📚 Year 7 OCR Statistics: Core Knowledge Review | Year 7 OCR 统计：核心知识点梳理
Statistics is the science of collecting, organising, displaying and interpreting data. In Year 7 OCR Mathematics, students learn to work with different types of data, choose appropriate diagrams to represent them, and calculate simple averages and measures of spread. A strong grasp of these core ideas builds confidence for handling real-life information and prepares you for more advanced topics later on. This article brings together all the essential concepts you need to master, with clear explanations and examples in both English and Chinese.
统计学是收集、整理、展示和解读数据的科学。在Year 7 OCR数学课程中，学生要学习处理不同类型的数据，选择合适的统计图表来表示它们，并计算简单的平均数和离散程度。牢固掌握这些核心概念，不仅能增强你处理现实信息的信心，也为以后学习更深入的内容打下基础。本文汇集了所有你需要掌握的重要知识点，并配有中英双语清晰的解释和例子。
1. What is Statistics? | 什么是统计？
Statistics helps us make sense of numbers and information we see every day. It involves asking a question, gathering data, organising it, presenting it in tables or diagrams, and then drawing conclusions. Whether looking at sports scores, survey results or weather patterns, statistics gives us tools to describe trends and make predictions.
统计学帮助我们理解每天看到的数字和信息。它包括提出问题、收集数据、整理数据、以表格或图表形式展示数据，然后得出结论。无论是看体育比分、调查结果还是天气模式，统计学都为我们提供了描述趋势和进行预测的工具。
A key idea in statistics is that data vary. Not everything is identical, and statistics allows us to explore those differences. When we collect data, we need to be clear about what we are measuring, how we will measure it and who or what we are collecting information from. This group is called the population, and a smaller part chosen for study is a sample.
统计学的一个核心观点是数据存在差异。事物并非完全一样，统计学让我们能够探索这些差异。当我们收集数据时，需要明确我们测量的是什么、如何测量，以及从谁或什么那里获取信息。这个整体被称为总体，从中选出来研究的一小部分叫做样本。
2. Types of Data | 数据的类型
Data can be grouped into two main types: qualitative and quantitative. Qualitative data describes qualities or categories, such as eye colour, favourite subject or type of pet. This is sometimes called categorical data. Quantitative data deals with numbers and amounts, like height, mass, number of siblings or marks in a test.
数据可以分为两大类：定性数据和定量数据。定性数据描述性质或类别，例如眼睛颜色、最喜欢的科目或宠物类型。这类数据有时也称为分类数据。定量数据涉及数字和数量，比如身高、体重、兄弟姐妹的数量或测验得分。
Quantitative data can be further split into discrete and continuous data. Discrete data can only take certain values, usually whole numbers. Examples are the number of pupils in a class or the number of goals scored in a match. Continuous data can take any value within a range and is usually measured. Height, time and temperature are continuous because they can be measured to finer and finer degrees if we have precise tools.
定量数据又可以进一步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或一场比赛中的进球数。连续数据可以在一个范围内取任何值，通常是测量得出的。高度、时间和温度都是连续数据，因为如果有足够精密的工具，我们可以测出越来越精细的数值。
3. Collecting Data | 收集数据
Before we can analyse data, we need a reliable way to collect it. Common methods include surveys, questionnaires, observations and experiments. When designing a survey, questions must be clear and fair, without leading the respondent towards a particular answer. A good question is specific and offers suitable response options.
在分析数据之前，我们需要可靠的方法来收集数据。常见的方法包括调查、问卷、观察和实验。设计调查时，问题必须清晰公正，不能引导受访者给出特定的答案。好的问题是具体的，并能提供合适的回答选项。
We also need to think about how to record data efficiently. A tally chart is a simple tool where we make a mark for each observation and group marks in fives to make counting easier. The fifth mark is drawn diagonally across the first four. At the end, we count the tallies to get frequencies.
我们还需要考虑如何高效地记录数据。计数表是一种简单的工具，我们为每个观测结果划一道标记，并按五个一组分组，使得计数更容易。第五划斜穿过前四划。最后，我们数出计数符号，得到频数。
4. Frequency Tables | 频数表
A frequency table organises data into categories or intervals and shows how many times each category occurs. It is one of the first steps in summarising data. The table should have clear headings, and the total frequency should equal the number of data items collected.
频率表将数据分成类别或区间，并显示每个类别出现的次数。这是汇总数据的最初步骤之一。表格应有清晰的标题，总频数应等于所收集的数据项数量。
Below is an example of a frequency table showing the favourite colours of 30 students.
下面是一个频率表示例，显示了30名学生最喜欢的颜色。
| Colour | 颜色 | Tally | 计数 | Frequency | 频数 |
|---|---|---|
| Blue | 蓝色 | NN N | 5 |
| Red | 红色 | NN NN II | 12 |
| Green | 绿色 | NN III | 8 |
| Yellow | 黄色 | N | 1 |
| Other | 其他 | IIII | 4 |
| Total | 总计 | 30 | |
Once we have a frequency table, it is much easier to find the total number of data points, see which category is most common, and start comparing groups.
有了频率表，我们就能更容易地求出数据点的总数，看到哪个类别最常出现，并开始比较不同组别。
5. Bar Charts and Pictograms | 条形图与象形图
Bar charts use rectangular bars to represent frequencies. The height or length of each bar shows how many items are in that category. Bars should be of equal width, and there must be gaps between them for categorical data. The chart needs a title, and both axes must be labelled.
条形图用矩形条表示频数。每个条形的高度或长度显示该类别中有多少个项目。条形宽度应相等，对于分类数据，条形之间应留有空隙。图表需要标题，两条轴都必须标注。
Pictograms use small pictures or symbols to show data. Each picture represents a certain number of items. A key is essential to explain what one picture stands for. Pictograms are especially useful for making data look more engaging, but they are less precise if a value is not a whole multiple of the symbol.
象形图使用小图片或符号来展示数据。每个图片代表一定数量的项目。图例至关重要，用来解释一个图片代表多少。象形图能让数据看起来更吸引人，但如果某个数值不是整个符号的整数倍，其精确度就会降低。
6. Pie Charts | 饼图
A pie chart is a circle divided into sectors, where each sector’s angle is proportional to the frequency it represents. The whole circle stands for the total frequency. Pie charts are excellent for comparing parts of a whole, showing how data is shared among categories in percentage terms.
饼图是一个被划分为扇区的圆，每个扇区的角度与其所代表的频数成正比。整个圆代表总频数。饼图非常适合比较整体中的各个部分，以百分比的形式显示数据在各类别间的分布。
To draw a pie chart, we calculate the angle for each category using the formula:
要绘制饼图，我们需要使用以下公式计算每个类别的角度：
Angle = (Frequency ÷ Total frequency) × 360°
角度 = (频数 ÷ 总频数) × 360°
We then use a protractor to measure and draw each sector. The sectors are labelled or a key is provided so the reader knows what each slice represents. Pie charts do not show exact frequencies unless values are written, so we often use them alongside a table.
然后我们使用量角器测量并画出每个扇区。各扇区需要标注或提供图例，让读者知道每份扇区代表什么。饼图一般不显示精确的频数，除非将数值也写上，所以我们通常将它与表格一起使用。
7. Line Graphs and Time Series | 折线图与时间序列
Line graphs are used to display data that changes over time. They plot points connected by straight lines. The horizontal axis usually represents time, and the vertical axis shows the quantity being measured. Line graphs help us spot trends, such as increases, decreases or periods of no change.
折线图用于显示随时间变化的数据。它们标出点并用直线连接。横轴通常代表时间，纵轴表示被测量的量。折线图帮助我们识别趋势，例如上升、下降或保持不变的时期。
When interpreting a line graph, we read values from the axes, identify peaks and troughs, and describe the overall pattern. It is important not to connect points if the data is discrete and does not make sense in between, but for continuous data like temperature, joining the points is appropriate.
解读折线图时，我们从坐标轴读取数值，找出波峰和波谷，并描述总体模式。如果数据是离散的，在数据点之间没有实际意义，就不应把点连起来；但对于像温度这样的连续数据，连接各个点是合适的。
8. Mode and Median | 众数和中位数
Averages summarise a data set with a single central value. The mode is the value that appears most often. It is the only average that can be used for categorical, non-numerical data, such as finding the most popular car colour. A data set can have one mode, more than one mode, or no mode at all if every value occurs equally often.
平均数用一个中心值来概括一组数据。众数是指出现次数最多的值。它是唯一可用于非数值的分类数据的平均数，例如找最受欢迎的汽车颜色。一组数据可以有一个众数、多个众数，或者如果每个值出现的次数一样多，就没有众数。
The median is the middle value when the data is arranged in order of size. If there is an odd number of values, the median is the central one. If there is an even number of values, the median is found by calculating the mean of the two middle numbers. The median is not affected by extremely large or small values, so it is often used for data like income.
中位数是将数据按大小顺序排列后中间的那个值。如果是奇数个数值，中位数就是正中间的那个。如果是偶数个数值，中位数是中间两个数的平均数。中位数不受极大或极小值的影响，因此常用于收入等数据。
9. The Mean | 平均数（均值）
The mean is the sum of all the values divided by the number of values. It is the most common average, often simply called ‘the average’. Because it uses every data point, the mean can be influenced by outlier values that are very different from the rest of the data.
均值是所有数值的总和除以数值的个数。这是最常见的平均数，通常就被叫做“平均”。因为它用到了每一个数据点，所以均值会受到异常值的影响，这些异常值与其余数据差别很大。
Mean = Sum of all data values ÷ Number of values
均值 = 所有数据值的总和 ÷ 数值的个数
For example, the marks 7, 8, 9, 10 and 10 have a sum of 44 and five values, so the mean is 44 ÷ 5 = 8.8. To find the total from the mean, multiply the mean by the number of values. This reverse calculation is very useful in problem solving.
例如，分数7、8、9、10和10的总和是44，共有5个值，因此均值为44 ÷ 5 = 8.8。要从均值求出总和，则用均值乘以数值的个数。这种逆向计算在解题中非常有用。
10. The Range | 极差
While averages give a central value, the range tells us how spread out the data is. The range is the difference between the largest and the smallest values in a data set. A small range means the data is closely bunched together; a large range indicates more variation.
平均数给出数据的中心值，而极差告诉我们数据的离散程度。极差是数据中最大值和最小值的差。极差小表示数据紧密集中在一起；极差大说明数据的变动性更大。
Range = Largest value – Smallest value
极差 = 最大值 – 最小值
For the data set 3, 7, 8, 5, 12, the largest is 12, the smallest is 3, so the range is 12 – 3 = 9. The range is easy to calculate but can be distorted by a single unusually high or low value. When comparing two sets of data, we often look at both an average and the range to describe centre and spread.
对于数据集 3、7、8、5、12，最大值为12，最小值为3，因此极差等于12 – 3 = 9。极差计算简单，但可能会被一个异常高或异常低的值扭曲。比较两组数据时，我们常常同时看平均数和极差，以描述数据的集中趋势和分散程度。
11. Introduction to Probability | 概率入门
Probability is the study of chance and how likely an event is to happen. It links closely with statistics because we often use data to estimate probabilities. Probability is measured on a scale from 0 to 1, where 0 means impossible and 1 means certain. A probability of 0.5 suggests an event is equally likely to happen or not happen.
概率研究的是机会以及一个事件发生的可能性有多大。它与统计学紧密相关，因为我们常常用数据来估计概率。概率的度量范围是从0到1，其中0表示不可能，1表示必定发生。概率为0.5意味着某个事件发生与不发生的可能性相等。
In everyday language we use words like impossible, unlikely, even chance, likely and certain. These can be placed on the probability scale. For instance, rolling a 7 on a normal six-sided die is impossible (probability 0), while rolling a number less than 7 is certain (probability 1).
在日常用语中，我们会用不可能、不太可能、等可能、很可能和必定等词语。它们都可以放在概率标尺上。例如，用一颗普通的六面骰子掷出7点是不可能的（概率为0），而掷出小于7的点数则是必定发生的（概率为1）。
12. Calculating Basic Probability | 计算基本概率
When all outcomes are equally likely, the probability of an event can be found using this formula:
当所有结果等可能出现时，一个事件的概率可以用以下公式求出：
Probability = Number of favourable outcomes ÷ Total number of possible outcomes
概率 = 有利结果的数量 ÷ 所有可能结果的总数
For example, in a bag with 3 red, 2 blue and 5 green counters, the probability of picking a red counter is 3 out of 10, or 3/10. We can write this as a fraction, decimal (0.3) or percentage (30%). The probability of an event not happening is 1 minus the probability that it does happen.
比如，在一个装有3个红色、2个蓝色和5个绿色筹码的袋子里，抽到红色筹码的概率是10分之3，即3/10。我们可以将其写成分数、小数（0.3）或百分比（30%）。一个事件不发生的概率等于1减去该事件发生的概率。
We must always check that the total number of outcomes makes sense and that the events we are looking at are truly equally likely. Probability calculations like these appear in many real-life situations, from weather forecasts to playing games.
我们必须始终确保可能结果的总数合理，并且我们所考虑的事件确实是等可能的。这样的概率计算出现在许多现实生活情境中，从天气预报到玩游戏都有涉及。
Published by TutorHao | Statistics Revision Series | aleveler.com
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