📚 GCSE Maths: Statistics Key Points | GCSE 数学：统计 考点精讲
Statistics at GCSE level builds the essential skills for handling, presenting, and interpreting data. This revision guide walks you through the key topics, from collecting data and calculating averages to drawing histograms, cumulative frequency graphs, and understanding probability basics. Clear worked examples and exam-style tips are included to help you master every concept.
GCSE 阶段的统计学习为你打下处理、展示和解释数据所需的核心技能。这份复习指南将带你梳理关键考点，从数据收集、计算平均数，到绘制直方图、累积频率图以及理解概率基础。每个概念都配有清晰的例题和考试风格的小贴士，帮助你牢固掌握。
1. Types of Data and Sampling | 数据类型与抽样
Data can be qualitative (non-numerical, e.g. colours, favourite subjects) or quantitative (numerical). Quantitative data is further split into discrete data (counted items, e.g. number of students) and continuous data (measured values, e.g. height, time).
数据可以是定性数据（非数值，如颜色、喜爱的科目）或定量数据（数值）。定量数据又分为离散数据（计数得到，如学生人数）和连续数据（测量得到，如身高、时间）。
The quality of a sample depends on how it is selected. Random sampling gives every member of the population an equal chance of being chosen. Stratified sampling divides the population into distinct groups (strata) and takes a random sample from each in proportion to its size. Systematic sampling selects every nth member from a list. A convenience sample selects the easiest members to reach, but often leads to bias.
样本的质量取决于选取方式。随机抽样使总体中每个个体被抽到的机会均等。分层抽样先将总体分成不同的层，然后按各层大小比例从每层随机抽样。系统抽样从名单中每隔一定间隔选取一个个体。方便抽样选取最容易接触到的个体，但往往会导致偏差。
When designing a survey, avoid leading questions and make sure the response categories cover all possibilities without overlap. A pilot survey helps test the questionnaire and improve it before the main data collection.
设计调查问卷时，应避免引导性问题，并确保选项覆盖所有可能且互不重叠。试点调查有助于在正式收集数据前测试和改善问卷。
2. Mean, Median, Mode | 均值、中位数与众数
The mean is the sum of all values divided by the number of values. For a list x₁, x₂, …, xₙ:
Mean = (∑ xᵢ) ÷ n
均值等于所有数值之和除以数值的个数。对于一组数据 x₁, x₂, …, xₙ：
均值 = (∑ xᵢ) ÷ n
The median is the middle value when the data are arranged in order. If there is an even number of values, the median is the mean of the two middle numbers. The mode is the value that appears most often. A set may have one mode, more than one mode, or no mode at all.
中位数是将数据按大小顺序排列后位于中间的值。如果数据个数为偶数，中位数是中间两个数的均值。众数是出现次数最多的值。一组数据可能有一个众数、多个众数，也可能没有众数。
To choose the best average: the mean uses all data but is affected by extreme values (outliers); the median is not affected by outliers and is better for skewed distributions; the mode is useful for categorical data.
选择最佳平均值：均值利用了所有数据，但易受极端值（异常值）影响；中位数不受异常值影响，对偏态分布更合适；众数适用于分类数据。
For a frequency table, multiply each value by its frequency, sum these products, and divide by the total frequency to find the mean.
Mean from frequency = ∑(f × x) ÷ ∑f
对于频率表，将每个值乘以对应的频数、求和后除以总频数即可得到均值。
频率表均值 = ∑(f × x) ÷ ∑f
3. Range and Interquartile Range | 极差与四分位距
The range is the difference between the largest and smallest values. It is a simple measure of spread but is heavily influenced by outliers.
极差（全距）是最大值与最小值的差。它是最简单的离散程度度量，但极易受异常值影响。
The quartiles divide an ordered data set into four equal parts: 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. The interquartile range (IQR) is Q₃ – Q₁. It measures the spread of the middle 50% and is resistant to outliers.
四分位数把有序数据分成四等份：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。四分位距 = Q₃ – Q₁。它度量中间 50% 数据的分散程度，且不受异常值影响。
To find quartiles from a list of n values: order the data, find the median (Q₂). If n is odd, exclude the median when splitting the data into lower and upper halves. For grouped data, use interpolation or cumulative frequency graphs.
从数据列表中求四分位数：将数据排序，找出中位数 Q₂。若 n 为奇数，在把数据分成上下两部分时排除中位数本身。对于分组数据，可使用插值法或借助累积频率图。
4. Box Plots | 箱线图
A box plot (box-and-whisker plot) displays the minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃), and maximum. The box spans from Q₁ to Q₃, with a line inside marking the median. Whiskers extend to the minimum and maximum (or to 1.5×IQR beyond the quartiles if showing outliers).
箱线图（箱形图）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。矩形盒从 Q₁ 到 Q₃，内部一条线标记中位数。须线延伸到最小值和最大值（或到 1.5×IQR 处，若标明异常值）。
Box plots are excellent for comparing distributions side by side. They allow you to visualise the central tendency, spread, and skewness without showing the raw data.
箱线图非常适合并排比较分布的特征。无需展示原始数据，就能直观看出集中趋势、离散程度和偏态。
Skewness: If the median is closer to Q₁, the data are positively skewed; closer to Q₃, negatively skewed. If the whiskers are unequal, the longer whisker points in the direction of the skew.
偏态：若中位数更靠近 Q₁，数据呈正偏态；更靠近 Q₃，则呈负偏态。若两侧须线长度不等，较长的一侧指向偏态方向。
When drawing a box plot, always use a labelled scale. The width of the box is arbitrary—what matters is the position of the five key points.
绘制箱线图时务必使用带刻度的数轴。盒的宽度无关紧要，关键在于五个关键点的位置。
5. Frequency Tables and Grouped Data | 频率表与分组数据
When raw data are numerous, we organise them into frequency tables. For discrete data, list each distinct value and its frequency. For continuous data, we group values into class intervals like 0 ≤ h < 10, 10 ≤ h < 20, etc.
当原始数据量较大时，我们会用频率表加以整理。对于离散数据，列出每个不同值及其频数。对于连续数据，我们将其分组为组距，如 0 ≤ h < 10, 10 ≤ h < 20 等。
For grouped data, we can estimate the mean by using the midpoint of each class interval. Multiply each midpoint by the class frequency, sum, and divide by total frequency. This is an estimate because we assume all values lie at the midpoint.
对于分组数据，我们可以利用组中值来估计均值。用每组的组中值乘以该组频数，求和后除以总频数。这是估计值，因为我们假设所有数据都落在组中值上。
The modal class is the class interval with the highest frequency. The median class is the interval that contains the median position (n/2). For exact quartiles, cumulative frequency graphs are used.
众数组是频数最高的组距。中位数组是包含第 n/2 个数据位置的组距。要求精确的四分位数，需借助累积频率图。
Always check that class intervals are continuous and have no gaps. If given as 0–9, 10–19, rewrite boundaries as 0 ≤ x < 10, 10 ≤ x < 20.
务必检查组距是否连续且无空隙。若给出的分组为 0–9, 10–19，应把边界重写为 0 ≤ x < 10, 10 ≤ x < 20。
6. Histograms and Frequency Density | 直方图与频率密度
A histogram is used to display grouped continuous data. Unlike a bar chart, the area of each bar represents frequency. When class widths are unequal, we must plot frequency density on the vertical axis:
Frequency density = Frequency ÷ Class width
直方图用于展示分组连续数据。与条形图不同，直方图中每个条形的面积代表频数。若组宽不等，纵轴必须表示频率密度：
频率密度 = 频数 ÷ 组宽
To draw a histogram: calculate the frequency density for each class, then draw bars whose heights equal the frequency density and widths equal the class width. The total area of all bars equals the total frequency.
绘制直方图：计算每组的频率密度，然后以频率密度为高、组宽为宽画出条形。所有条形的总面积等于总频数。
To find frequency from a histogram, multiply the frequency density by the class width. This is a common exam skill—watch for bars that are partially shaded or questions that ask for the number of items in a given interval.
从直方图中求频数，用频率密度乘以组宽。这是考试中的常见考点——注意可能出现的部分阴影条形，或要求计算某一区间内数据个数的题目。
Comparisons: bar charts are for categorical or discrete data with equal gaps between bars; histograms have no gaps and are for continuous data with varying widths.
对比：条形图用于分类或离散数据，条形间有空隙；直方图用于连续数据，条形间无空隙且宽度可变。
7. Cumulative Frequency | 累积频率
A cumulative frequency table adds up the frequencies as you move through the classes. The cumulative frequency for a class is the total frequency up to the upper boundary of that class.
累积频率表在遍历各组时将频数累加。某一组的累积频率是从第一组到该组上限为止的总频数。
Plot a cumulative frequency graph with the upper class boundary on the horizontal axis and cumulative frequency on the vertical axis. The points are joined with a smooth curve (not straight lines).
绘制累积频率图时，横轴为组上限，纵轴为累积频率。将各点用平滑曲线连接（不用折线）。
Use the graph to estimate the median (find the value at half the total cumulative frequency), the lower quartile (at 1/4 of total), the upper quartile (at 3/4 of total), and the interquartile range (Q₃ – Q₁). You can also read off any percentile.
利用累积频率图可以估计中位数（在总累积频率的 1/2 处读取）、下四分位数（1/4 处）、上四分位数（3/4 处）以及四分位距（Q₃ – Q₁）。也可以读出任意百分位数。
A steep section of the curve indicates a high frequency density—many data points in a narrow interval. A flat section indicates low frequency density.
曲线陡峭的部分表明频率密度高——在狭窄区间内数据密集。平缓部分表明频率密度低。
Always label axes, plot points neatly, and show how you read off values with dotted lines. Accuracy is essential.
务必标注坐标轴，清晰地描点，并用虚线示意读数过程。准确性至关重要。
8. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph plots paired numerical data to explore the relationship between two variables. The independent variable goes on the x-axis, the dependent on the y-axis.
散点图用于描画成对数值数据，以探索两个变量之间的关系。自变量放在 x 轴，因变量放在 y 轴。
Correlation describes the direction and strength of the relationship. Positive correlation means as one variable increases, the other tends to increase. Negative correlation means as one increases, the other tends to decrease. No correlation means no apparent pattern.
相关性描述关系的方向和强度。正相关指一个变量增大时另一个也趋于增大。负相关指一个增大时另一个趋于减小。零相关指无明显规律。
Draw a line of best fit (trend line) by eye. It should pass through the middle of the points with roughly equal numbers of points above and below the line. Use it to estimate one variable given the other (interpolation within the data range; extrapolation beyond—but extrapolation is less reliable).
目测画出最佳拟合线（趋势线）。它应穿过点的中心，使线上方和下方的点数大致相等。利用该线可根据一个变量估计另一个变量（内插在数据范围内；外推超出范围——但外推可靠性较低）。
Do not force the line through the origin unless the context suggests a direct proportion. Always write ‘line of best fit’ or ‘trend line’ and avoid joining points dot-to-dot.
除非情景暗示为正比例关系，不要强迫直线穿过原点。需注明“最佳拟合线”或“趋势线”，且不要点对点连接。
9. Time Series and Moving Averages | 时间序列与移动平均值
A time series graph plots data collected at regular time intervals. The time (e.g. months, years) goes on the horizontal axis, and the variable on the vertical axis. Plot points and connect them with straight lines.
时间序列图描画按固定时间间隔收集的数据。时间（如月份、年份）在横轴，变量在纵轴。描点后用直线连接。
Time series often show a trend (general long-term movement) and seasonal variations (regular patterns within a year). A moving average helps smooth out short-term fluctuations and highlight the trend.
时间序列常呈现出趋势（长期总体变化）和季节变动（一年内的规律模式）。移动平均值有助于平滑短期波动、突出趋势。
To calculate a moving average, find the mean of a fixed number of consecutive data points (e.g. a 4-point moving average). Plot the moving averages at the centre of the time periods they cover. Use the smoothed line to predict future values, but always state the assumption that the trend continues.
计算移动平均值时，求固定个数连续数据点的均值（如 4 点移动平均值）。将移动平均值标在它们所覆盖时间段的中心位置。用平滑后的曲线预测未来值，但需说明前提是趋势延续。
10. Probability Basics | 概率基础
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes:
P(event) = number of favourable outcomes ÷ total number of outcomes
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) = 有利结果数 ÷ 所有可能结果总数
The sum of probabilities for all possible outcomes of an experiment is 1. The probability of an event not happening is 1 – P(event).
一次试验所有可能结果的概率之和为 1。事件不发生的概率 = 1 – P(事件)。
Use sample space diagrams, two-way tables, and tree diagrams to list outcomes systematically. In a tree diagram, multiply probabilities along branches for combined events (AND), and add probabilities for different branches leading to the same outcome (OR). For independent events: P(A and B) = P(A) × P(B).
使用样本空间图表、双向表和树状图系统列举结果。在树状图中，沿分支相乘得到组合事件的概率（AND），将不同分支得到同一结果的概率相加（OR）。对于独立事件：P(A 且 B) = P(A) × P(B)。
Watch for conditional probability questions where the probability of the second event depends on the first. Check that probabilities on branches from the same point sum to 1, and always write probabilities as fractions, decimals or percentages in their simplest form.
注意条件概率题，其中第二个事件的概率依赖于第一个事件。检查同一点出发的各分支概率之和是否为 1，并始终将概率化为最简分数、小数或百分数。
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