📚 GCSE Edexcel Maths: Statistics – Key Concepts and Revision Guide | GCSE Edexcel 数学：统计考点精讲
Statistics in GCSE Edexcel Mathematics focuses on collecting, representing, analysing and interpreting data, as well as understanding probability. This guide covers all essential topics, from types of data and sampling methods to cumulative frequency histograms and conditional probability. Each section pairs clear explanations in English with their Chinese equivalents, mirroring the structure commonly found in bilingual revision materials. Use this to strengthen both your mathematical thinking and your bilingual grasp of statistical terminology.
GCSE Edexcel 数学中的统计部分关注数据的收集、呈现、分析和解释，以及概率的理解。本指南涵盖所有核心考点，从数据类型与抽样方法，到累积频率直方图和条件概率。每个小节都提供英语解释与对应的中文翻译，帮助你巩固数学思维，同时掌握统计术语的双语表达。
1. Types of Data | 数据类型
Data can be classified as qualitative (categorical) or quantitative (numerical). Quantitative data is further split into discrete and continuous. Discrete data can only take specific values, usually whole numbers, while continuous data can take any value within a range.
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和连续型。离散数据只能取特定的值，通常是整数；连续数据则可以在某个区间内取任意值。
Knowing the data type is crucial when choosing an appropriate chart or statistical measure. Bar charts suit categorical data, while histograms are used for grouped continuous data.
了解数据类型对于选择合适的图表或统计量至关重要。条形图适合分类数据，而直方图则用于分组的连续数据。
A primary source of data is collected first-hand for a specific purpose; secondary data already exists. Both can be biased if sampling is not representative.
一手数据是为特定目的而亲自收集的；二手数据是已有的数据。如果抽样不具代表性，两者都可能产生偏差。
2. Sampling Methods | 抽样方法
A population is the whole group being studied. A sample is a subset used to draw conclusions about the population. Random sampling gives every member an equal chance of being selected, reducing bias. Simple random sampling can be done using random number generators or a lottery method.
总体是被研究的整个群体。样本是用来推断总体结论的一个子集。随机抽样让每个成员有相等的被选中机会，从而减少偏差。简单随机抽样可使用随机数生成器或抽签法进行。
Stratified sampling divides the population into groups (strata) and takes a proportional random sample from each stratum. This guarantees representation of key subgroups. The formula for the number sampled from a stratum is: (stratum size / population size) × sample size.
分层抽样将总体划分为若干层，并从每层按比例随机抽样，从而保证关键子组的代表性。从某层抽取的样本量公式为：（层的大小 / 总体大小）× 样本量。
Systematic sampling selects every k-th member after a random start. It is easier than simple random sampling but can introduce bias if there is a hidden pattern. Quota sampling is non-random; interviewers fill quotas for certain characteristics, which may lead to selection bias.
系统抽样在随机起点后每隔 k 个个体选取一个。它比简单随机抽样简单，但如果存在隐藏模式，会引入偏差。配额抽样是非随机抽样，访员按特定特征配额选取，可能导致选择偏差。
3. Averages: Mean, Median & Mode | 平均值：平均数、中位数与众数
The mean is calculated by summing all values and dividing by the number of values. For a list x₁, x₂, …, xₙ, mean x̄ = (Σxᵢ)/n. For grouped data, we use midpoints of class intervals and estimate the mean.
平均数通过将所有数值相加再除以数值个数来计算。对于列表 x₁, x₂, …, xₙ，平均数 x̄ = (Σxᵢ)/n。对于分组数据，我们使用组中值来估计平均数。
The median is the middle value when data is ordered. If n is odd, it is the (n+1)/2 th value; if n is even, it is the average of the n/2 th and (n/2 + 1)th values. The median is unaffected by extreme outliers, unlike the mean.
中位数是将数据排序后的中间值。如果 n 为奇数，中位数是第 (n+1)/2 个值；如果 n 为偶数，则是第 n/2 个与第 (n/2 + 1) 个值的平均数。中位数不受极端异常值的影响，这与平均数不同。
The mode is the value that occurs most often. A data set may have one mode (unimodal), two (bimodal) or more. For grouped data, the modal class is the interval with the highest frequency.
众数是出现次数最多的值。数据集可以有一个众数（单峰）、两个（双峰）或更多。对于分组数据，众数组是频率最高的区间。
4. Measures of Spread: Range & Interquartile Range | 离散度量：全距与四分位距
The range = maximum value – minimum value. It is simple but highly influenced by outliers. A more robust measure is the interquartile range (IQR) = upper quartile (Q3) – lower quartile (Q1).
全距 = 最大值 – 最小值。它计算简单，但极易受异常值影响。更稳健的度量是四分位距（IQR）= 上四分位数（Q3）– 下四分位数（Q1）。
The lower quartile Q1 is the median of the lower half of the data; the upper quartile Q3 is the median of the upper half. When calculating quartiles from a list, use the (n+1)/4 th and 3(n+1)/4 th positions. The IQR covers the middle 50% of the data.
下四分位数 Q1 是数据下半部分的中位数；上四分位数 Q3 是数据上半部分的中位数。从列表计算四分位数时，使用第 (n+1)/4 和 3(n+1)/4 个位置。IQR 涵盖了中间 50% 的数据。
Comparing data sets: the median shows typical values, while the IQR shows consistency. A smaller IQR indicates less variation. Always pair a measure of centre with a measure of spread when describing distributions.
比较数据集时：中位数显示典型值，而 IQR 显示数据的一致性。IQR 越小，变异越小。描述分布时，一定要将中心度量与离散度量配对使用。
5. Box Plots | 箱线图
A box plot (box-and-whisker plot) displays the five-number summary: minimum, Q1, median, Q3, maximum. It is drawn on a scale, with a box from Q1 to Q3 and a line at the median. The whiskers extend to the minimum and maximum, unless outliers are identified separately.
箱线图（盒须图）展示五数概括：最小值、Q1、中位数、Q3、最大值。它在刻度上绘制，盒子从 Q1 到 Q3，中位数处画一条线。须线延伸至最小值和最大值，除非异常值被单独标出。
Outliers can be defined as values that lie more than 1.5 × IQR below Q1 or above Q3. These are often marked with dots. Box plots are excellent for comparing distributions side by side, highlighting skewness and spread.
异常值可以定义为低于 Q1 – 1.5×IQR 或高于 Q3 + 1.5×IQR 的值。这些通常用点标记。箱线图非常适合并排比较分布情况，突出偏态和离散程度。
If the median is closer to Q1, the data is positively skewed (higher values spread out); if closer to Q3, it is negatively skewed. A symmetric box plot suggests a roughly symmetric distribution.
如果中位数更靠近 Q1，则数据呈正偏态（高值分散）；如果更靠近 Q3，则为负偏态。箱线图对称表明分布大致对称。
6. Cumulative Frequency | 累积频率
Cumulative frequency is the running total of frequencies. To construct a cumulative frequency table, add each frequency to the sum of previous frequencies. The final total equals the total number of data points.
累积频率是频率的累加。构建累积频率表时，将每个频率加到之前所有频率的总和上。最终总和等于数据点的总数。
A cumulative frequency graph (ogive) is plotted with the upper class boundary on the x-axis and cumulative frequency on the y-axis. Points are joined with a smooth curve. Use the graph to estimate the median (50th percentile), quartiles (25th and 75th percentiles) and any other percentiles.
累积频率图（折线图）以组上限为 x 轴，累积频率为 y 轴。点用平滑曲线连接。利用图表可估计中位数（第 50 百分位数）、四分位数（第 25 和 75 百分位数）及其他百分位数。
The interquartile range can be read directly from the graph: IQR = Q3 – Q1. The cumulative frequency graph also allows you to find the number of items above or below a given value. Remember to plot against upper boundaries for continuous data.
四分位距可直接从图中读出：IQR = Q3 – Q1。累积频率图还能让你找出高于或低于某个给定值的项目数量。记住，对于连续数据，要用组上限进行描点。
7. Histograms | 直方图
Histograms represent grouped continuous data. Unlike bar charts, the area of each bar represents frequency. This is essential when class widths are unequal. The vertical axis is frequency density = frequency / class width.
直方图表示分组的连续数据。与条形图不同，每个直条的面积代表频率。当组距不相等时，这一点至关重要。纵轴是频率密度 = 频率 / 组距。
To draw a histogram, calculate the frequency density for each class and draw bars with the class width as the base. The height of the bar is the frequency density. To find a frequency from the graph, multiply the frequency density by the class width.
绘制直方图时，为每个组计算频率密度，并以组距为底边画矩形条。矩形条的高度即频率密度。要从图中求频率，用频率密度乘以组距。
Interpreting histograms: the modal class is the one with the highest frequency density, not necessarily the tallest bar if widths differ. You can estimate the total frequency by summing the areas of all bars.
解读直方图：众数组是频率密度最高的组，如果组距不同，不一定是最高的矩形条。你可以通过将所有矩形条的面积相加来估计总频率。
8. Scatter Graphs & Correlation | 散点图与相关性
A scatter graph plots bivariate data to show the relationship between two variables. Correlation describes the strength and direction of a 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.
散点图绘制双变量数据，以显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。正相关表示一个变量增大，另一个也倾向于增大。负相关表示一个变量增大，另一个倾向于减小。
Correlation is not causation. The correlation coefficient r ranges from -1 to 1. At GCSE, you describe correlation as strong, moderate or weak positive/negative, or no correlation. You may need to draw a line of best fit by eye, balancing points above and below.
相关性不代表因果关系。相关系数 r 的范围从 -1 到 1。在 GCSE 阶段，你只需将相关性描述为强、中或弱正/负相关，或无相关。你可能需要凭观察画一条最佳拟合线，让线上方和下方的点大致平衡。
The line of best fit can be used to estimate unknown values. Interpolation is estimating within the range of existing data (reliable). Extrapolation is estimating outside the range (unreliable as the trend may not continue).
最佳拟合线可用于估计未知值。内插法是在已有数据范围内估计（可靠）。外推法是在范围之外估计（不可靠，因为趋势可能不会延续）。
9. Basic Probability | 基础概率
Probability is a measure of how likely an event is to occur, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal or percentage. For equally likely outcomes, P(event) = number of favourable outcomes / total number of outcomes.
概率是对事件发生可能性的一种度量，范围从 0（不可能）到 1（必然）。可用分数、小数或百分比表示。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。
The sum of probabilities of all possible outcomes is 1. The complement rule: P(not A) = 1 – P(A). In a Venn diagram, the sample space is represented by a rectangle, and events by circles. The intersection (A ∩ B) is where both occur; the union (A ∪ B) is where either or both occur.
所有可能结果的概率之和为 1。补集规则：P(非 A) = 1 – P(A)。在文氏图中，样本空间用矩形表示，事件用圆表示。交集 (A ∩ B) 表示两者同时发生；并集 (A ∪ B) 表示至少一个发生。
Mutually exclusive events cannot happen at the same time, so P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B). Independent events have no influence on each other: P(A and B) = P(A) × P(B).
互斥事件不能同时发生，因此 P(A ∩ B) = 0 且 P(A ∪ B) = P(A) + P(B)。独立事件互不影响：P(A 与 B) = P(A) × P(B)。
10. Tree Diagrams & Conditional Probability | 树状图与条件概率
Tree diagrams help visualise compound events, especially when there are multiple stages. Each branch shows the probability of an outcome. Probabilities on branches from the same point must sum to 1. To find the probability of a sequence, multiply along the branches.
树状图有助于可视化复合事件，特别是多阶段事件。每条分支显示一个结果的概率。同一点发出的分支上的概率之和必须为 1。要求一个事件序列的概率，沿分支相乘即可。
Conditional probability is the probability of event A given that event B has occurred, written as P(A|B) = P(A ∩ B) / P(B). On a tree diagram, the probabilities on the second set of branches are conditional on the first outcome.
条件概率是指在事件 B 已发生的条件下事件 A 发生的概率，写作 P(A|B) = P(A ∩ B) / P(B)。在树状图中，第二层分支上的概率以第一层结果为条件。
For independent events, P(A|B) = P(A) because B does not affect A. When events are not independent, you must use the given conditional probabilities. Venn diagrams can also show conditional probability by restricting the sample space to the part where the given event has occurred.
对于独立事件，P(A|B) = P(A)，因为 B 不影响 A。当事件不独立时，必须使用给出的条件概率。文氏图也可以通过将样本空间限定在给定事件已发生的部分来展示条件概率。
Exam questions often ask for the probability of ‘at least one’ success. The easiest method is 1 – P(none). A tree diagram clearly shows all paths, helping you identify the relevant combinations and sum their probabilities.
考试中常要求计算“至少一次”成功的概率。最简单的方法是 1 – P(全不)。树状图能清晰地显示所有路径，帮助你识别相关组合并将其概率相加。
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