📚 Year 11 WJEC Statistics: Core Knowledge Recap | WJEC 统计：核心知识点梳理
Welcome to this comprehensive revision guide for Year 11 WJEC Statistics. We will walk through the essential topics you need to master for your exam, from data collection and probability to correlation and index numbers. Each section is designed to clarify key concepts with concise explanations and practical examples.
欢迎阅读这篇 Year 11 WJEC 统计的综合复习指南。我们将梳理考试必须掌握的核心知识点，涵盖数据收集、概率、相关性及指数等主题。每个小节都通过简洁的解释和实用示例帮助你理清重要概念。
1. Types of Data | 数据类型
Data can be classified as qualitative (non-numerical, e.g. colours or favourite subjects) or quantitative (numerical). Quantitative data is further split into discrete (countable, e.g. number of students) and continuous (measurable, e.g. height, time). Knowing the data type helps you choose appropriate charts and calculations.
数据可分为定性数据（非数值型，如颜色、喜欢的科目）和定量数据（数值型）。定量数据又分为离散型（可计数，如学生人数）和连续型（可测量，如身高、时间）。了解数据类型有助于选择合适的图表和计算方法。
Another important distinction is between primary data (collected by the researcher for a specific purpose) and secondary data (obtained from existing sources). Primary data can be more relevant but is often costlier and more time-consuming to collect.
另一个重要区别是原始数据（研究者为特定目的直接收集）和二手数据（从现有来源获取）。原始数据可能更具针对性，但收集成本更高、更耗时。
2. Sampling Methods | 抽样方法
When studying a population, we often use a sample. Common random sampling methods include simple random sampling, where every member has an equal chance of being selected, and stratified sampling, where the population is split into groups (strata) and a proportional random sample is taken from each.
在研究总体时，我们通常会使用样本。常见的随机抽样方法包括简单随机抽样（每个成员被选中的机会均等）和分层抽样（将总体分成若干层，并从每层中按比例随机抽取样本）。
Non-random methods include convenience sampling (selecting individuals who are easy to reach) and quota sampling (interviewing a set number of people from each category). These can introduce bias because not every individual has a known chance of selection.
非随机方法包括便利抽样（选择容易接触到的人）和配额抽样（从每个类别中采访设定数量的人）。这些方法可能会引入偏差，因为并非每个个体都有已知被选中的概率。
3. Charts and Diagrams | 图表与图形
Bar charts display categorical data with rectangular bars; the height or length represents frequency. Pie charts show proportions of a whole, where each slice’s angle is proportional to the frequency. For continuous data, histograms group data into intervals and the area of each bar is proportional to frequency – the frequency density formula is key: frequency density = frequency ÷ class width.
条形图用长方形条展示分类数据，高度或长度表示频数。饼图展示整体的比例，每个扇形的角度与频数成正比。对于连续数据，直方图将数据分组为区间，每个条形的面积与频数成正比——频率密度的公式很关键：频率密度 = 频数 ÷ 组距。
Frequency polygons are formed by joining the midpoints of the tops of histogram bars, and stem-and-leaf diagrams preserve the original data while displaying its shape. Always label axes clearly and include a key where needed.
频率多边形通过连接直方图各条形顶端的中点形成，茎叶图则在展示数据分布的同时保留了原始数值。务必清晰地标注坐标轴，并在需要时提供图例。
4. Averages and Measures of Spread | 平均数与离散度量
The three common averages are the mean (sum of values ÷ number of values, denoted x̄ for a sample), the median (middle value when data is ordered, or the average of the two middle values for an even number of observations), and the mode (most frequent value). The mean is affected by outliers, while the median is resistant.
三种常见的平均数是：平均数（数值总和 ÷ 数值个数，样本记为 x̄）、中位数（数据排序后的中间值，若数据个数为偶数则取中间两值的平均数）和众数（出现频率最高的值）。平均数受异常值影响，而中位数具有抗干扰性。
Measures of spread include the range (maximum – minimum), the interquartile range IQR = Q3 – Q1 (the range of the middle 50%), and standard deviation. For a sample, standard deviation s = √[Σ(x – x̄)² ÷ (n – 1)]. A larger standard deviation indicates greater variability.
离散度的度量包括极差（最大值 – 最小值）、四分位数间距 IQR = Q3 – Q1（中间50%数据的范围）以及标准差。对于样本，标准差 s = √[Σ(x – x̄)² ÷ (n – 1)]。标准差越大，表示数据变异性越大。
5. Cumulative Frequency and Box Plots | 累积频率与箱线图
Cumulative frequency diagrams plot the running total of frequencies against the upper class boundaries. They allow you to estimate the median, quartiles and percentiles by reading off the graph at certain cumulative frequencies (e.g. median at 50% of the total frequency).
累积频率图将频数的累计总和相对于组上界绘制成图。通过从图上读取特定累积频率对应的数值（例如中位数在总频数的50%处），可以估算中位数、四分位数和百分位数。
Box plots (box-and-whisker diagrams) show the minimum, lower quartile Q1, median Q2, upper quartile Q3 and maximum. They are useful for comparing distributions and identifying skewness. Whiskers typically extend to the minimum and maximum that are not outliers.
箱线图（盒须图）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。箱线图便于比较分布和识别偏态。须通常延伸至非异常值的最小值和最大值。
6. Probability Basics and Tree Diagrams | 概率基础与树状图
Probability is measured on a scale from 0 (impossible) to 1 (certain). The probability of an event A, P(A), is the number of successful outcomes divided by the total number of equally likely outcomes. For mutually exclusive events, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).
概率的度量范围从 0（不可能）到 1（必然）。事件 A 的概率 P(A) 等于成功结果的数量除以所有等可能结果的总数。对于互斥事件，P(A 或 B) = P(A) + P(B)。对于独立事件，P(A 且 B) = P(A) × P(B)。
Tree diagrams help visualise combined events. Multiply along branches for ‘and’ probabilities, and add the probabilities of relevant branches for ‘or’. Always check that probabilities on branches from the same point sum to 1.
树状图有助于可视化复合事件。沿分支相乘得到“且”的概率，将相关分支的概率相加得到“或”的概率。务必检查同一点分出的各分支概率之和为 1。
7. Conditional Probability | 条件概率
Conditional probability, written P(A|B), means the probability of event A occurring given that event B has already occurred. The formula is P(A|B) = P(A and B) ÷ P(B). This appears in tree diagrams when the second set of branches changes depending on the first outcome.
条件概率，记作 P(A|B)，表示在事件 B 已经发生的情况下，事件 A 发生的概率。公式为 P(A|B) = P(A 且 B) ÷ P(B)。当树状图的第二层分支根据第一层结果而变化时，就会用到条件概率。
Questions often involve ‘with replacement’ (independent events, probabilities unchanged) and ‘without replacement’ (dependent events, probabilities change). Always recognise which scenario applies.
题目常涉及“放回”（独立事件，概率不变）和“不放回”（相依事件，概率改变）两种情况。务必辨别题目属于哪种情形。
8. Correlation and Regression | 相关与回归
Correlation describes the strength and direction of a linear relationship between two variables. It can be positive (as one increases, the other tends to increase), negative, or zero. The product moment correlation coefficient, r, ranges from -1 to +1; values close to 1 or -1 indicate strong correlation.
相关描述两个变量之间线性关系的强度和方向。可以是正相关（一个变量增加，另一个也倾向于增加）、负相关或无相关。积矩相关系数 r 的取值范围是从 -1 到 +1；接近 1 或 -1 的值表示强相关。
Regression lines are used to model the relationship and make predictions. The equation of the least squares regression line is usually given as y = a + bx, where b is the gradient. Always draw the regression line on the scatter graph and interpolate only within the data range; extrapolation can be unreliable.
回归线用于建模关系并进行预测。最小二乘回归线的方程常表示为 y = a + bx，其中 b 是斜率。务必在散点图上画出回归线，并且只对数据范围内进行插值；外推可能不可靠。
9. Time Series and Moving Averages | 时间序列与移动平均
A time series is a set of data collected at regular intervals over time. It typically contains four components: the trend (long-term movement), seasonal variation (regular pattern within each year), cyclic variation, and random variation.
时间序列是随时间以固定间隔收集的一组数据。它通常包含四个组成部分：趋势（长期走向）、季节性变动（每年内的规律模式）、周期性变动和随机变动。
Moving averages smooth out short-term fluctuations and highlight the trend. For quarterly data, a 4-point moving average is used, then often centred. Seasonal variation is then calculated by subtracting the trend from the actual value, and seasonal effects can be used for forecasting.
移动平均能抚平短期波动、突显趋势。对于季度数据，使用 4 点移动平均，随后通常进行中心化处理。然后从实际值中减去趋势值计算季节性变动，并可利用季节效应进行预测。
10. Index Numbers | 指数
Index numbers show how a value changes over time compared with a base period. The base period value is set to 100. An index number for a later period is (value ÷ base value) × 100. They are widely used to track prices, inflation (e.g. Consumer Price Index) and economic output.
指数用于显示数值相对于基期随时间的变化。基期的值设为 100。其后时期的指数 = (当期值 ÷ 基期值) × 100。指数被广泛用于跟踪价格、通胀（例如消费者价格指数）和经济产出。
Weighted index numbers give more importance to some items. For example, the Retail Price Index uses weights based on household spending patterns. A simple weighted prices index is calculated as Σ(price relative × weight) ÷ Σ weights.
加权指数赋予某些项目更高的重要性。例如，零售价格指数使用基于家庭支出模式的权重。简单加权价格指数计算为 Σ(价格比 × 权重) ÷ Σ 权重。
11. Binomial Distribution | 二项分布
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p. If X ~ B(n, p), the probability of getting exactly r successes is:
二项分布用于描述固定次数独立试验中成功的次数，每次试验的成功概率 p 相同。若 X ~ B(n, p)，则恰好得到 r 次成功的概率为：
P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ
where ⁿCᵣ is the binomial coefficient. Mean of X = np, variance = np(1-p). Use binomial tables or the formula for calculations. The distribution assumes trials are independent and the probability remains constant.
其中 ⁿCᵣ 是二项式系数。X 的均值 = np，方差 = np(1-p)。计算时可使用二项分布表或公式。此分布假定各次试验独立且概率保持不变。
12. Statistical Diagrams: Histograms and Comparative Charts | 统计图表：直方图与对比图
Histograms require finding frequency density for unequal class widths. The vertical axis is labelled ‘Frequency density’. To draw a histogram, calculate frequency density = frequency ÷ class width, then draw bars with appropriate width and height. The area of each bar equals the frequency.
直方图在组距不等时需要计算频率密度。纵轴标注为“频率密度”。绘制直方图时，计算频率密度 = 频数 ÷ 组距，然后按相应宽度和高度画出条形。每个条形的面积等于频数。
Comparative charts such as multiple bar charts, composite bar charts and population pyramids allow you to compare two or more datasets visually. Always include a clear key and use consistent scales. When interpreting charts, comment on shapes, central tendency, spread and any anomalies.
复合条形图、堆积条形图和人口金字塔等对比图表能够让你直观地比较两个或多个数据集。务必包含清晰的图例并使用一致的刻度。在解读图表时，要评述形状、集中趋势、离散程度以及任何异常。
Published by TutorHao | Statistics Revision Series | aleveler.com
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