📚 KS3 CIE Statistics: Core Concepts Revision | KS3 CIE 统计：核心知识点梳理
Statistics is all about collecting, organising, presenting and interpreting data to make sense of the world around us. At KS3 CIE level, you will learn how to handle different types of data, draw and read charts, find averages and spread, and start working with probability. This article pulls together all the key ideas you need to master.
统计学是关于收集、整理、展示和解读数据，从而理解我们周围世界的一门学问。在 KS3 CIE 阶段，你将学习如何处理不同类型的数据，绘制并阅读图表，计算平均数和离散程度，以及初步接触概率。本文汇总了你需要掌握的所有核心概念。
1. What is Statistics? | 什么是统计？
Statistics involves gathering information, known as data, and turning it into useful summaries. The process usually starts with a question, then collecting data, presenting it in tables or graphs, and finally drawing conclusions.
统计学涉及收集信息（也就是数据）并将其转化为有用的总结。这个过程通常从一个问题开始，然后收集数据，用表格或图形展示，最后得出结论。
Data can be collected through surveys, experiments or observations. For example, asking classmates about their favourite fruit gives you a data set. Understanding how statistics works helps you spot patterns and make sensible predictions.
数据可以通过调查、实验或观察来收集。例如，询问同学们最喜欢的水果就能获得一组数据。理解统计学的原理有助于你发现规律并做出合理的预测。
2. Types of Data | 数据的类型
Data can be split into two main categories: qualitative and quantitative. Qualitative data describes qualities or categories, like eye colour or favourite subject. Quantitative data involves numbers and can be counted or measured.
数据可以分为两大类：定性数据和定量数据。定性数据描述属性或类别，例如眼睛颜色或最喜欢的科目。定量数据则涉及数字，是可以计数或测量的。
Quantitative data is further divided into discrete and continuous. Discrete data can only take certain values, often whole numbers – for instance, the number of students in a class. Continuous data can take any value within a range, such as height or time.
定量数据又分为离散数据和连续数据。离散数据只能取特定的值，通常是整数，例如班级里的学生人数。连续数据可以在一个范围内取任意值，比如身高或时间。
- Qualitative: hair colour, type of pet
- 定性数据：头发颜色、宠物类型
- Quantitative discrete: number of goals scored, shoe size
- 定量离散数据：进球数、鞋码
- Quantitative continuous: temperature, mass, length
- 定量连续数据：温度、质量、长度
3. Frequency Tables | 频率表
A frequency table is one of the simplest ways to organise data. It shows each possible value or category alongside its frequency – the number of times it appears.
频率表是整理数据最简单的方法之一。它列出每一个可能的取值或类别，旁边写出它的频数——也就是它出现的次数。
Tally marks are often used when collecting data to keep a quick count. Once the tally is complete, you add up the marks to find the frequencies. A well-structured frequency table makes it easy to spot the mode.
在收集数据时经常使用画记法（正字计数）来快速计数。画记完成后，把记号加起来就得到了频数。结构清晰的频率表可以帮助你轻松找出众数。
| Fruit (水果) | Tally (画记) | Frequency (频数) |
|---|---|---|
| Apple | 卌 卌 || | 12 |
| Banana | 卌 ||| | 8 |
| Orange | 卌 6 | 6 |
4. Bar Charts and Line Graphs | 条形图和折线图
Bar charts are used to display categorical data or discrete numerical data. Each bar’s height or length represents the frequency. Bars should be of equal width with even gaps between them.
条形图用于展示分类数据或离散的数值数据。每个条形的高度或长度代表频数。条形应宽度一致，条形之间应有均匀的间距。
Line graphs are ideal for showing how data changes over time. The horizontal axis often shows time intervals, and points are joined with straight lines. You can spot trends, such as increasing temperature or falling sales, very quickly.
折线图非常适合展示数据随时间的变化。横轴通常表示时间间隔，各个点用直线连接起来。你可以很快看出趋势，比如温度上升或销售下降。
When drawing any chart, always label the axes, give a title, and choose a sensible scale. Misleading scales can distort the message of the data.
绘制任何图表时，都要标注坐标轴、给出标题，并选择合适的刻度。误导性的刻度会歪曲数据想要传达的信息。
5. Pie Charts | 饼图
A pie chart shows proportions of a whole. The whole circle represents the total frequency, and each slice represents a category. The angle of a slice is calculated by multiplying the fraction of the total by 360°.
饼图展示整体中的各部分比例。整个圆代表总频数，每个扇形代表一个类别。扇形的角度可以通过分数乘以 360° 来计算。
For instance, if 10 out of 30 students prefer cycling, the angle is (10/30) × 360° = 120°. Pie charts are best when you want to compare parts to the whole and there are not too many categories.
例如，如果 30 名学生中有 10 人喜欢骑自行车，那么扇形的角度就是 (10/30) × 360° = 120°。饼图最适合用于比较部分与整体的关系，而且类别不宜过多。
Remember to use a protractor to measure angles accurately and label each slice clearly, sometimes adding percentages to make comparison easier.
记得用量角器准确测量角度，并清晰地标注每个扇形，有时可以加上百分比让比较更加直观。
6. Scatter Graphs and Correlation | 散点图与相关性
Scatter graphs display paired numerical data to see if there is a relationship, or correlation, between two variables. Each point on the graph represents one pair of values.
散点图用来展示成对的数值数据，以观察两个变量之间是否存在关系，也就是相关性。图上的每个点代表一对数值。
Correlation can be positive (as one increases, the other tends to increase), negative (as one increases, the other tends to decrease), or zero (no clear pattern). If points lie close to a straight line, the correlation is strong; if spread out, it is weak.
相关性可以是正相关（一个增加，另一个也往往增加）、负相关（一个增加，另一个往往减少）或零相关（没有明显的模式）。如果点靠近一条直线，相关性就强；如果点很分散，相关性就弱。
A line of best fit can be drawn through the points to summarise the trend. It does not have to go through all points but should roughly pass through the middle of the cloud of points. You can use this line to estimate unknown values.
可以在点之间画一条最佳拟合线来概括趋势。它不必经过所有点，但应大致穿过点云的中部。你可以利用这条线来估计未知值。
7. Mean, Median and Mode | 平均数、中位数和众数
Measures of central tendency summarise a data set with a typical value. The three main averages are mean, median and mode.
集中趋势量数用一个典型值来概括一组数据。三个主要的平均数是平均数、中位数和众数。
Mode is the value that appears most often. A data set can have one mode, more than one mode (bimodal or multimodal), or no mode at all.
众数是出现次数最多的值。一组数据可以有一个众数、多个众数（双峰或多峰），甚至没有众数。
Median is the middle value when data are arranged in order. For an odd number of values, it is the central number; for an even number, it is the mean of the two middle numbers.
中位数是将数据按顺序排列后位于中间的值。当数据个数为奇数时，它就是正中间的那个数；当个数为偶数时，则是中间两个数的平均数。
Mean is calculated by adding all values together and dividing by the number of values. It is the most commonly used average but can be affected by extreme values (outliers).
平均数是把所有数值加起来再除以数值的个数。它是最常用的平均数，但容易受到极端值（异常值）的影响。
Mean = Σx / n
For grouped frequency tables, mean is estimated using midpoints of class intervals multiplied by frequencies, then dividing by total frequency.
对于分组频率表，平均数是利用组中值乘以频数，再除以总频数来估算的。
8. Range | 极差
Range is a simple measure of spread. It tells you how spread out the data are by subtracting the smallest value from the largest value.
极差是一种简单的离散程度量数。它通过最大值减去最小值告诉你数据分散的程度。
Range = Highest value – Lowest value
A large range means the data are very spread out; a small range means the values are close together. Range is easy to compute but does not give any information about the distribution between the extremes.
极差大说明数据很分散；极差小说明数值彼此接近。极差容易计算，但不提供关于极端值之间分布情况的信息。
When comparing two sets of data, looking at both an average and the range gives a more complete picture. For example, two classes might have the same mean score but very different ranges.
比较两组数据时，同时观察平均数和极差能提供更完整的图像。例如，两个班级可能有相同的平均分，但分数的极差却大不相同。
9. Stem-and-Leaf Diagrams | 茎叶图
A stem-and-leaf diagram is a way of showing the shape and distribution of numerical data without losing the original values. Each number is split into a stem (the leading digit or digits) and a leaf (the last digit).
茎叶图是一种既能展示数值数据的形状和分布，又不会丢失原始数值的方法。每个数字被分成茎（前一位或多位数字）和叶（最后一位数字）。
For example, the number 42 has stem 4 and leaf 2. The stems are listed vertically in order, and the leaves are written in a row next to their stem, usually in ascending order. A key must be given to explain the representation.
例如，数字 42 的茎是 4，叶是 2。茎垂直按顺序列出，叶则写在对应茎的右边那一行，通常按升序排列。必须给出图例来解释这种表示方式。
| Stem | Leaf |
|---|---|
| 2 | 0, 3, 7 |
| 3 | 1, 5, 5, 9 |
| 4 | 2, 6 |
Key: 2|0 means 20
图例：2|0 表示 20
Stem-and-leaf diagrams make it easy to find the median and mode directly, and they also show the shape of the data similarly to a bar chart.
茎叶图可以让你轻松地直接找到中位数和众数，同时它还能像条形图那样显示数据的形态。
10. Probability Scale | 概率尺度
Probability measures how likely an event is to happen. It is given a value between 0 and 1, where 0 means impossible and 1 means certain. You can also express probability as a fraction, decimal or percentage.
概率衡量一个事件发生的可能性有多大。它的取值在 0 到 1 之间，0 表示不可能发生，1 表示必然发生。概率也可以用分数、小数或百分比来表达。
Words such as ‘impossible’, ‘unlikely’, ‘even chance’, ‘likely’ and ‘certain’ correspond to positions on the probability scale. Understanding this scale helps you assign numerical probabilities to simple events.
像“不可能”“不太可能”“对等机会”“很可能”“必然”这些词语对应着概率尺度上的不同位置。理解这个尺度有助于你为简单事件赋予数值概率。
The probability of an event not happening is 1 minus the probability that it does happen. This is called the complement rule.
一个事件不发生的概率等于 1 减去它发生的概率。这就是互补规则。
P(not A) = 1 – P(A)
11. Experimental Probability | 实验概率
Experimental probability (also called relative frequency) is found by carrying out an experiment or survey repeatedly. It is calculated using the formula:
实验概率（也叫相对频率）是通过反复进行实验或调查得到的。它的计算公式是：
Experimental Probability = Number of successful trials / Total number of trials
When you toss a coin many times, the experimental probability of getting heads may not be exactly 0.5 at first, but it tends to get closer to 0.5 as the number of trials increases. This is known as the law of large numbers.
当你多次抛一枚硬币时，起初得到正面的实验概率可能并不是恰好 0.5，但随着试验次数的增加，它往往会趋近于 0.5。这被称作大数定律。
Theoretical probability assumes all outcomes are equally likely, while experimental probability is based on actual results. Comparing the two helps you understand fairness and randomness.
理论概率假设所有结果是等可能的，而实验概率则基于实际结果。将两者进行比较有助于你理解公平性和随机性。
Expected frequency can be found by multiplying the theoretical probability by the number of trials. For example, if you roll a fair dice 60 times, you expect a six about 60 × (1/6) = 10 times.
期望频数可以通过用理论概率乘以试验次数得出。例如，如果你掷一个公平的骰子 60 次，你期望出现 6 的次数约为 60 × (1/6) = 10 次。
Published by TutorHao | Statistics Revision Series | aleveler.com
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