📚 Year 9 CIE Statistics: Core Concepts Review | Year 9 CIE 统计：核心知识点梳理
This article provides a comprehensive summary of the key statistical concepts covered in Year 9 CIE Statistics. It will help you revise data handling, averages, spread, graphical representations, and probability, giving you a solid foundation for IGCSE and beyond.
本文全面梳理了 Year 9 CIE 统计课程中的核心知识点，涵盖数据处理、平均数与离散程度、图形表示以及概率等内容，帮助大家高效复习，为 IGCSE 及更高阶段的学习打下坚实基础。
1. Types of Data | 数据的类型
Data can be classified as qualitative (categorical) or quantitative (numerical). Qualitative data describe non‑numerical characteristics, such as favourite colour, type of pet, or blood group. Quantitative data are numerical measurements or counts, like height, weight, or test scores.
数据可分为定性（类别）数据和定量（数值）数据。定性数据描述非数字的特征，例如最喜欢的颜色、宠物种类或血型。定量数据是数字测量值或计数，如身高、体重或考试分数。
Quantitative data can be further split into discrete and continuous. Discrete data can only take specific, separate values—usually whole numbers—such as the number of books on a shelf or shoes in a cupboard. Continuous data can take any value within a given range, for instance temperature, length, or time.
定量数据可进一步分为离散数据和连续数据。离散数据只能取特定的、分离的值——通常是整数——例如书架上的书本数量或鞋柜里的鞋子数目。连续数据在给定范围内可取任意值，例如温度、长度或时间。
Knowing the data type helps you choose the most suitable graph or chart, and the correct statistical measures. For example, a pie chart works well for categorical data, while a histogram suits continuous grouped data.
了解数据类型有助于选择最合适的图表和正确的统计量。例如，饼图适合类别数据，而直方图适合连续的分组数据。
2. Organising Data: Tables and Charts | 整理数据：表格与图表
A frequency table records how often each value or category appears. It is the foundation for most statistical diagrams. Tally marks are often used during data collection to make counting easier.
频数表记录每个数值或类别出现的次数，是大多数统计图表的基础。数据收集时常用划记符号（正字）使计数更方便。
Bar charts display categorical data, with gaps between the bars. Each bar’s height represents the frequency of the category. You can also use a bar‑line graph, where a vertical line replaces the bar.
柱状图（条形图）用于展示类别数据，各条形之间有间隙。每个条形的高度代表该类别的频数。你也可以使用柱线图，用垂直线段代替条形。
Pie charts show proportions of a whole. The angle of each sector is calculated as (frequency ÷ total frequency) × 360°. They are excellent for showing percentages visually.
饼图展示整体中的比例。每个扇区的角度计算公式为 (频数 ÷ 总频数) × 360°。饼图非常适合直观显示百分比。
Line graphs are used when data are collected over time or when the independent variable is quantitative and continuous. The points are joined in order, revealing trends and patterns.
折线图用于按时间收集的数据，或自变量为定量连续变量的情况。按顺序连接各点可以揭示趋势和模式。
3. Stem‑and‑Leaf Plots | 茎叶图
A stem‑and‑leaf plot organises numerical data while preserving the original values. The ‘stem’ represents the leading digit(s), and the ‘leaf’ is the final digit. For example, the number 47 would have stem 4 and leaf 7. A key must always be shown, e.g. ‘4 | 7 means 47’.
茎叶图在整理数值数据的同时保留了原始数值。“茎”代表前导数字，“叶”是最后一位数字。例如，47 的茎为 4，叶为 7。必须给出图例，如“4 | 7 表示 47”。
Stem‑and‑leaf plots make it easy to find the median, mode, and range. An ordered stem‑and‑leaf plot simply arranges the leaves in ascending order. Back‑to‑back stem‑and‑leaf plots are useful for comparing two datasets with the same stem, such as boys’ and girls’ test scores.
茎叶图便于找出中位数、众数和极差。有序茎叶图将叶片从小到大排列。背靠背茎叶图可用于比较两组共享相同茎的数据，如男生和女生的考试分数。
4. Measures of Central Tendency | 集中趋势度量
The mean (average) is the sum of all values divided by the number of values. From a frequency table, mean = Σ(f × x) ÷ Σf, where f is frequency and x is the data value.
平均数（均值）等于所有数值之和除以数值的个数。在频数表中，平均数 = Σ(f × x) ÷ Σf，其中 f 为频数，x 为数据值。
The median is the middle value when data are arranged in order. For n values, the median’s position is (n + 1) ÷ 2. If there are two middle values, the median is their average. The median is unaffected by extreme values (outliers).
中位数是将数据排序后位于中间的值。对于 n 个数据，中位数的位置为 (n + 1) ÷ 2。如果有两个中间值，中位数取两者的平均。中位数不受极端值（异常值）的影响。
The mode is the value that appears most often. A dataset can have one mode (unimodal), two modes (bimodal), or no mode at all. The mode is particularly useful for categorical 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）或没有众数。众数对于类别数据特别有用。
Choosing the best average depends on the data distribution. When outliers are present, the median often gives a better indication of the centre than the mean.
选择最佳平均数取决于数据分布。存在异常值时，中位数往往比平均数更能代表数据的中心。
5. Measures of Dispersion | 离散程度度量
Dispersion describes how spread out the data are. The simplest measure is the range: Range = maximum value – minimum value. It is easy to calculate but can be heavily affected by a single extreme value.
离散程度描述数据的分散情况。最简单的度量是极差：极差 = 最大值 – 最小值。极差易于计算，但极易受单一极端值影响。
The interquartile range (IQR) is a more robust measure. It is the difference between the upper quartile (Q₃) and the lower quartile (Q₁): IQR = Q₃ – Q₁. To find quartiles, first arrange data in order. Q₂ is the median; Q₁ is the median of the lower half; Q₃ is the median of the upper half.
四分位距（IQR）是更稳健的度量。它是上四分位数（Q₃）与下四分位数（Q₁）之差：IQR = Q₃ – Q₁。求四分位数时先将数据排序：Q₂ 为中位数；Q₁ 为下半部分数据的中位数；Q₃ 为上半部分数据的中位数。
The IQR measures the spread of the middle 50% of the data, therefore it is not affected by outliers. Together with the median, it gives a good summary of a dataset’s centre and variability.
四分位距衡量中间 50% 数据的分散程度，因此不受异常值影响。与中位数配合使用，可以很好地概括数据集的中心和变异程度。
6. Grouped Data and Histograms | 分组数据与直方图
When continuous data are recorded in class intervals, we use histograms. Unlike bar charts, there are no gaps between the bars because the data are continuous. The frequency density is used to draw the bars when class widths are unequal: frequency density = frequency ÷ class width.
当连续数据以组距记录时，我们使用直方图。与柱状图不同，直方形之间没有间隙，因为数据是连续的。当组距不相等时，需用频数密度绘制直方：频数密度 = 频数 ÷ 组距。
To estimate the mean from a grouped frequency table, we use the midpoint of each class interval. Estimated mean = Σ(f × midpoint) ÷ Σf. This is an approximation because we do not know the exact values within each group.
从分组频数表估算平均数时，使用每组的组中值。估算平均数 = Σ(f × 组中值) ÷ Σf。这是一个近似值，因为我们不知道每组内各数据的具体数值。
The modal class is the class interval with the highest frequency. The class containing the median can be found by locating the (n ÷ 2)th value on a cumulative frequency curve or by calculation.
众数组是频数最高的组区间。包含中位数的组可以通过累积频数曲线定位第 (n ÷ 2) 个值或用计算求得。
7. Cumulative Frequency and Box Plots | 累积频数与箱线图
A cumulative frequency table adds up frequencies as you move through the data. Plotting cumulative frequency against the upper class boundary gives a smooth S‑shaped curve (ogive).
累积频数表在遍历数据时将频数依次累加。以累积频数为纵轴、组上限为横轴描点并连成光滑的 S 形曲线（卵形线）。
From a cumulative frequency curve you can read off the median (Q₂, at the ½ total frequency mark), the lower quartile (Q₁, at ¼ total frequency) and the upper quartile (Q₃, at ¾ total frequency). The interquartile range can then be found.
从累积频数曲线上可以读出中位数（Q₂，位于总频数的 ½ 处）、下四分位数（Q₁，位于总频数的 ¼ 处）和上四分位数（Q₃，位于总频数的 ¾ 处），并可由此求出四分位距。
A box‑and‑whisker plot (box plot) uses the five‑number summary: minimum, Q₁, median, Q₃, maximum. The box spans from Q₁ to Q₃, with a line at the median. Whiskers extend to the minimum and maximum (or to 1.5 × IQR beyond the quartiles, with outliers plotted separately). Box plots are excellent for comparing distributions.
箱线图（箱形图）使用五数概括：最小值、Q₁、中位数、Q₃、最大值。箱体从 Q₁ 延伸到 Q₃，中位数处画一条线。触须延伸至最小值和最大值（或至四分位数 ± 1.5 × IQR 处，异常值单独标出）。箱线图非常利于比较数据分布。
8. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph displays the relationship between two variables, usually plotted on the x‑axis and y‑axis. Each point represents a paired observation. The pattern of points reveals whether there is a correlation.
散点图展示两个变量之间的关系，通常以 x 轴和 y 轴表示。每个点代表一对观测数据。点的分布模式可以揭示是否存在相关关系。
Positive correlation means as one variable increases, the other tends to increase (e.g., height and arm span). Negative correlation means as one variable increases, the other tends to decrease (e.g., car age and value). No correlation shows no clear pattern.
正相关表示一个变量增大时另一个变量也趋于增大（如身高与臂展）。负相关表示一个变量增大时另一个趋于减小（如车龄与价值）。无相关则没有明显模式。
A line of best fit can be drawn through the points, roughly passing through the centre of the data. It should have roughly equal numbers of points above and below the line. The line can be used to estimate unknown values (interpolation within the data range is more reliable than extrapolation beyond it).
可以在各点间画一条最佳拟合线，大致穿过数据中心，并使线上方和下方的点数大致相等。该线可用于估算未知值（在数据范围内进行内插比外推更可靠）。
9. Basic Probability Concepts | 概率基础
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It can be expressed as a fraction, decimal, or percentage. Theoretical probability is based on equally likely outcomes: P(event) = number of favourable outcomes ÷ total number of outcomes.
概率用于衡量事件发生的可能性，范围从 0（不可能）到 1（必然）。概率可用分数、小数或百分数表示。理论概率基于等可能的结果：P(事件) = 有利结果数 ÷ 总结果数。
Sample space is the set of all possible outcomes. Listing sample spaces systematically (e.g., using tables or lists) helps avoid missing outcomes. For two dice, a sample space table shows all 36 ordered pairs.
样本空间是所有可能结果的集合。系统列出样本空间（例如使用表格或列表）有助于避免遗漏。掷两枚骰子时，样本空间表可显示全部 36 个有序数对。
Experimental probability is found by conducting an experiment or survey: relative frequency = number of times event occurs ÷ total number of trials. As the number of trials increases, the experimental probability tends to get closer to the theoretical probability.
实验概率通过实验或调查获得：相对频数 = 事件发生次数 ÷ 试验总次数。随着试验次数增加，实验概率往往趋近于理论概率。
10. Tree Diagrams for Combined Events | 组合事件的树状图
Tree diagrams show all possible outcomes of two or more combined events, with branches representing each outcome and its probability. The probabilities on branches from the same point must sum to 1. To find the probability of a sequence of events, multiply the probabilities along the branches.
树状图展示两个或多个组合事件的所有可能结果，每条分支表示一个结果及其概率。从同一点出发的各分支概率之和必须等于 1。求一系列事件发生的概率时，将路径上的概率相乘。
For independent events, e.g., flipping a coin and rolling a die, the probability of each branch does not depend on previous outcomes. P(A and B) = P(A) × P(B). For conditional events (where one event influences the next), the second set of branches shows probabilities that change depending on the first outcome.
对于独立事件，如抛硬币和掷骰子，每条分支的概率不依赖于之前的结果，P(A 且 B) = P(A) × P(B)。对于条件事件（一个事件影响后续事件），第二组分枝展示的概率随第一个结果而变化。
P(A and B) = P(A) × P(B|A)
P(A 且 B) = P(A) × P(B|A)
Tree diagrams can also be used to find probabilities of ‘at least one’ or ‘none’ by adding the relevant final probabilities. They are a powerful tool for solving probability problems step by step.
树状图还可通过相加相关的最终概率来求解“至少一次”或“全不发生”的概率。它是逐步解决概率问题的强大工具。
Published by TutorHao | Statistics Revision Series | aleveler.com
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