📚 Year 10 CIE Statistics: Core Knowledge Points Review | 核心知识点梳理
Statistics in CIE Year 10 provides the foundational tools for collecting, presenting, analysing, and interpreting data. Mastering these core concepts will not only prepare you for IGCSE exams but also build critical thinking skills essential for further study. This article systematically organizes the major topics you need to know, with clear English and Chinese explanations side by side.
CIE 十年级统计课程提供了收集、展示、分析和解释数据的基本工具。掌握这些核心概念不仅能帮助你为 IGCSE 考试做好准备，还能培养进一步学习所需的关键思维能力。本文将系统梳理你必须了解的主要专题，并以清晰的中英文对照进行讲解。
1. Types of Data | 数据类型
Data can be classified as qualitative (categorical) or quantitative (numerical). Qualitative data describe qualities, such as colour or brand, while quantitative data involve numbers. Quantitative data is further split into discrete data, which can only take specific values (like the number of students), and continuous data, which can take any value within a range (like height or time). Understanding the data type determines which charts and statistics are appropriate.
数据可分为定性（分类）数据和定量（数值）数据。定性数据描述性质，如颜色或品牌；定量数据则涉及数字。定量数据进一步分为离散数据（只能取特定值，如学生人数）和连续数据（可在一定范围内取任意值，如身高或时间）。了解数据类型有助于选择合适的图表和统计量。
- Qualitative/Categorical: favourite subject, marital status
- 定量/分类数据：最喜欢的科目、婚姻状况
- Quantitative Discrete: number of cars in a household, shoes size
- 定量离散数据：家庭汽车数量、鞋码
- Quantitative Continuous: weight, temperature, length of a leaf
- 定量连续数据：体重、温度、叶子的长度
2. Data Collection and Sampling Methods | 数据收集与抽样方法
Data can be gathered through a census (every member of the population) or a sample (a subset). A sample is often more practical, but it must be representative. Common sampling methods include random sampling (every member has an equal chance), stratified sampling (population divided into strata, then random samples from each), systematic sampling (choosing every kth member), and convenience sampling (using readily available subjects). Bias can occur if the sample is not selected properly, such as when only friends are surveyed or when a questionnaire has leading questions.
数据可以通过普查（调查总体中的每一个成员）或抽样（选取一个子集）来收集。抽样通常更实际，但样本必须具有代表性。常见的抽样方法包括随机抽样（每个成员被抽中的机会均等）、分层抽样（按层划分总体后再从各层随机选取）、系统抽样（每隔 k 个成员选取一个）和便利抽样（使用容易获得的个体）。如果样本选择不当（例如只调查朋友或问卷中有诱导性问题），就会产生偏差。
- Random Sampling: drawing names from a hat
- 随机抽样：从帽子里抽取名字
- Stratified Sampling: dividing school by year groups then sampling proportionally
- 分层抽样：按年级分组后按比例抽取
- Bias examples: asking only pet owners about favourite animals; a survey with 'Don't you agree that...'
- 偏差示例：仅向宠物主人询问最喜爱的动物；问卷中包含“难道你不认为……”
3. Frequency Tables and Charts | 频数表与图表
us data, we group the values into intervals (classes) and record the frequency in each class. A histogram represents grouped continuous data: the area of each bar is proportional to the frequency, and for unequal class widths, we use frequency density = frequency ÷ class width. Bar charts are for discrete or categorical data with equal widths. A pie chart shows proportions as sectors of a circle, with each angle = (frequency / total) × 360°. A frequency polygon can be constructed by joining the midpoints of the tops of histogram bars.
将原始数据整理成频数表是分析的第一步。对于离散数据，表格列出每个取值和相应的频数。对于连续数据，我们将数值分组到区间（组距）中，并记录每个区间的频数。直方图用于分组的连续数据：每个长方形的面积与频数成正比；当组距不相等时，需要使用频数密度 = 频数 ÷ 组距。条形图用于等宽离散或分类数据。饼图将比例表示为圆中的扇形，每个扇形的圆心角 = (频数 ÷ 总数) × 360°。连接直方图各长方形顶部中点即可得到频数多边形。
| Mass (kg) | Frequency |
|---|---|
| 40 < x ≤ 50 | 5 |
| 50 < x ≤ 60 | 12 |
| 60 < x ≤ 80 | 8 |
Frequency density for 60 < x ≤ 80: class width = 20, so density = 8/20 = 0.4
60 < x ≤ 80 区间的频数密度：组距 = 20，密度 = 8/20 = 0.4
4. Measures of Central Tendency | 集中趋势测量
The three main measures are mean, median, and mode. The mean (x̄) is the arithmetic average: x̄ = Σx / n for raw data, or for grouped data x̄ ear interpolation from a cumulative frequency graph. The mode is the most frequent value. The mean is sensitive to outliers, while the median is robust. Choose the best average depending on the data distribution and the presence of extreme values.
三种主要的测量是平均数、中位数和众数。平均数（x̄）是算术平均值：对于原始数据，x̄ = Σx / n；对于分组数据，x̄ = Σ(f × 组中值) / Σf。中位数是数据排序后处于中间位置的数值；未分组数据的位置为 (n+1)/2，分组数据则通过累积频率图进行线性插值估算。众数是出现次数最多的数值。平均数对异常值敏感，而中位数较为稳健。应根据数据分布和是否存在极端值来选择最合适的平均值。
Mean example: data 3, 5, 8, 9, 10 → mean = (3+5+8+9+10)/5 = 7
平均数示例：数据 3, 5, 8, 9, 10 → 平均数 = 7
Median for 6 values: 2, 3, 5, 7, 8, 12 → median = (5+7)/2 = 6
6 个值的中位数：2, 3, 5, 7, 8, 12 → 中位数 = (5+7)/2 = 6
5. Measures of Dispersion | 离散程度测量
he difference between the largest and smallest values. The interquartile range (IQR) is Q₃ – Q₁, covering the middle 50% asures the average distance of each data point from the mean. For a sample, s = √[ Σ(x - x̄ statistics are essential for comparing data sets more thoroughly than just using averages.
离散程度描述数据的分散情况。极差即最大值与最小值之差。四分位距（IQR）= Q₃ – Q₁，涵盖中间 50% 的数据，不受极端值影响。标准差（σ 或 s）衡量每个数据点与平均值的平均距离。对于样本，s = √[ Σ(x - x̄)² / (n - 1) ]。标准差越大，表示数据越分散。方差是标准差的平方。这些统计量对于更全面地比较数据集至关重要，而不仅仅是使用平均值。
, 4, 6, 8, 10 → mean = 6; deviations: -4, -2, 0, 2, 4; squared sum = 16+4+0+4+16 = 40; s = √40/4 = √10 ≈ 3.16
样本数据：2, 4, 6, 8, 10 → 平均数 = 6；离差平方和 = 40；s = √40/4 ≈ 3.16
6. Cumulative Frequency and Box-and-Whisker Plots | 累积频率与箱线图
ary against the cumulative frequency. From the graph, you can estimate the median (50th percentile), lower quartile Q₁ (25th percentile), and upper quartile Q₃ (75th percentile). A box-and-whisker plot displays these five-number summaries (minimum, Q₁, median, Q₃, maximum) and highlights skewness and outliers. The interquartile range is found from the box length, and any value more than 1.5 × IQR beyond the quartiles is considered an outlier.
累积频数表是按组依次累加频数得到的。累积频率曲线（拱形图）将组的上限对累积频数作图。从图中可以估计中位数（第 50 百分位数）、下四分位数 Q₁（第 25 百分位数）和上四分位数 Q₃（第 75 百分位数）。箱线图用五个数概括数据（最小值、Q₁、中位数、Q₃、最大值），并能显示出偏态和异常值。四分位距由箱体长度得出，任何超出 Q₁ 或 Q₃ 1.5 × IQR 的值被视为异常值。
cy diagram: if total frequency = 80, median at 40, Q₁ at 20, Q₃ at 60 → IQR = 60 - 20 = 40
由累积频率图：若总频数 = 80，中位数在 40 处，Q₁ 在 20 处，Q₃ 在 60 处 → IQR = 40
7. Probability Fundamentals | 概率基础
of favourable outcomes / total number of outcomes. The sum of probabilities of all possible outcomes is 1. The complement rule states P(not A) = 1 – re unknown. Two events are mutually exclusive if they cannot happen at the same time; for such events, P(A or B) = P(A) + P(B).
概率衡量事件发生的可能性，范围从 0（不可能）到 1（必然）。对于等可能的结果，P(事件) = 有利结果的数量 / 所有可能结果的总数。所有可能结果的概率之和为 1。互补规则：P(非 A) = 1 – P(A)。在 n 次试验中的期望频数为 n × P(A)。当理论概率未知时，可用实验中的相对频率来估计概率。若两个事件不能同时发生，则它们互斥；对于互斥事件，P(A 或 B) = P(A) + P(B)。
Tossing a fair die: P(even) = 3/6 = 1/2; P(not 6) = 5/6
掷一枚公正骰子：P(偶数) = 3/6 = 1/2；P(非 6) = 5/6
8. Tree Diagrams and Conditional Probability | 树图与条件概率
Tree diagrams are essential for organising multi-stage probability experiments. Each branch represents a possible outcome with its probability written along it. To find the probability of a combined path, multiply the probabilities along the branches. When events are dependent (e.g., drawing without replacement), probabilities on the second layer change based on the first outcome. Conditional probability, P(A|B), is the probability of A occurring given that B has already occurred, calculated by P(A ∩ B) / P(B). Tree diagrams help visualise and compute these probabilities clearly.
树图是组织多阶段概率实验的重要工具。每个分支代表一种可能的结果，旁边写有概率。要计算某条组合路径的概率，就将路线上各分支的概率相乘。当事件相互依赖时（如不放回抽取），第二层的概率会根据第一层结果而变化。条件概率 P(A|B) 表示在 B 已发生的条件下 A 发生的概率，计算公式为 P(A|B) = P(A ∩ B) / P(B)。树图能帮助清晰可视化和计算这些概率。
lue. Pick two without replacement. P(first red and second blue) = (3/5)×(2/4) = 6/20 = 0.3
袋子：3 红，2 蓝。不放回摸两次。P(先红后蓝) = (3/5)×(2/4) = 0.3
9. Binomial Distribution | 二项分布
ditions are: a fixed number of trials n, each trial has two outcomes (success/failure), trials are independent, and p remains constant. The probability of exactly r successes is given by P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ, where q = 1 – ituations include tossing a coin multiple times or checking a sample of bolts for defects.
二项分布描述在固定次数的独立试验中成功的次数，每次试验的成功概率 p 不变。条件为：试验次数 n 固定，每次试验只有两个结果（成功/失败），试验互相独立，且 p 保持不变。恰好获得 r 次成功的概率为 P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ，其中 q = 1 – p，ⁿCᵣ 为二项系数。二项分布的均值为 np，方差为 npq。典型的二项情形包括多次抛硬币或检查一批螺栓的次品情况。
X ~ B(4, 0.3): P(X=2) = ⁴C₂ × (0.3)² × (0.7)² = 6 × 0.09 × 0.49 = 0.2646
X ~ B(4, 0.3): P(X=2) = 6 × 0.09 × 0.49 = 0.2646
10. Scatter Diagrams and Correlation | 散点图与相关性
A scatter diagram plots bivariate data (pairs of variables) on a coordinate grid to reveal the relationship between them. Correlation describes the strength and direction of the linear relationship. Positive correlation means as one variable increases, the other tends to increase; negative correlation means as one increases, the other tends to decrease. If no clear pattern exists, there is zero or no correlation. A line of best fit can be drawn by eye to pass through the middle of the points, with roughly equal points above and below. This line can be used to estimate one variable given the other, but extrapolation beyond the data range is unreliable. Note that correlation does not imply causation.
散点图将双变量数据（成对变量）绘制在坐标网格上，以揭示它们之间的关系。相关性描述线性关系的强度和方向。正相关意味着一个变量增加时，另一个也倾向于增加；负相关则相反。若没有明显模式，则为零或无相关。最佳拟合线可通过目测画出，使之穿过点的中心，并使线上方和下方的点数量大致相等。这条线可用于根据一个变量估计另一个变量，但超出数据范围的外推不可靠。注意，相关并不代表因果。
- Example: Temperature and ice cream sales → positive correlation
- 示例：气温与冰淇淋销量 → 正相关
- Example: Age of car and its value → negative correlation
- 示例：车龄与车辆价值 → 负相关
11. Time Series and Moving Averages | 时间序列与移动平均
A time series records data at regular intervals over time. It often shows four components: trend (long-term movement), seasonal variation (regular pattern over a short period), cyclical variation (longer-term oscillations), and random fluctuations. To smooth out short-term fluctuations and reveal the underlying trend, we calculate moving averages. For example, a 3-point moving average replaces each value with the average of that point and its neighbours. Plotting the moving averages helps identify the trend line, which can be extended to make forecasts, though forecasts become less reliable the further ahead they go. Seasonal effects are then estimated by comparing actual data to the trend.
时间序列按固定的时间间隔记录数据。它通常显示出四种成分：趋势（长期走向）、季节变动（短期内的规律模式）、周期变动（更长期的波动）和随机波动。为了消除短期波动并揭示潜在趋势，我们计算移动平均值。例如，三点移动平均用该点及其相邻点的平均值替换原值。绘制移动平均有助于识别趋势线，并可延伸进行预测，但预测时间越远，可靠性越低。而后通过比较实际数据与趋势线来估计季节效应。
Sales: 5, 7, 6, 10 → 3-point moving average: (5+7+6)/3 = 6, (7+6+10)/3 ≈ 7.67
销售额：5, 7, 6, 10 → 3-点移动平均：6, 7.67
Published by TutorHao | Statistics Revision Series | aleveler.com
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