📚 Year 7 CAIE Statistics: Key Concepts | CAIE Year 7 统计核心知识点梳理
Welcome to your key revision guide for Year 7 CAIE Statistics. This article brings together the core concepts you need to understand data handling, averages, charts and basic probability. Each topic is explained with clear examples to help you build confidence in interpreting and presenting data.
欢迎来到 Year 7 CAIE 统计核心知识点梳理。本文将你需要掌握的数据处理、平均数、图表以及基础概率等核心概念汇总在一起。每个主题都配有清晰的解释和示例，帮助你自信地解读和展示数据。
1. What is Statistics? Data Collection | 统计是什么？数据收集
Statistics is the study of collecting, organising, analysing and interpreting data. Data is any information we gather to answer a question. For example, to find the favourite fruit in your class, you might design a survey and record everyone’s response. The process always starts with a clear question: ‘What do we want to know?’
统计学是研究数据收集、整理、分析和解释的学科。数据是为了回答某个问题而收集的任何信息。例如，为了了解班级里最受欢迎的水果，你可以设计一份调查并记录每个人的回答。这个过程总是从一个清晰的问题开始：“我们想知道什么？”
Data can be collected in two main ways. Primary data is information you collect yourself, such as conducting a survey or measuring leaves from a tree. Secondary data is information gathered by someone else, like statistics published on government websites or data from a textbook. Year 7 students often work with primary data to learn the full data cycle.
收集数据主要有两种方式。一手数据是你自己收集的信息，比如进行一次调查或测量树上的叶子。二手数据是由他人收集的信息，如政府网站上发布的统计数据或教科书中的数据。Year 7 学生经常使用一手数据来学习完整的数据处理循环。
2. Types of Data | 数据类型
Understanding the type of data you have is vital because it determines which chart and average are suitable. Data falls into two broad categories: qualitative and quantitative. Qualitative (categorical) data describes qualities or groups, for example hair colour, favourite sport or brand of shoe. Quantitative data refers to numerical measurements, such as height in centimetres, number of pets or time taken to run 100 metres.
理解你所拥有的数据类型至关重要，因为它决定了哪种图表和平均数才是合适的。数据可以分为两大类：定性数据和定量数据。定性（类别）数据描述性质或群体，例如头发颜色、最喜爱的运动或鞋子品牌。定量数据指的是数值型的测量值，比如以厘米为单位的身高、宠物的数量或跑100米所用时间。
Quantitative data can be further split into discrete and continuous. Discrete data can only take certain, often whole-number values – think of the number of students in a room or the number of books on a shelf. Continuous data can take any value within a range, usually involving measurements like mass, temperature or length, where decimals matter.
定量数据可以进一步分为离散数据和连续数据。离散数据只能取某些特定的、通常是整数的值——比如房间里学生的人数或书架上的书本数量。连续数据可以在一个范围内取任意值，通常涉及质量、温度或长度等测量，小数部分也很重要。
3. Recording Data: Frequency Tables and Tally Marks | 记录数据：频率表与计数符号
Once data is collected, it needs to be organised. A frequency table shows how often each value or category occurs. Tally marks are a quick way to count during data collection. Each vertical stroke ‘|’ represents one observation, and the fifth tally is drawn diagonally across the previous four to make groups of five easy to count, as in ‘||||’.
数据收集之后，需要进行整理。频率表显示了每个值或类别出现的次数。计数符号（划记法）是在数据收集过程中快速计数的方法。每一条竖线“|”代表一次观察，第五条计数斜线会划过前四条，形成五个一组，便于统计，如“卌”。
Here is an example of a frequency table for the colours of 20 cars passing a school gate:
下面是一张统计经过学校门口的20辆汽车颜色的频率表示例：
| Colour (颜色) | Tally (计数) | Frequency (频数) |
|---|---|---|
| Red | |||| | 4 |
| Blue | || | 2 |
| White | 卌 || | 7 |
| Black | 卌 | | 6 |
| Other | | | 1 |
Always check that the total frequency matches the number of observations. In this case 4 + 2 + 7 + 6 + 1 = 20, which is correct.
请务必检查总频数是否与观察次数吻合。本例中 4 + 2 + 7 + 6 + 1 = 20，结果正确。
4. Bar Charts | 条形图
A bar chart is used to display the frequency of categorical data. Each category has a rectangular bar of equal width, and the height of the bar corresponds to the frequency. Bars are drawn with gaps between them to show that the categories are separate, not connected.
条形图用于展示类别数据的频数。每个类别都有一个等宽的矩形条，条的高度对应频数。条形之间留有间隙，表明它们是相互独立的类别，而不是连续的。
When drawing a bar chart by hand, always use a ruler, label both axes clearly, and give the chart a title. The horizontal axis lists the categories (e.g. colours, sports), while the vertical axis shows the frequency scale, starting from zero. Many mistakes happen when the vertical scale does not start at zero, so be careful with that.
手工绘制条形图时，一定要使用直尺，清晰标注两条坐标轴，并为图表加上标题。横轴列出类别（如颜色、运动），纵轴显示从零开始的频数刻度。纵轴刻度若没有从零开始，很容易引起误读，因此需要格外注意。
5. Pie Charts | 饼图
Pie charts show how a total is divided into parts. Each slice, or sector, represents a category, and the angle of the sector is proportional to the frequency of that category. The entire circle is 360°, which represents the total frequency.
饼图展示了一个整体如何被划分为各个部分。每一块扇形代表一个类别，扇形的角度与该类别的频数成比例。整个圆是360°，代表总频数。
The formula to calculate the sector angle is:
计算扇形角度的公式为：
Sector angle = (Frequency of category ÷ Total frequency) × 360°
For example, if a survey of 30 students shows that 12 chose drama as their favourite activity, the drama sector angle = (12 ÷ 30) × 360° = 144°. Once all angles are calculated, use a protractor to draw the sectors accurately. Remember to label each sector or provide a key.
例如，如果一项针对30名学生的调查显示12人选择了戏剧作为最喜爱的活动，那么戏剧的扇形角度 = (12 ÷ 30) × 360° = 144°。计算出所有角度后，用量角器准确画出各个扇形。记得为每个扇形添加标签或提供图例。
6. Pictograms | 象形图
A pictogram uses small pictures or symbols to represent a certain number of data items. For example, a book symbol could represent 5 books read by a student. The key is essential – it tells the reader what each symbol stands for, such as ‘☺ = 2 people’. Without a key, a pictogram cannot be interpreted accurately.
象形图使用小图片或符号来表示一定数量的数据项。例如，一个书本符号可以代表一位学生读过的5本书。图例是必不可少的——它告诉读者每个符号代表什么，比如“☺ = 2 人”。没有图例，象形图就无法被准确解读。
When drawing or reading a pictogram, look carefully at how fraction symbols are shown. If each full circle represents 4 goals and a half circle is shown, that half counts as 2 goals. Counting symbols systematically ensures you do not misinterpret the totals.
在绘制或阅读象形图时，要仔细观察分数符号是如何表示的。如果每个完整的圆代表4个进球，并出现一个半圆，那么这个半圆就代表2个进球。系统性地数符号能确保你不会误解总数。
7. The Mean | 均值
The mean is the average you get by sharing the total equally. To calculate the mean, add up all the data values and then divide by the number of values.
均值是通过均分总量得到的平均数。计算均值时，先将所有数据值相加，再除以数据的个数。
Mean = (Sum of all data values) ÷ (Number of data values)
For example, the heights of five students are 142 cm, 148 cm, 155 cm, 138 cm and 152 cm. The sum is 735 cm, and there are 5 values, so the mean height = 735 ÷ 5 = 147 cm. The mean is useful but can be affected by extremely high or low values, called outliers.
例如，五名学生的身高分别为 142 cm、148 cm、155 cm、138 cm 和 152 cm。总和为 735 cm，数据个数为5，因此平均身高 = 735 ÷ 5 = 147 cm。均值很有用，但会受到极高或极低数值（称为异常值）的影响。
8. The Median | 中位数
The median is the middle value when all the data are arranged in order from smallest to largest. It splits the data into two halves, with an equal number of values above and below it. The median is not affected by outliers, making it a good average for skewed data.
中位数是将所有数据按照从小到大的顺序排列后，位于中间位置的那个值。它将数据分成数量相等的两部分，各有半数数据位于以上和以下。中位数不受异常值的影响，因此是偏态数据的一种理想平均数。
If there is an odd number of values, the median is simply the middle value. For the set 3, 7, 8, 12, 15, the median is 8. When there is an even number of values, the median is the mean of the two middle numbers. For example, in the data set 4, 6, 9, 11, the two middle numbers are 6 and 9, so the median = (6 + 9) ÷ 2 = 7.5.
如果数据个数为奇数，中位数就是正中间的值。例如在数据集 3, 7, 8, 12, 15 中，中位数是8。如果数据个数为偶数，中位数是中间两个数的均值。例如在数据集 4, 6, 9, 11 中，中间两个数为6和9，因此中位数 = (6 + 9) ÷ 2 = 7.5。
9. The Mode | 众数
The mode is the value that appears most often in a data set. A set can have one mode (unimodal), two modes (bimodal), or more than two modes (multimodal). If no value repeats, we say there is no mode. The mode is the only average that can be used for categorical data, such as the most frequently observed hair colour.
众数是数据集中出现次数最多的值。一个数据集可以有一个众数（单峰）、两个众数（双峰）或多于两个众数（多峰）。如果没有数值重复出现，我们说该数据集没有众数。众数是唯一可用于类别数据的平均数，例如观察到的最常见头发颜色。
Consider the following set of quiz scores: 7, 8, 7, 10, 7, 9, 8. The number 7 appears three times, 8 appears twice, and the others once. So the mode is 7. Always check the frequency of each value before deciding the mode.
考虑以下测验成绩：7, 8, 7, 10, 7, 9, 8。数字7出现了三次，8出现了两次，其他各出现一次，所以众数是7。在确定众数之前，务必检查每个值出现的频数。
10. The Range | 极差
The range measures how spread out the data is. It is the difference between the largest and smallest values. A small range indicates the data points are close together, while a large range suggests they are more spread out.
极差衡量数据分散的程度，它是最大值与最小值之间的差值。极差较小表示数据点比较集中，极差较大则说明数据分布更广。
Range = Largest value − Smallest value
For the data set 23, 45, 18, 37, 56, the largest value is 56 and the smallest is 18, so the range = 56 − 18 = 38. The range does not tell us about the shape of the distribution, but it is a simple measure of variability that pairs well with the median.
对于数据集 23, 45, 18, 37, 56，最大值为56，最小值为18，因此极差 = 56 − 18 = 38。极差不能说明数据分布的形状，但它是一个简单的离散度量，通常与中位数搭配使用。
11. Interpreting Statistical Diagrams | 解读统计图表
Being able to read and interpret charts is as important as drawing them. When analysing any statistical diagram, first check the title, axis labels and key. Then identify what the graph is telling you: which category is the most or least frequent, what the total number of observations is, and whether any unusual patterns appear.
能够阅读和解读图表与绘制图表同样重要。分析任何统计图表时，首先要检查标题、坐标轴标签和图例。然后找出图表传达的信息：哪个类别频数最高或最低、观察总数是多少，以及是否出现任何异常模式。
Comparing data across categories is a common task. For instance, a dual bar chart can show the favourite subjects of boys and girls side by side. From such a chart you might conclude that science is more popular among boys in this sample, while art is the clear favourite for girls. Always use the numbers shown on the axis to support your conclusion.
跨类别比较数据是一项常见任务。例如，双条形图可以并列展示男生和女生最喜爱的学科。从这样的图表中，你可能得出结论：在该样本中科学在男生中更受欢迎，而艺术显然是女生的最爱。始终要借助坐标轴上显示的数字来支持你的结论。
12. Introduction to Probability | 概率入门
Probability is the branch of mathematics that deals with chance. It gives us a way to measure how likely an event is to happen. Probability is always a number between 0 and 1: 0 means an event is impossible, 1 means it is certain, and a value of 0.5 means it has an even chance of occurring.
概率是数学中研究随机现象的分支。它为我们提供了一种衡量事件发生可能性的方式。概率总是介于0到1之间的一个数字：0表示事件不可能发生，1表示事件必然发生，而0.5表示事件发生和不发生的可能性相等。
When all outcomes are equally likely, the probability of an event is given by:
当所有结果等可能时，一个事件的概率由下式给出：
P(event) = Number of favourable outcomes ÷ Total number of possible outcomes
For example, when you roll a fair six-sided die, the probability of rolling a 5 is 1 out of 6, which we write as 1/6. The probability of rolling an even number (2, 4 or 6) is 3/6, which simplifies to 1/2. Probability often uses fractions, but decimals and percentages are also acceptable.
例如，掷一个均匀的六面骰子，出现数字5的概率是1/6。掷出偶数（2, 4 或 6）的概率是 3/6，可简化为 1/2。概率常用分数表示，但小数和百分数也是可以接受的。
Expectation combines probability with the number of trials. If the probability of getting a head when tossing a coin is 1/2, and you toss it 100 times, you would expect heads about 50 times. Remember, this is an estimate – actual results can vary in the short term.
期望值将概率与试验次数结合。如果抛一枚硬币得到正面朝上的概率是1/2，那么抛掷100次，你可以期望得到大约50次正面。请记住，这只是一个估计值——短期内的实际结果可能会有所波动。
Published by TutorHao | Statistics Revision Series | aleveler.com
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