📚 IGCSE Maths: Statistics Key Points Revision | IGCSE 数学：统计 考点精讲
Statistics is a core component of the IGCSE Mathematics syllabus, requiring you to collect, display, analyse and interpret data. This article covers all essential statistical concepts, from data types to probability, with clear English explanations and matching Chinese translations to help you master every topic efficiently.
统计是 IGCSE 数学的核心组成部分，要求你会收集、展示、分析和解读数据。本文涵盖所有重要的统计概念，从数据类型到概率，每一点都用清晰的英文解释配合对应的中文翻译，帮助你高效掌握每一个主题。
1. Types of Data and Data Collection | 数据类型与数据收集
Data can be qualitative (descriptive) such as colours or names, or quantitative (numerical) such as heights or test scores. Quantitative data can be further divided into discrete data, which can only take certain values (e.g. number of students), and continuous data, which can take any value within a range (e.g. weight). Understanding these types helps you choose appropriate charts and calculations.
数据可以是定性的（描述性），比如颜色或名字；也可以是定量的（数值型），比如身高或考试成绩。定量数据又分为离散数据，只能取特定值（如学生人数），以及连续数据，可以在一个范围内取任意值（如体重）。理解这些类型有助于你选择合适的图表和计算方法。
Data can be collected through surveys, experiments or observations. A population is the whole group you want to study, while a sample is a smaller group selected from the population. A random sample ensures every member has an equal chance of being chosen, reducing bias. Avoid convenience sampling where only easy-to-reach individuals are asked, as it may not represent the whole population fairly.
数据可以通过调查、实验或观察来收集。总体是你想研究的整个群体，而样本是从总体中选出的一个较小的群体。随机抽样确保每个成员都有相同的机会被选中，从而减少偏差。应避免便利抽样，即只询问容易接触到的个体，因为这可能不能公平地代表整个总体。
2. Bar Charts and Pictograms | 柱状图与象形图
A bar chart uses rectangular bars to represent the frequency of different categories. The bars have equal width and there are gaps between them to show that the data are separate categories. The height of each bar corresponds to the frequency. Always label the axes and give the chart a title. Bar charts can be drawn vertically or horizontally.
柱状图用矩形条表示不同类别的频数。条形的宽度相等，并且条形之间有间隙，以表明数据是独立的类别。每个条形的高度代表频数。务必标记坐标轴并为图表加上标题。柱状图可以纵向或横向绘制。
A pictogram uses pictures or symbols to represent data. Each symbol stands for a certain number of items. A key must be provided, e.g. one house symbol equals 10 houses. If a frequency is not an exact multiple, part of a symbol is drawn. Pictograms are visually appealing but must be drawn accurately, with symbols kept the same size and properly aligned.
象形图用图画或符号表示数据。每个符号代表一定数量的项目。必须提供图例，例如一个房子符号代表 10 栋房子。如果频数不是精确的倍数，则画出符号的一部分。象形图视觉上吸引人，但必须精确绘制，符号大小保持一致并对齐。
3. Pie Charts | 饼图
A pie chart displays data as sectors of a circle, where the angle of each sector is proportional to the frequency it represents. The total frequency corresponds to 360°. To calculate the angle for a category, use the formula: angle = (frequency ÷ total frequency) × 360°. Then use a protractor to draw the sectors.
饼图用圆的扇形显示数据，每个扇形的角度与其所代表的频数成正比。总频数对应 360°。要计算某个类别的角度，使用公式：角度 = (频数 ÷ 总频数) × 360°。然后用量角器画出扇形。
Always label each sector with the category name or provide a legend. To find the frequency from a pie chart, reverse the formula: frequency = (angle ÷ 360°) × total frequency. Pie charts are best for showing proportions of a whole, but can be hard to compare if there are many small sectors.
务必在每个扇形上标明类别名称或提供图例。要从饼图中求出频数，反向使用公式：频数 = (角度 ÷ 360°) × 总频数。饼图最适合显示整体的比例，但如果有很多细小的扇形，则难以比较。
4. Histograms and Frequency Density | 直方图与频数密度
For grouped continuous data, a histogram is used. Unlike a bar chart, the bars touch each other to show continuous intervals. The area of each bar represents the frequency. If the class intervals are unequal, you must use frequency density on the vertical axis. Frequency density = frequency ÷ class width. Then the area of each bar = frequency density × class width = frequency.
对于分组的连续数据，使用直方图。与柱状图不同，条形彼此接触以表示连续区间。每个条形的面积代表频数。如果组距不相等，则必须在纵轴上使用频数密度。频数密度 = 频数 ÷ 组距。这样每个条形的面积 = 频数密度 × 组距 = 频数。
To draw a histogram, first find the class width for each group. For equal class widths, you can plot frequency directly on the y-axis. But if class widths differ, always calculate frequency density. On the graph, the bars are drawn with their bases covering the class interval and heights equal to the frequency density.
要画直方图，首先求出每组的组距。如果组距相等，可以直接在 y 轴上标出频数。但是，如果组距不同，就必须计算频数密度。在图上，条形的底边覆盖组距区间，高度等于频数密度。
To interpret a histogram, remember: frequency = frequency density × class width. Look at the area, not just the height. This is a common IGCSE exam trap where a bar might be taller yet represent a smaller frequency because its class width is narrower.
解读直方图时记住：频数 = 频数密度 × 组距。要看面积，而不仅仅是高度。这是 IGCSE 考试中常见的陷阱：一个条形可能更高，但因为组距较窄，代表的频数反而更小。
5. Frequency Polygons | 频数多边形
A frequency polygon is a line graph that can be drawn from a histogram or on its own. To create one, plot a point at the midpoint of each class interval against its frequency (or frequency density if class widths differ). Connect the points with straight lines. Extend the lines to the horizontal axis at the midpoints of the classes before the first and after the last to form a closed polygon.
频数多边形是一种折线图，可以从直方图画出，也可以单独绘制。绘制时，在每个组距的中点处标出对应频数（如果组距不同则标频数密度）的点。用直线将这些点连接起来。把折线延伸到第一个组之前和最后一个组之后的中点位置与横轴相交，形成一个封闭的多边形。
Frequency polygons are useful for comparing two or more distributions on the same axes. For grouped data, midpoints are calculated as (lower bound + upper bound) ÷ 2. Always use a clear key when overlaying multiple polygons.
频数多边形便于在同一坐标轴上比较两个或多个分布。对于分组数据，中点计算公式为 (下界 + 上界) ÷ 2。当叠加多个多边形时，一定要用清晰的图例。
6. Mean, Median and Mode | 平均数、中位数与众数
The mean is the average, calculated by adding all values and dividing by the number of values. For a list: mean = ∑x ÷ n. For a frequency table: mean = ∑(fx) ÷ ∑f, where x is the data value and f is its frequency. For grouped data, use the midpoint of each class as an estimate of x. The mean can be affected by extreme values (outliers).
平均数就是平均值，计算方法为所有数值相加后除以数值的个数。对于列表：平均数 = ∑x ÷ n。对于频数表：平均数 = ∑(fx) ÷ ∑f，其中 x 是数据值，f 是频数。对于分组数据，使用每个组的中点作为 x 的估计值。平均数可能受极端值（异常值）的影响。
Mean = (Sum of all values) ÷ (Number of values)
平均数 = (所有数值之和) ÷ (数值的个数)
The median is the middle value when the data is arranged in order. For an odd number of data points, it is the exact middle. For an even number, it is the mean of the two middle values. The median is not affected by outliers, making it a better measure of central tendency for skewed distributions.
中位数是将数据按顺序排列后位于中间的值。如果数据个数是奇数，就是正中间的那个；如果是偶数，则是中间两个数的平均值。中位数不受异常值影响，因此对于偏态分布来说，是更好的集中趋势度量。
The mode is the value that appears most often. A data set can have one mode (unimodal), two modes (bimodal) or more. For grouped data, the modal class is the class interval with the highest frequency. While easy to find, the mode may not be near the centre of the data.
众数是出现次数最多的值。一组数据可以有一个众数（单峰），两个众数（双峰）或更多。对于分组数据，众数组是频数最高的组距。众数容易找到，但可能不在数据的中心附近。
7. Range and Quartiles | 范围与四分位数
The range is the difference between the largest and smallest values. It shows the spread of the data but can be greatly affected by a single outlier. A more robust measure of spread is the interquartile range (IQR).
范围是最大值与最小值之差。它显示了数据的分布宽度，但极易受单个异常值影响。更稳健的散布度量是四分位距 (IQR)。
Quartiles divide ordered data into four equal parts. The lower quartile (Q₁) is the median of the lower half of the data, not including the overall median if the data size is odd. The upper quartile (Q₃) is the median of the upper half. The median is Q₂. The interquartile range = Q₃ − Q₁. It measures the spread of the middle 50% of the data.
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，如果数据个数为奇数则不包括整个数据的中位数。上四分位数 (Q₃) 是上半部分的中位数。中位数即为 Q₂。四分位距 = Q₃ − Q₁。它衡量中间 50% 数据的分布宽度。
To find quartiles from a list: order the data, find the median, then find the median of the lower and upper halves. For a large frequency table, use cumulative frequencies to locate the positions: Q₁ at (n+1)/4, Q₂ at (n+1)/2, Q₃ at 3(n+1)/4. (Some specifications may use n/4, etc., so follow your syllabus.)
从列表中找四分位数：将数据排序，找出中位数，然后找出下半部分和上半部分的中位数。对于大型频数表，使用累积频数来定位：Q₁ 在 (n+1)/4 的位置，Q₂ 在 (n+1)/2，Q₃ 在 3(n+1)/4。（某些考试大纲可能用 n/4 等，所以请按照你的教学大纲执行。）
8. Box Plots (Box-and-Whisker Diagrams) | 箱线图
A box plot displays the five-number summary: minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃) and maximum. Draw a scale first. Draw a box from Q₁ to Q₃, with a vertical line inside the box at the median. Extend whiskers from the box to the minimum and maximum values, unless there are outliers.
箱线图显示五数概括：最小值、下四分位数 (Q₁)、中位数 (Q₂)、上四分位数 (Q₃) 和最大值。先画好刻度。从 Q₁ 到 Q₃ 画一个长方形盒子，在盒子内部中位数处画一条竖线。从盒子引出线段（须线）到最小值和最大值，除非有异常值。
Outliers are usually defined as values smaller than Q₁ − 1.5 × IQR or larger than Q₃ + 1.5 × IQR. These are plotted as individual crosses or dots, and the whisker extends to the next non-outlier value. A box plot makes it easy to compare distributions, comment on skewness and identify outliers.
异常值通常定义为小于 Q₁ − 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的数值。这些点用单独的叉或点表示，须线则延伸到下一个非异常值。箱线图易于比较分布、评价偏态和识别异常值。
From a box plot, you can say: the median close to the centre of the box suggests symmetry; a long whisker on one side suggests skewness. Compare two box plots by referencing medians and IQRs. The distribution with a smaller IQR is more consistent.
从箱线图中可以看出：中位数接近盒子中心表明对称分布；某一侧须线较长表明偏态。通过比较中位数和 IQR 来对比两个箱线图。IQR 较小的分布更稳定一致。
9. Cumulative Frequency Graphs | 累积频数图
A cumulative frequency table adds up frequencies as you move through the classes. The cumulative frequency for a class is the total number of data points less than or equal to its upper class boundary. Plot points at each upper class boundary against the cumulative frequency. Join the points with a smooth curve to create a cumulative frequency graph (ogive).
累积频数表在遍历各组时不断累加频数。某一组的累积频数是指小于或等于该组上界的所有数据点的总数。在每个组的上界处，画出对应累积频数的点。用光滑曲线连接这些点，即得到累积频数图（肩形图）。
To estimate the median from the graph, go to half the total frequency on the cumulative frequency axis, draw a horizontal line to the curve, then a vertical line down to read the value. To find Q₁ and Q₃, use one-quarter and three-quarters of the total frequency. The IQR can be found directly from the graph. You can also estimate the number of values below a certain threshold, or find a percentile.
从图中估算中位数：在累积频数轴上取总频数的一半处，画一条水平线与曲线相交，然后垂直向下读取数值。要找出 Q₁ 和 Q₃，则分别采用总频数的四分之一和四分之三处。IQR 可以直接从图上求出。你还可以估计低于某个阈值的数据个数，或求出百分位数。
Always use a ruler to draw lines on the graph. Label the axes clearly and show your working on the graph, as examiners award marks for using the correct method.
在图上画线时务必用直尺。清楚地标记坐标轴，并在图上展示你的过程，因为考官会根据正确的方法给分。
10. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph shows the relationship between two sets of data. Each point represents a pair of values (x, y). If the points tend to go upwards from left to right, it is positive correlation. If downwards, negative correlation. If no pattern, there is no correlation. Correlation does not imply causation; a strong correlation means a linear relationship exists, not that one variable causes the other.
散点图显示两组数据之间的关系。每个点代表一对数值 (x, y)。如果点从左上到右下大致呈上升趋势，则是正相关；如果下降，则是负相关。如果没有规律，则无相关。相关不代表因果；强相关意味着存在线性关系，但不意味着一个变量导致另一个变化。
A line of best fit can be drawn by eye, passing through as many points as possible with about equal numbers of points above and below the line. Use a ruler for a straight line. Do not force it through the origin unless the data suggests so. The line can be used to estimate unknown values within the data range (interpolation), but extrapolation beyond the range can be unreliable.
最佳拟合线可以通过目测画出，尽量穿过尽可能多的点，并使线上方的点和下方的点数量大致相等。画直线时用直尺。除非数据表明应当通过原点，否则不要硬性要求通过原点。这条线可以用于估计数据范围内的未知值（内插），但超出范围的外推可能不可靠。
11. 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, probability = number of favourable outcomes ÷ total number of outcomes. Probabilities can be written as fractions, decimals or percentages.
概率衡量一个事件发生的可能性，范围从 0（不可能）到 1（必然）。对于等可能结果，概率 = 有利结果的数量 ÷ 总结果数量。概率可以用分数、小数或百分数表示。
P(Event) = Number of favourable outcomes ÷ Total number of outcomes
P(事件) = 有利结果数 ÷ 总结果数
The probability of an event not happening is 1 minus the probability that it does happen. The sum of probabilities of all mutually exclusive outcomes of an experiment is 1. For two 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).
一个事件不发生的概率等于 1 减去它发生的概率。一个实验中所有互斥结果的概率之和为 1。对于两个互斥事件 A 和 B，P(A 或 B) = P(A) + P(B)。对于独立事件，P(A 与 B) = P(A) × P(B)。
Sample space diagrams list all possible outcomes, helping to calculate probabilities. Tree diagrams are used for two or more sequential events, multiplying along branches and adding for combined events. Use frequency trees or two-way tables for organising data before calculating probabilities.
样本空间图列出所有可能的结果，有助于计算概率。树状图用于两个或以上连续事件，沿分支相乘，对不同组合相加。在计算概率之前，可用频数树或双向表整理数据。
12. Summary and Exam Tips | 总结与考试技巧
Statistics in IGCSE Mathematics tests your ability to interpret and present data clearly. Always read the question carefully: check whether data is discrete or continuous, whether class widths are equal, and what type of average or measure of spread is required. Show your working for calculations of mean, median and quartiles. When drawing graphs, use a pencil and ruler, label axes, and give titles. For probability, be precise with fractions and remember to simplify if possible. Practise past paper questions to become familiar with common graph interpretations and to avoid common mistakes such as confusing bar charts with histograms.
IGCSE 数学中的统计考查你清晰解读和展示数据的能力。务必仔细读题：检查数据是离散的还是连续的，组距是否相等，需要求哪种平均数或散布度量。计算平均数、中位数和四分位数时要展示过程。画图时用铅笔和直尺，标记坐标轴，加上标题。概率题中，分数要精确，记得化简。多练习历年真题，熟悉常见的图表解读，避免常见错误，比如混淆柱状图和直方图。
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