📚 Year 11 Cambridge Statistics: Core Knowledge Checklist | Year 11 剑桥统计：核心知识点梳理
Statistics is about collecting, organising, analysing, and interpreting data to make informed decisions. In Year 11 Cambridge IGCSE Statistics (0479), you will build on earlier skills and explore probability distributions, correlation, time series, and index numbers. This article covers the essential topics you must master for the exam, presented in a clear, bilingual format for easy revision.
统计学是关于收集、整理、分析和解释数据以做出明智决策的学科。在 Year 11 剑桥 IGCSE 统计 (0479) 课程中，你将拓展已有技能，并学习概率分布、相关分析、时间序列和指数等核心内容。本文梳理了考试必会的关键知识点，以清晰的中英双语形式呈现，方便复习。
1. Types of Data | 数据类型
Data can be classified as qualitative (categorical) or quantitative (numerical). Qualitative data describes qualities, like eye colour or types of car, and can be nominal or ordinal. Quantitative data involves numbers: discrete data take only certain values (e.g. number of students), while continuous data can take any value within a range (e.g. height).
数据可分为定性（分类）数据与定量（数值）数据。定性数据描述属性，如眼睛颜色或汽车类型，可以是名义或有序数据。定量数据涉及数字：离散数据仅取特定值（如学生人数），连续数据可在某个范围内取任意值（如身高）。
Choosing the right graph and summary statistics depends on the data type. For qualitative data, bar charts and pie charts are suitable; for quantitative data, histograms, cumulative frequency graphs and box‑and‑whisker plots are more common.
选择合适的图表和汇总统计量取决于数据类型。对于定性数据，可使用条形图和饼图；对于定量数据，直方图、累积频数图和箱线图更为常用。
2. Data Collection Methods | 数据收集方法
Primary data is collected first‑hand by the researcher through experiments, surveys, or observations. Secondary data is obtained from existing sources such as government reports, websites, or journals. Primary data is often more relevant but time‑consuming, while secondary data is cheaper but may be less accurate.
原始数据由研究者通过实验、调查或观察等一手方式收集。二手数据则来自现有资料，如政府报告、网站或期刊。原始数据通常更切题但耗时，而二手数据成本低但可能不够精确。
Questionnaires must be carefully designed to avoid bias: questions should be clear, not leading, and offer a balanced range of response options. Pilot studies and randomisation help improve data quality.
问卷设计需细致以避免偏差：问题应清晰、不具诱导性，并提供均衡的选项范围。试点研究和随机化有助于提升数据质量。
3. Sampling Techniques | 抽样方法
A sample is a subset of a population selected for a study. Random sampling ensures every member has an equal chance, reducing bias. Simple random sampling, stratified sampling, and systematic sampling are common methods. Stratified sampling divides the population into groups (strata) and samples proportionally to ensure representation.
样本是从总体中选出用于研究的子集。随机抽样确保每个成员有平等被选中的机会，减少偏差。简单随机抽样、分层抽样和系统抽样是常用方法。分层抽样将总体分成层，并按比例抽取样本，以保证代表性。
Non‑random methods, like convenience or quota sampling, are quicker but can introduce selection bias. The sample size should be large enough to give reliable estimates; larger samples reduce sampling error.
非随机方法，如方便抽样或配额抽样，速度较快但可能引入选择偏差。样本量应足够大以提供可靠的估计；较大的样本能降低抽样误差。
4. Frequency Distributions & Histograms | 频数分布与直方图
A frequency distribution table organises data into classes (intervals) with corresponding frequencies. For continuous data, class boundaries and class widths are important. The class width is the upper boundary minus the lower boundary. The modal class is the interval with the highest frequency.
频数分布表将数据分为组（区间）并列出对应频数。对于连续数据，组界和组距至关重要。组距等于上界减下界。众数所在组是频数最高的区间。
Histograms display frequency distributions: bar area is proportional to frequency. When class widths are unequal, frequency density must be used: Frequency density = Frequency ÷ Class width. The vertical axis then shows frequency density, not raw frequency.
直方图展示频数分布：条形面积与频数成正比。当组距不等时，需使用频数密度：频数密度 = 频数 ÷ 组距。此时纵轴表示频数密度，而非原始频数。
5. Measures of Central Tendency | 集中趋势度量
The mean, median, and mode summarise a typical value. For ungrouped data, mean x̄ = Σx/n. For grouped data, estimate the mean using the midpoints of each class: x̄ = Σ(f × midpoint) / Σf. The median is the middle value when data are ordered; for a frequency table, use cumulative frequency to locate the median class or exact median.
平均数、中位数和众数概括了典型值。对于未分组数据，平均数 x̄ = Σx/n。对于分组数据，使用各组中点值估算平均数：x̄ = Σ（f × 中点）/ Σf。中位数是排序后数据的中间值；在频数表中，可用累积频数定位中位数组或精确中位数。
The mode is the most frequent value; for grouped data, the modal class is given. The choice of average depends on the data shape: the mean is affected by outliers, while the median is robust.
众数是出现最多的值；对于分组数据，给出众数所在组。选用哪种平均数取决于数据分布形态：平均数受异常值影响，而中位数更为稳健。
6. Measures of Dispersion | 离散程度度量
Spread tells us how data vary. The range is the simplest measure: Maximum – Minimum. The interquartile range (IQR = Q₃ – Q₁) measures the middle 50% spread and is less affected by outliers. Quartiles are found from an ordered list or by using cumulative frequency graphs.
离散程度反映数据的变异情况。极差是最简单的度量：最大值 － 最小值。四分位距 (IQR = Q₃ – Q₁) 衡量中间 50% 数据的分散情况，不易受异常值影响。四分位数可从排序列表中求出，或利用累积频数图读取。
Variance and standard deviation measure average deviation from the mean. For a population, variance σ² = Σ(x − μ)²/N; for a sample, s² = Σ(x − x̄)²/(n − 1). Standard deviation is the square root of variance, providing a measure in original units.
方差和标准差衡量数据偏离平均数的平均程度。对于总体，方差 σ² = Σ(x − μ)²/N；对于样本，s² = Σ(x − x̄)²/(n − 1)。标准差是方差的平方根，以原始单位度量离散程度。
7. Probability Fundamentals | 概率基础
Probability is a measure of how likely an event is to occur, ranging from 0 (impossible) to 1 (certain). For equally likely outcomes, P(A) = Number of favourable outcomes / Total number of outcomes. The probability of an event not occurring is P(A′) = 1 − P(A).
概率是衡量事件发生可能性的数值，介于 0（不可能）到 1（必然）之间。对于等可能结果，P(A) = 有利结果数 / 总结果数。事件不发生的概率 P(A′) = 1 − P(A)。
Combined events use the addition rule: P(A or B) = P(A) + P(B) − P(A and B). Mutually exclusive events cannot occur together, so P(A and B) = 0. Independent events satisfy P(A and B) = P(A) × P(B). Tree diagrams are helpful for multi‑stage experiments.
组合事件使用加法法则：P(A 或 B) = P(A) + P(B) − P(A 且 B)。互斥事件不能同时发生，故 P(A 且 B) = 0。独立事件满足 P(A 且 B) = P(A) × P(B)。树状图常用于多步骤试验。
8. The Binomial Distribution | 二项分布
A binomial distribution arises from a fixed number, n, of independent trials, each with two outcomes (success/failure) and constant probability of success p. The random variable X is the number of successes. The probability of exactly r successes is given by:
二项分布源于固定次数 n 的独立试验，每次试验有两种结果（成功/失败），且成功概率 p 恒定。随机变量 X 表示成功次数。恰好 r 次成功的概率由下式给出：
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ
The mean of a binomial distribution is μ = np, and the variance is σ² = np(1 − p). Recognising binomial situations and using tables or the formula to calculate probabilities is key.
二项分布的均值为 μ = np，方差为 σ² = np(1 − p)。识别二项分布情境，并运用公式或表格计算概率是关键。
9. The Normal Distribution | 正态分布
The normal distribution is a continuous, symmetric, bell‑shaped curve defined by its mean μ and standard deviation σ. About 68% of data lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. This is the empirical rule.
正态分布是一种连续、对称的钟形曲线，由其均值 μ 和标准差 σ 决定。约 68% 的数据落在均值 ±1σ 内，95% 落在 ±2σ 内，99.7% 落在 ±3σ 内。这便是经验法则。
In IGCSE, you are not required to perform complex calculations; you need to interpret the shape, understand the area as proportion, and compare distributions. The normal curve models many natural phenomena, such as heights or exam scores.
在 IGCSE 中，不需要进行复杂计算；你需要解读分布形态、理解面积代表比例，并比较不同分布。正态曲线可模拟许多自然现象，如身高或考试成绩。
10. Scatter Diagrams & Correlation | 散点图与相关
A scatter diagram plots paired data (x, y) to reveal a relationship. Correlation describes the direction and strength of the linear association. Positive correlation means as x increases, y tends to increase; negative correlation means as x increases, y tends to decrease.
散点图绘制成对数据 (x, y) 以展示关系。相关描述线性关联的方向和强度。正相关表示 x 增大时 y 趋于增大；负相关表示 x 增大时 y 趋于减小。
Correlation does not imply causation. A line of best fit can be drawn by eye or using the method of least squares. The equation of the regression line y = a + bx can be used for prediction, but extrapolation beyond the data range is unreliable.
相关关系不等于因果关系。最佳拟合线可凭眼力绘制，也可用最小二乘法求出。回归直线方程 y = a + bx 可用于预测，但超出数据范围的外推不可靠。
11. Time Series & Moving Averages | 时间序列与移动平均
A time series is a set of data recorded at regular time intervals. It often contains four components: trend, seasonal variation, cyclical variation, and random fluctuation. Moving averages smooth out short‑term fluctuations to reveal the underlying trend.
时间序列是按固定时间间隔记录的一组数据。它通常包含四个成分：趋势、季节变动、循环变动和随机波动。移动平均法可消除短期波动，揭示潜在趋势。
To calculate a moving average, average a fixed number of consecutive periods. For example, a 4‑point moving average is the mean of every four consecutive data points. Plotting moving averages against midpoints helps draw the trend line.
计算移动平均时，对连续固定数量的时间段求平均。例如，4 点移动平均是每连续四个数据点的均值。将移动平均值对应中间点绘制，有助于画出趋势线。
12. Index Numbers | 指数
Index numbers compare the value of a variable to a base value, usually expressed as a percentage. The base period has an index of 100. A simple price index is given by: Index = (Current value / Base value) × 100. They help track changes in prices, quantities, or other economic indicators over time.
指数是变量值与基期值的比较，通常以百分比表示。基期指数为 100。简单价格指数的公式为：指数 =（当前值 / 基期值）× 100。指数可用于追踪价格、数量或其他经济指标随时间的变化。
Weighted index numbers, such as the Laspeyres or Paasche indices, assign different importance to items. In IGCSE, you focus on calculating simple aggregate price indexes and interpreting their meaning.
加权指数，如拉氏指数或帕氏指数，会赋予各项目不同的重要程度。在 IGCSE 中，重点在于计算简单综合价格指数并理解其含义。
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