📚 Core Concepts of WJEC GCSE Statistics | Yr 11 统计核心知识点梳理
This article provides a structured revision guide to the essential topics in the WJEC GCSE Statistics specification for Year 11. It covers everything from data types to the normal distribution, offering clear bilingual explanations to support both English and Chinese learners. Each section introduces a key concept with worked ideas, followed by parallel Chinese translations, making it ideal for GCSE exam preparation.
本文按照 WJEC 统计课程核心知识点系统梳理，为 Year 11 学生提供完整的双语复习指南。涵盖数据类型、图表展示、集中趋势、概率分布等关键内容，每部分均以英文要点和对应中文解析呈现，帮助中英双语学习者高效备考。
1. Types of Data and Data Collection | 数据类型与数据收集
Data can be classified as qualitative (categorical) or quantitative (numerical). Quantitative data is either discrete (countable) or continuous (measurable). Data collection methods include surveys, experiments, observations, and using secondary sources. The reliability of data depends on how it is gathered and whether biases are controlled.
数据可分为定性（分类）数据和定量（数值）数据。定量数据又分为离散型（可数）和连续型（可测量）。收集数据的方式包括调查、实验、观察和二手资料。数据的可靠性取决于收集方法以及是否控制了偏差。
Primary data is collected first-hand for a specific purpose, while secondary data is obtained from existing records. When designing a questionnaire, questions should be clear, unbiased, and appropriate to the target population. A pilot study helps to refine the data collection tool.
一手数据是为特定目的直接收集的，二手数据来自已有的记录。设计问卷时，问题应清晰、无偏见并适合目标人群。试调查有助于改进数据收集工具。
2. Sampling Methods | 抽样方法
A population is the entire group being studied, while a sample is a subset selected to represent it. Random sampling gives every member an equal chance of selection and reduces bias. In simple random sampling, each individual is chosen by chance, often using random number tables or generators.
总体是研究对象的全体，样本是选取用以代表总体的子集。随机抽样让每个成员被选中的机会均等，可减少偏差。在简单随机抽样中，通常借助随机数表或生成器来抽取样本。
Stratified sampling divides the population into distinct groups (strata) and selects a proportional sample from each. Systematic sampling picks every k-th item from a list. Convenience sampling selects readily available individuals but may introduce significant bias.
分层抽样先把总体分成不同的层，再从各层按比例抽取样本。系统抽样从名单中每隔 k 个抽取一个。便利抽样选择最容易接触到的人，但可能带来较大偏差。
A sample frame is a list of all members of the population. If the sampling frame is incomplete, the sample may suffer from undercoverage. The sampling method directly affects the validity of conclusions drawn from the data.
抽样框是总体所有成员的名单。如果抽样框不完整，样本可能存在覆盖不全问题。抽样方法直接影响从数据中得出结论的有效性。
3. Presenting Data: Charts and Graphs | 数据展示：图表与图形
Effective data presentation helps reveal patterns and comparisons. Bar charts are used for categorical or discrete data, with gaps between bars. A pie chart shows proportions of a whole, where each sector angle = (frequency ÷ total) × 360°.
有效的数据展示有助于揭示模式和对比。条形图用于分类或离散数据，条与条间有空隙。饼图展示各部分在整体中的占比，每个扇形的角度 = (频数 ÷ 总计) × 360°。
For continuous data, a histogram displays frequency density. In a histogram, the area of each bar is proportional to frequency, and class width may be unequal. Frequency density = frequency ÷ class width.
对于连续数据，直方图展示频率密度。在直方图中，每个矩形的面积与频数成正比，且组距可以不相等。频率密度 = 频数 ÷ 组距。
Pictograms use symbols to represent a number of items, while stem-and-leaf diagrams retain the original data values and show the distribution shape.
象形图用符号表示物品数量；茎叶图则保留了原始数据值，并能展示分布形状。
4. Measures of Central Tendency | 集中趋势的度量
Central tendency describes the typical value in a data set. The mode is the most frequent value. The median is the middle value when data are ordered. The mean is the sum of all values divided by the number of values:
x̄ = Σx / n
集中趋势描述数据集的典型值。众数是出现次数最多的值。中位数是将数据排序后处于中间位置的值。均值是所有数值之和除以数值的个数：
x̄ = Σx / n
For grouped data, the mean is estimated using midpoints. The modal class is the group with the highest frequency density. The median can be found from a cumulative frequency curve by locating the ½nth value.
对于分组数据，均值通过组中值进行估算。众数组是频率密度最高的组。中位数可通过累积频率曲线找到第 ½n 个值来确定。
Each measure has advantages: the mean uses all data but is sensitive to outliers; the median is robust to outliers; the mode is quick to identify and useful for categorical data.
每种度量都有优点：均值用到了所有数据但对异常值敏感；中位数不受异常值影响；众数易于识别，对于分类数据特别有用。
5. Measures of Spread | 离散程度的度量
Spread quantifies how data values are dispersed. The range = maximum – minimum, but it is easily affected by outliers. The interquartile range (IQR) is the difference between the upper quartile (Q₃) and the lower quartile (Q₁): IQR = Q₃ − Q₁, measuring the middle 50 % of data.
离散程度衡量数据值的分散情况。极差 = 最大值 − 最小值，但易受异常值影响。四分位距 (IQR) 是上四分位数 Q₃ 与下四分位数 Q₁ 之差：IQR = Q₃ − Q₁，代表了中间 50% 数据的跨度。
The standard deviation gives a more comprehensive measure of spread around the mean. For a sample, the formula is:
s = √[ Σ(x − x̄)² / (n − 1) ]
标准差提供了数据在均值周围离散程度的更全面的度量。对于样本，公式为：
s = √[ Σ(x − x̄)² / (n − 1) ]
A small standard deviation indicates data are clustered near the mean; a large one shows wide spread. Along with the mean, the standard deviation is used to compare distributions effectively.
标准差小表示数据集中在均值附近；标准差大则表示分散。标准差与均值一起使用，能有效比较不同的分布。
6. Box Plots and Cumulative Frequency | 箱线图与累积频率
A box plot displays the five-number summary: minimum, Q₁, median, Q₃, maximum. It quickly shows the central location, spread, and any skewness or outliers. The box spans from Q₁ to Q₃, with the median marked inside.
箱线图展示五数概括：最小值、Q₁、中位数、Q₃、最大值。它能迅速显示数据中心、离散程度、偏态以及异常值。箱体从 Q₁ 跨至 Q₃，中位数在箱体内部标出。
Outliers are often defined as values more than 1.5 × IQR below Q₁ or above Q₃. Whiskers extend to the furthest data point that is not an outlier.
异常值通常定义为低于 Q₁ − 1.5×IQR 或高于 Q₃ + 1.5×IQR 的值。箱线图的须线延伸到非异常值的最远数据点。
A cumulative frequency graph plots the running total of frequencies against the upper class boundary. It can be used to estimate the median, quartiles, and percentiles. The steeper the curve, the higher the frequency density in that region.
累积频率图将频数的累计值对应组上限绘制成图。可用来估计中位数、四分位数和百分位数。曲线越陡，说明该区域频率密度越高。
7. Probability Basics | 概率基础
Probability measures the chance of an event occurring, ranging from 0 (impossible) to 1 (certain). The theoretical probability P(A) = number of favourable outcomes / total number of equally likely outcomes.
概率衡量事件发生的可能性，范围从 0（不可能）到 1（确定）。理论概率 P(A) = 有利结果数 / 所有等可能结果总数。
For two events A and B, the addition rule is 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).
对于两个事件 A 和 B，加法法则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。如果 A 与 B 互斥，则 P(A ∩ B) = 0，所以 P(A ∪ B) = P(A) + P(B)。
Tree diagrams help to visualise combined events and conditional probabilities. Multiply along branches for ‘and’ probabilities; add relevant outcomes for ‘or’ probabilities. Conditional probability P(A|B) = P(A ∩ B) / P(B).
树状图有助于可视化复合事件和条件概率。沿分支相乘以求“且”的概率；将相关结果相加以求“或”的概率。条件概率 P(A|B) = P(A ∩ B) / P(B)。
Relative frequency from an experiment can be used as an estimate of probability, especially when outcomes are not equally likely. The larger the number of trials, the more reliable the estimate.
通过试验得到的相对频率可用作概率的估计，尤其在结果不等可能时。试验次数越多，估计越可靠。
8. The 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), then P(X = r) = ⁿCᵣ × pʳ × (1 − p)ⁿ⁻ʳ.
二项分布描述在固定次数的独立试验中成功次数的分布，每次试验的成功概率 p 相同。若 X ~ B(n, p)，则 P(X = r) = ⁿCᵣ × pʳ × (1 − p)ⁿ⁻ʳ。
Conditions for a binomial distribution: fixed number of trials n, trials are independent, only two outcomes (success/failure), and p remains constant. ⁿCᵣ is the binomial coefficient, calculated as n! / (r!(n − r)!).
二项分布的条件：试验次数 n 固定、各次独立、只有两个结果（成功/失败）、每次 p 不变。 ⁿCᵣ 是二项式系数，等于 n! / (r!(n − r)!)。
The mean of a binomial distribution is μ = np, and the variance is σ² = np(1 − p). When n is large and p is close to 0.5, the binomial can be approximated by a normal distribution.
二项分布的均值为 μ = np，方差 σ² = np(1 − p)。当 n 较大且 p 接近 0.5 时，可用正态分布近似二项分布。
9. The Normal Distribution | 正态分布
The normal distribution is a continuous, bell-shaped distribution symmetrical about the mean μ. It is defined completely by μ and standard deviation σ, written N(μ, σ²). About 68 % of data lie within 1σ of the mean, 95 % within 2σ, and 99.7 % within 3σ (the empirical rule).
正态分布是一种连续、钟形、以均值 μ 为对称轴的分布，完全由 μ 和标准差 σ 定义，记作 N(μ, σ²)。大约 68% 的数据落在均值 ±1σ 内，95% 在 ±2σ 内，99.7% 在 ±3σ 内（经验法则）。
Many natural measurements (heights, weights, test scores) approximate a normal distribution. To find probabilities, the standard normal distribution Z ~ N(0, 1) is used, where Z = (X − μ) / σ. Tables or technology give the cumulative probability Φ(z).
许多自然测量值（身高、体重、考试分数）近似服从正态分布。为求概率，需转换为标准正态分布 Z ~ N(0, 1)，公式 Z = (X − μ) / σ。利用标准正态表或技术可查出累积概率 Φ(z)。
The area under the normal curve represents probability. For a given z-value, the table provides P(Z < z). Probabilities for intervals are found by subtracting appropriate cumulative areas.
正态曲线下的面积代表概率。对于给定的 z 值，表格提供 P(Z < z)。区间概率可通过相减相应的累积面积来求得。
10. Standardised Scores (z-scores) | 标准化分数（Z值）
A standardised score (z-score) indicates how many standard deviations a value lies above or below the mean: z = (x − μ) / σ. It allows comparisons across different datasets with different means and standard deviations.
标准化分数（z 值）表示某个数值高于或低于均值多少个标准差：z = (x − μ) / σ。它可以在具有不同均值和标准差的数据集之间进行比较。
A positive z-score means the value is above the mean; negative means below. Z-scores near 0 are typical, while values beyond ±2 or ±3 are unusual. When the population parameters are unknown, sample estimates are used: z = (x − x̄) / s.
正 z 值表示该值高于均值，负值表示低于均值。接近 0 的 z 值较为常见，而超出 ±2 或 ±3 则比较罕见。当总体参数未知时，使用样本估计值：z = (x − x̄) / s。
Standardised scores are useful in educational testing, quality control, and finance to assess relative performance or detect outliers.
标准化分数在教育测量、质量控制和金融等领域非常有用，可用于评估相对表现或识别异常值。
11. Scatter Graphs, Correlation and Regression | 散点图、相关性及回归
A scatter graph shows the relationship between two variables. Correlation describes the strength and direction of a linear relationship. Positive correlation means both variables increase together; negative correlation means one increases as the other decreases.
散点图展示两个变量之间的关系。相关性描述线性关系的强度和方向。正相关表示两个变量一同增加；负相关表示一个增加而另一个减少。
Correlation is not causation. The product-moment correlation coefficient (r) measures linear correlation, ranging from −1 (perfect negative) to +1 (perfect positive). A value near 0 suggests no linear correlation.
相关性不等于因果关系。积矩相关系数 (r) 衡量线性相关程度，范围从 −1（完全负相关）到 +1（完全正相关）。接近 0 则表明没有线性相关。
A line of best fit (regression line) can be drawn by eye or calculated as y = a + bx, where b = S_xy / S_xx. This line minimises the sum of squared residuals and can be used to make predictions within the range of the data (interpolation). Extrapolation beyond the data range is unreliable.
最佳拟合线（回归直线）可通过目测绘制，也可用最小二乘法求出：y = a + bx，其中 b = S_xy / S_xx。该直线使残差平方和最小，可用于在数据范围内进行预测（内插）。超出数据范围的外推是不可靠的。
12. Time Series and Moving Averages | 时间序列与移动平均
A time series is a set of data recorded over time, often at regular intervals. It can show trends (long-term movement), seasonal variations (regular fluctuations), and irregular fluctuations.
时间序列是按时间（通常等间隔）记录的一组数据，可以显示出趋势（长期动向）、季节性变动（有规律的波动）以及不规则波动。
Moving averages help smooth out short-term fluctuations to reveal the underlying trend. For quarterly data, a 4-point moving average is typical: average the first four values, then drop the first and add the fifth, and so on. The centred moving average aligns the trend with original time periods.
移动平均有助于平滑短期波动，揭示潜在的趋势。对于季度数据，通常计算 4 点移动平均：求前四个值的平均值，然后去掉第一个加入第五个，依此类推。中心化移动平均将趋势对齐到原始时间点上。
Seasonal variation can be estimated by subtracting the trend from the original values. The average seasonal effect for each season is used to adjust forecasts, improving the accuracy of predictions for future periods.
季节性变差可通过原始值减去趋势值来估计。每个季节的平均季节性影响可用于调整预测值，从而提高对未来时期预测的准确性。
Published by TutorHao | Statistics Revision Series | aleveler.com
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