📚 Cambridge Year 10 Statistics: Core Concepts Review | 剑桥 Year 10 统计：核心知识点梳理
Welcome to the comprehensive review of core statistical concepts for Cambridge Year 10. This guide covers essential topics including data types, graphical representations, measures of central tendency and spread, probability, and an introduction to correlation and regression. By mastering these fundamentals, students will build a strong foundation for their IGCSE Statistics exam.
欢迎阅读剑桥 Year 10 统计核心知识点梳理。本指南涵盖数据类型、图形表示、集中趋势和离散程度的度量、概率以及相关与回归简介等重要主题。掌握这些基础知识，学生将为 IGCSE 统计考试打下坚实基础。
1. Types of Data | 数据类型
Data can be classified as categorical (qualitative) or numerical (quantitative). Numerical data are further divided into discrete (countable, e.g. number of students) and continuous (measurable, e.g. height). Recognising the data type is the first step in choosing suitable graphs and summary statistics.
数据可分为分类数据（定性数据）和数值数据（定量数据）。数值数据又分为离散数据（可数的，如学生人数）和连续数据（可测量的，如身高）。识别数据类型是选择合适图表和汇总统计量的第一步。
Categorical data are often displayed using bar charts or pie charts, while discrete data are shown with bar charts (gaps between bars) and continuous data with histograms (no gaps) or line graphs.
分类数据通常用条形图或饼图展示，离散数据用条形图（条间有间隔）展示，连续数据则用直方图（无间隔）或折线图展示。
2. Data Collection and Sampling | 数据收集与抽样
Data can be collected through surveys, experiments, or observations. It is essential to design unbiased questions and record data accurately. A population is the entire group of interest, while a sample is a subset selected for study.
数据可以通过调查、实验或观察收集。设计无偏问题并准确记录数据至关重要。总体是所关注的整个群体，而样本是为研究选出的子集。
Common sampling methods include simple random sampling, stratified sampling (ensuring proportional representation of subgroups), systematic sampling (selecting every kth item), and quota sampling. A sample should be representative to avoid bias and allow valid conclusions.
常见抽样方法包括简单随机抽样、分层抽样（确保子组比例代表）、系统抽样（每第 k 个抽取一个）和配额抽样。样本应具有代表性，以避免偏差并得出有效结论。
3. Frequency Distributions and Charts | 频率分布与图表
A frequency table organises raw data into classes or groups, showing the number of observations in each class. For continuous data, class intervals must be clearly defined (e.g. 10 ≤ h < 20).
频率表将原始数据整理成组或类别，显示每组的观测数。对于连续数据，组距必须明确定义（例如 10 ≤ h < 20）。
Histograms represent grouped continuous data where the area of each bar is proportional to the frequency. For unequal class widths, frequency density = frequency ÷ class width must be used to ensure fair representation.
直方图用于表示分组连续数据，每个直条的面积与频率成比例。对于不等宽的组距，必须使用频率密度（频率 ÷ 组距）以确保公平表示。
For discrete data, bar charts show frequencies with gaps between bars. Pie charts display proportions of a whole, with each sector angle = (frequency/total) × 360°.
对于离散数据，条形图在条间有间隔地显示频数。饼图显示整体的各比例，每个扇形的角度 = (频数/总数) × 360°。
4. Measures of Central Tendency | 集中趋势的度量
The mean (x̄) is the arithmetic average, calculated by summing all data values and dividing by the number of values: x̄ = Σx / n. The mean is sensitive to extreme values (outliers).
均值（x̄）是算术平均数，计算方法是将所有数据值相加再除以数据个数：x̄ = Σx / n。均值对极端值（异常值）敏感。
The median is the middle value when data are ordered. If n is even, it is the mean of the two middle values. The median is resistant to outliers and is preferred for skewed distributions.
中位数是将数据排序后的中间值。如果 n 为偶数，则为两个中间值的均值。中位数对异常值不敏感，适用于偏态分布。
The mode is the value that occurs most frequently. A dataset can have one mode (unimodal), two (bimodal), or more. The mode is the only measure suitable for categorical data.
众数是出现频率最高的值。一个数据集可以有一个众数（单峰）、两个众数（双峰）或更多。众数是唯一适用于分类数据的度量。
5. Measures of Spread | 离散程度的度量
Range = maximum – minimum is the simplest measure of spread, but it is heavily influenced by outliers. Interquartile range (IQR) = Q₃ – Q₁ gives the spread of the middle 50% and is more robust.
极差 = 最大值 – 最小值是最简单的离散度量，但极易受异常值影响。四分位距 (IQR) = Q₃ – Q₁ 反映中间 50% 数据的分散程度，更具鲁棒性。
Variance and standard deviation measure how far each value is from the mean. For a sample, variance s² = Σ(x – x̄)² / (n – 1), and standard deviation s = √[Σ(x – x̄)² / (n – 1)]. A larger spread indicates greater variability.
方差和标准差衡量每个值与均值的距离。对于样本，方差 s² = Σ(x – x̄)² / (n – 1)，标准差 s = √[Σ(x – x̄)² / (n – 1)]。离散程度越大，表明变异性越大。
Percentiles split the data into 100 equal parts. The kth percentile is the value below which k% of the data fall. Q₁ is the 25th percentile, the median is the 50th, and Q₃ is the 75th.
百分位数将数据分为 100 等份。第 k 百分位数是指有 k% 的数据低于该值的值。Q₁ 是第 25 百分位数，中位数是第 50，Q₃ 是第 75。
6. Cumulative Frequency and Box Plots | 累积频率与箱线图
A cumulative frequency table adds frequencies sequentially. The cumulative frequency curve (ogive) plots upper class boundaries against cumulative frequency. It is used to estimate the median, quartiles, and percentiles by interpolation.
累积频率表按顺序累加频数。累积频率曲线（卵形线）以组上限为横坐标、累积频率为纵坐标绘制，用于通过插值估计中位数、四分位数和百分位数。
A box plot (box-and-whisker plot) displays the five-number summary: minimum, Q₁, median, Q₃, maximum. The box represents the IQR, the line inside shows the median, and the whiskers extend to the minimum and maximum, or up to 1.5 × IQR from the quartiles.
箱线图（盒须图）展示五数概括：最小值、Q₁、中位数、Q₃、最大值。盒子代表 IQR，内部线条表示中位数，须线延伸到最小值和最大值，或从四分位数起最远至 1.5 × IQR。
7. Stem-and-Leaf Diagrams | 茎叶图
A stem-and-leaf diagram organises numerical data while preserving each original value. The ‘stem’ is the leading digit(s), and the ‘leaf’ is the final digit. It shows the shape of the distribution and allows quick identification of the median and mode.
茎叶图在组织数值数据的同时保留每一个原始值。“茎”是前置数字，“叶”是末位数字。它能展示分布形态，并能快速识别中位数和众数。
Back-to-back stem-and-leaf diagrams compare two related datasets sharing a common stem, with leaves extending left and right. They are useful for comparing distributions such as test scores of two classes.
背靠背茎叶图用于比较两个相关数据集，共享同一根茎，叶子分别向左右延伸。它们适用于比较分布，如两个班级的考试分数。
8. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph plots bivariate data to show the relationship between two variables. Points are plotted on a coordinate grid with the independent variable on the x-axis and the dependent variable on the y-axis.
散点图将双变量数据绘制成点，以显示两个变量之间的关系。点绘制在坐标网格上，自变量在 x 轴，因变量在 y 轴。
Correlation describes the direction and strength of a linear relationship. It can be positive (as x increases, y tends to increase), negative (as x increases, y tends to decrease), or none. Strength is described as strong, moderate, or weak.
相关性描述线性关系的方向和强度。可以是正相关（x 增加，y 倾向于增加）、负相关（x 增加，y 倾向于减少）或无相关。强度描述为强、中等或弱。
A line of best fit can be drawn by eye to pass as near as possible to all points, balancing points above and below. It is used to predict values within the range of data (interpolation); predicting outside (extrapolation) can be unreliable.
可以通过目测画出最佳拟合线，使其尽可能接近所有点，平衡线上方和下方的点。它用于预测数据范围内的值（内插）；范围外的预测（外推）可能不可靠。
9. Basic Probability | 基本概率
Probability is a measure of the likelihood of an event, expressed as a number between 0 (impossible) and 1 (certain). For equally likely outcomes, P(A) = number of favourable outcomes / total number of outcomes.
概率是事件发生可能性的度量，用 0（不可能）到 1（必然）之间的数字表示。对于等可能结果，P(A) = 有利结果数 / 总结果数。
The sum of probabilities of all possible outcomes is 1. The complement rule states: P(not A) = 1 – P(A). This is useful for finding the probability of an event not occurring.
所有可能结果的概率之和为 1。互补规则为：P(非 A) = 1 – P(A)。这可用于求事件不发生的概率。
Combined events: For mutually exclusive events A and B, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B). Probability tree diagrams help visualise successive independent events.
组合事件：对于互斥事件 A 和 B，P(A 或 B) = P(A) + P(B)。对于独立事件，P(A 且 B) = P(A) × P(B)。概率树图有助于直观地展示连续的独立事件。
10. Introduction to Regression | 回归简介
Regression extends the line of best fit to model the linear relationship between two variables. The equation of the line is y = a + bx, where b is the slope (rate of change) and a is the y-intercept (value of y when x = 0).
回归将最佳拟合线延伸为两个变量之间线性关系的模型。直线方程为 y = a + bx，其中 b 是斜率（变化率），a 是 y 截距（x = 0 时的 y 值）。
In Year 10, the line is often estimated from a scatter graph by selecting two points on the line. The slope b = (change in y) / (change in x) = Δy / Δx. The intercept a is read from the graph or calculated.
在 Year 10，通常通过选择散点图上直线上的两点来估计该直线。斜率 b = (y 的变化) / (x 的变化) = Δy / Δx。截距 a 可从图中读出或计算得出。
The regression line minimises the sum of the squares of the vertical distances (residuals) from the points. It is used to predict y for a given x, but predictions are most reliable near the centre of the data.
回归直线最小化点到直线的垂直距离（残差）的平方和。它用于给定 x 预测 y，但在数据中心附近预测最可靠。
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