📚 Year 8 SQA Statistics: Core Knowledge Summary | Year 8 SQA 统计核心知识点梳理
Statistics is a branch of mathematics that focuses on gathering, organising, analysing and presenting data. It helps us make sense of numbers and draw meaningful conclusions from real‑world information. In Year 8 SQA Statistics, you will explore how to handle different types of data, display them using charts and graphs, calculate averages and spread, and begin to understand probability.
统计学是数学的一个分支，专注于收集、整理、分析和呈现数据。它帮助我们理解数字，并从现实信息中得出有意义的结论。在 Year 8 SQA 统计课程中，你将探索如何处理不同类型的数据、使用图表展示数据、计算平均值和离散程度，并开始理解概率。
1. Types of Data | 数据类型
Data can be divided into qualitative and quantitative types. Qualitative data describes qualities or categories that are not numerical, such as eye colour, favourite food or type of pet. Quantitative data involves numbers and measurements, for example height, test scores or temperature.
数据可分为定性数据和定量数据。定性数据描述非数字的品质或类别，例如眼睛颜色、最喜欢的食物或宠物类型。定量数据涉及数字和测量，例如身高、考试分数或温度。
Quantitative data can be further split into discrete and continuous. Discrete data can only take specific values, usually whole numbers. The number of students in a class is discrete because you cannot have half a student. Continuous data can take any value within a given range, such as weight, time or length. You could measure a time of 12.3 seconds, for example.
定量数据又可细分为离散型和连续型。离散型数据只能取特定值，通常是整数。班级中的学生人数是离散的，因为你不可能有半个学生。连续型数据在一个给定范围内可取任意值，例如重量、时间或长度。比如你可以量得 12.3 秒的时间。
2. Data Collection | 数据收集
Data is often gathered through surveys, experiments or observations. A survey uses a set of carefully designed questions to obtain information from a group of people. When designing a survey, you must avoid biased or leading questions that might influence the answers.
数据通常通过调查、实验或观察来收集。调查使用一组精心设计的问题从一群人那里获取信息。设计调查时，必须避免可能影响答案的偏见性或引导性问题。
There are different sampling methods to select a group from a larger population. In random sampling, every member of the population has an equal chance of being chosen. Stratified sampling divides the population into smaller groups (strata) based on characteristics such as age or gender, and then random samples are taken from each group in proportion to its size. These methods help ensure the sample represents the whole population fairly.
从较大的总体中选择一组人有着不同的抽样方法。在随机抽样中，总体中的每个成员都有相等的被选中机会。分层抽样根据年龄或性别等特征将总体划分为较小的组（层），然后按比例从每组中随机抽取样本。这些方法有助于确保样本公平地代表整个总体。
3. Frequency Tables | 频数表
A frequency table organises raw data by showing how often each value or group of values occurs. It typically includes a tally column and a frequency column. Tally marks are grouped in fives to make counting easier.
频数表通过显示每个值或每组值出现的频率来整理原始数据。它通常包含一个划记栏和一个频数栏。划记符号按五个一组记录，便于计数。
When dealing with a large set of continuous data, you can create a grouped frequency table. Data is divided into equal class intervals, for example 0-9, 10-19, 20-29 for test scores. The class width should remain consistent across all groups unless there is a special reason to change it. Grouped frequency tables help to summarise data and identify patterns more clearly.
处理大量连续数据时，可以创建分组频数表。数据被划分为相等的组距，例如考试分数的 0-9、10-19、20-29。除非有特殊原因，所有组的组距宽度应保持一致。分组频数表有助于总结数据并更清晰地识别模式。
4. Bar Charts and Pictograms | 条形图与象形图
A bar chart is used to display categorical data. Rectangular bars are drawn with lengths proportional to the values they represent. The bars can be drawn vertically or horizontally, and there must be gaps between them to show that the categories are separate and not continuous.
条形图用于展示类别数据。绘制的矩形条的长度与它们所代表的值成正比。条形可以垂直或水平绘制，它们之间必须有间隙，以表明各分类是独立的且不连续。
A pictogram uses small pictures or icons to represent data. Each picture stands for a fixed number of items. For example, one smiley face could represent 5 students. A half or part of a picture can be used to show smaller amounts. A clear key must always be included so that the reader understands what each symbol represents.
象形图使用小图片或图标来代表数据。每个图片代表固定数量的项目。例如，一个笑脸可能代表 5 名学生。可用半个或部分图片表示较小的数量。必须始终包含清晰的图例，以便读者理解每个符号的含义。
5. Pie Charts and Line Graphs | 饼图与折线图
A pie chart is a circular graph divided into sectors, where each sector represents a proportion of the whole data set. The size of each sector is calculated by finding the angle that corresponds to the category’s frequency. The formula is:
饼图是一个被划分为多个扇区的圆形图，每个扇区代表整个数据的一部分。每个扇区的大小通过计算该类别频数对应的角度得出。公式为：
Angle = (Category frequency / Total frequency) × 360°
Line graphs show how a quantity changes over time or in response to another variable. Data points are plotted and then joined with straight line segments. They are especially useful for displaying trends, such as the rise and fall of temperature during a day or the change in a plant’s height over several weeks.
折线图显示数量随时间或随另一变量的变化情况。先绘制数据点，然后用直线段连接。它们特别适用于展现趋势，例如一天内温度的升降或几周内植物高度的变化。
6. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph (or scatter plot) is used to investigate the relationship between two numerical variables. Each point on the graph represents a pair of values. By looking at the pattern of points, you can describe the correlation: if the points tend to rise from left to right, there is a positive correlation; if they fall from left to right, a negative correlation; and if no clear pattern exists, there is no correlation.
散点图（或散点图）用于研究两个数值变量之间的关系。图上的每个点代表一对值。通过观察点的分布模式，你可以描述相关性：如果点从左到右趋于上升，则存在正相关；如果从左到右下降，则存在负相关；如果没有明显规律，则没有相关性。
It is essential to remember that correlation does not imply causation. Just because two variables are correlated does not mean that one causes the other. You can draw a line of best fit through the points on a scatter graph to help make predictions about one variable based on the other.
务必记住相关性并不意味着因果关系。两个变量存在相关性，并不代表一个变量导致了另一个变量的变化。你可以在散点图的点之间画一条最佳拟合线，以便根据一个变量预测另一个变量。
7. Mean, Median and Mode | 平均数、中位数和众数
The three main measures of central tendency are the mean, median and mode. They summarise a data set with a single typical value.
三个主要的集中趋势量数是平均数、中位数和众数。它们用一个单一的典型值来概括一组数据。
The mean is calculated by adding all the data values together and dividing by the number of values. It is often referred to as the average.
平均数的计算方法是：将所有数据值相加，再除以数据的个数。它通常被称为平均值。
Mean = (x₁ + x₂ + … + xₙ) / n
The median is the middle value when the data is arranged in order from smallest to largest. If there is an even number of values, the median is the mean of the two middle numbers.
中位数是将数据按从小到大的顺序排列后的中间值。如果有偶数个数值，中位数则是中间两个数的平均值。
The mode is the value that appears most frequently in the data set. A set of data can have one mode, more than one mode (bimodal or multimodal), or no mode if all values occur with the same frequency.
众数是数据集中出现频率最高的值。一组数据可以有一个众数、多个众数（双众数或多众数），或者当所有值都出现相同次数时没有众数。
The table below shows a quick comparison of these three measures.
下表显示了这三种度量方式的快速比较。
| Measure | Definition | When to use |
|---|---|---|
| Mean | Sum / number of values | Best when data is evenly spread without extreme outliers |
| Median | Middle value in ordered list | Good when data is skewed or contains outliers |
| Mode | Most frequent value | Useful for categorical data or finding the most popular choice |
8. Range | 范围
The range is a simple measure of spread. It tells us how much the data varies by calculating the difference between the largest and smallest values.
范围是一种简单的离散度量。它通过计算最大值与最小值之间的差值，告诉我们数据的差异有多大。
Range = Maximum value – Minimum value
A small range means the data points are clustered closely together, while a large range indicates that the values are widely spread out. The range is especially useful for quickly comparing the consistency of two data sets.
较小的范围意味着数据点聚集得很紧密，而较大的范围则表示数值分布很广。范围对于快速比较两组数据的一致性特别有用。
9. Introduction to Probability | 概率入门
Probability measures how likely an event is to happen. It is always a number between 0 and 1 inclusive. A probability of 0 means the event is impossible, while a probability of 1 means it is certain. Probabilities can be written as fractions, decimals or percentages.
概率衡量事件发生的可能性有多大。它始终是一个介于 0 和 1 之间的数（含 0 和 1）。概率为 0 表示事件不可能发生，而概率为 1 表示事件必然发生。概率可以用分数、小数或百分比表示。
The basic formula for theoretical probability is:
理论概率的基本公式是：
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 4 is 1/6 because there is one favourable outcome and six possible outcomes. The probabilities of all possible outcomes in a scenario always add up to 1. Events that cannot happen at the same time are called mutually exclusive events.
例如，当你掷一个均匀的六面骰子时，掷出 4 的概率是 1/6，因为有一个有利结果和六个可能的结果。在某个情境中，所有可能结果的概率之和总是为 1。不可能同时发生的事件称为互斥事件。
10. Experimental vs Theoretical Probability | 实验概率与理论概率
Theoretical probability is worked out using reasoning and the assumption that all outcomes are equally likely. You do not need to perform an experiment; you can use the formula above directly.
理论概率是通过推理计算出来的，并假定所有结果的可能性均等。你不需要进行实验；可以直接使用上述公式。
Experimental probability, also known as relative frequency, is determined by actually carrying out trials or observing events. It uses real data from an experiment or historical records.
实验概率，也称为相对频率，是通过实际进行试验或观察事件来确定的。它使用实验或历史记录中的真实数据。
Experimental probability = Number of times the event occurs / Total number of trials
An interesting property is that as the number of trials increases, the experimental probability tends to get closer and closer to the theoretical probability. This is known as the Law of Large Numbers. For example, if you flip a coin many times, the proportion of heads will approach ½.
一个有趣的性质是，随着试验次数的增加，实验概率往往会越来越接近理论概率。这就是所谓的大数定律。例如，如果你多次抛掷一枚硬币，正面朝上的比例将趋近于 ½。
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