，线性相关越强。务必记住相关关系不意味着因果关系。
10. Linear Regression | 线性回归
Linear regression finds the equation of the straight line that best fits the data on a scatter plot. The regression line of y on x is written as y = a + b x, where b is the slope and a is the intercept. The slope b represents the estimated change in y for a unit increase in x.
线性回归找到最适合散点图数据的直线方程。y 对 x 的回归线写作 y = a + b x，其中 b 是斜率，a 是截距。斜率 b 表示 x 每增加一个单位时 y 的估计变化量。
The least squares method is used to estimate a and b by minimising the sum of squared vertical deviations. You can use summary statistics or calculator functions to find these values. The regression equation can be used to make predictions for y given x, but extrapolation beyond the data range should be avoided as it may be unreliable.
最小二乘法通过最小化垂直偏差的平方和来估计 a 和 b。可以使用汇总统计量或计算器功能求出这些值。回归方程可用于根据 x 预测 y，但应避免超出数据范围的外推，因为这可能不可靠。
y = a + b x, with b = Σ(x – x̄)(y – ȳ) / Σ(x – x̄)², a = ȳ – b x̄
回归线：y = a + b x，其中 b = Σ(x – x̄)(y – ȳ) / Σ(x – x̄)², a = ȳ – b x̄
Published by TutorHao | Statistics Revision Series | aleveler.com
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