📚 Year 8 CCEA Statistics: Formula & Theorem Quick Reference Guide | Year 8 CCEA 统计：公式定理速查手册
Welcome to your quick reference guide for Year 8 CCEA Statistics. This handbook gathers all the essential formulas, theorems, and key concepts you need to master at this stage. From calculating averages and range to interpreting scatter graphs and probability, each section is presented with clear statements and worked examples. Use this guide alongside your class notes and revision to build confidence in statistical thinking.
欢迎使用 Year 8 CCEA 统计学科快速参考手册。本手册汇集了现阶段需要掌握的所有基本公式、定理和核心概念。从计算平均数与极差，到解读散点图与概率，每个部分都以清晰的陈述和示例呈现。结合课堂笔记和复习使用，帮助你建立统计思维的自信心。
1. Types of Data | 数据类型
In statistics, we classify data into types to decide how to display and analyse it. Data can be categorical (qualitative) or numerical (quantitative). Numerical data can be further split into discrete and continuous data.
在统计学中，我们需要先对数据进行分类，以便选择正确的展示和分析方法。数据可以是分类的（定性数据）或数值的（定量数据）。数值数据又可以进一步细分为离散数据和连续数据。
Qualitative data describes qualities or categories that cannot be measured numerically, such as eye colour, favourite subject, or car brands.
定性数据描述的是无法用数字衡量的性质或类别，例如眼睛颜色、最喜欢的学科或汽车品牌。
Quantitative data consists of numbers that can be measured or counted. Discrete data can only take certain values, usually whole numbers (e.g. number of students in a class). Continuous data can take any value within a range, including decimals (e.g. height, weight, time).
定量数据由可以测量或计数的数字组成。离散数据只能取某些特定值，通常是整数（例如教室里的学生人数）。连续数据可以在一定范围内取任意值，包含小数（例如身高、体重、时间）。
2. Mean, Median, Mode | 平均数、中位数、众数
These are measures of central tendency, telling us where the centre of a data set lies. Each is calculated differently and gives a slightly different perspective.
这些是集中趋势的度量，告诉我们数据集的中心在哪里。每一种计算方式不同，并能提供略微不同的视角。
Mean: Add up all the values and divide by the number of values.
平均数：将所有数值相加，再除以数值的个数。
Mean = (Sum of all data values) ÷ (Number of data values)
平均数 = (所有数据值之和) ÷ (数据值的个数)
Median: Order the data from smallest to largest. The median is the middle value. If there are two middle values, median is the mean of those two.
中位数：将数据从小到大排序。中位数就是中间的那个值。如果有两个中间值，则取这两个数的平均数。
Mode: The value that appears most often. A data set can have one mode, more than one mode, or no mode at all.
众数：出现次数最多的数值。一组数据可以有一个众数、多个众数，或者没有众数。
Example: For the data set 3, 5, 5, 7, 9, the mean is (3+5+5+7+9)÷5 = 5.8, median is 5, and mode is 5.
示例：对于数据集 3、5、5、7、9，平均数为 (3+5+5+7+9)÷5 = 5.8，中位数为 5，众数为 5。
3. Range | 极差
Range is a simple measure of spread, showing how spread out the data values are. It is the difference between the largest and smallest values.
极差是一种简单的离散程度度量，表示数据值的分布范围。它是最大值与最小值的差。
Range = Largest value − Smallest value
极差 = 最大值 − 最小值
A larger range indicates greater variability in the data, while a small range suggests the data values are closely packed together.
极差越大，表示数据的变化程度越大；极差小则说明数据值较为集中。
4. Frequency Tables | 频率表
A frequency table organises data by showing how often each value or category occurs. It helps to summarise large data sets and calculate the mean from grouped data.
频率表通过显示每个值或类别出现的次数来整理数据。它有助于汇总大型数据集，并能从分组数据中计算平均数。
To find the mean from a frequency table, multiply each value by its frequency, add these products, then divide by the total frequency.
要从频率表中计算平均数，先将每个值乘以其频率，再将所有乘积相加，最后除以总频率。
Mean = (Sum of (Value × Frequency)) ÷ (Total Frequency)
平均数 = (值 × 频率 之和) ÷ (总频率)
For grouped frequency tables, use the midpoint of each class interval as the value.
对于分组频率表，使用每个组区间的中点作为该组的代表值。
5. Bar Charts and Pie Charts | 条形图与饼图
Bar charts and pie charts are common ways to display categorical data. Bar charts use the height of bars to represent frequency, while pie charts use sectors of a circle.
条形图和饼图是展示分类数据的常用方式。条形图用条形的高度表示频率，饼图则用扇区表示各部分的比例。
In a bar chart, each category has a bar of equal width. The vertical axis shows frequency. Gaps between bars remind us that the data are categorical.
在条形图中，每个类别拥有相同宽度的条形。纵轴显示频率。条形之间的空隙提醒我们数据是分类性质的。
To draw a pie chart, we calculate the angle for each sector using the formula:
绘制饼图时，我们需要使用以下公式计算每个扇区的角度：
Angle = (Frequency of category ÷ Total frequency) × 360°
角度 = (该类别频率 ÷ 总频率) × 360°
Always check that the angles sum to 360° and label each sector clearly.
始终要检查所有角度之和是否等于 360°，并为每个扇区清晰地贴上标签。
6. Scatter Graphs and Correlation | 散点图与相关性
Scatter graphs plot paired numerical data to see if there is a relationship between two variables. Each point represents a pair of values (x, y).
散点图将成对的数值数据绘制出来，以观察两个变量之间是否存在关系。每个点代表一对数值 (x, y)。
Correlation describes the direction and strength of the relationship:
相关性描述这种关系的方向和强度：
Positive correlation: as one variable increases, the other tends to increase. The points slope upwards.
正相关：当一个变量增大时，另一个变量也趋向增大。数据点呈向上倾斜趋势。
Negative correlation: as one variable increases, the other tends to decrease. The points slope downwards.
负相关：当一个变量增大时，另一个变量趋向减小。数据点呈向下倾斜趋势。
No correlation: there is no clear pattern; the points are scattered randomly.
无相关：没有明显的规律；数据点随机分布。
Correlation does not imply causation — just because two variables move together does not mean one causes the other.
相关性并不意味着因果关系——两个变量同步变化，并不代表一个导致了另一个。
7. Introduction to Probability | 概率基础
Probability measures how likely an event is to happen. It is expressed as a number between 0 (impossible) and 1 (certain), or as a fraction, decimal, or percentage.
概率衡量某事件发生的可能性大小。它用一个介于 0（不可能）和 1（必然）之间的数字表示，也可以用分数、小数或百分数来表达。
The probability scale:
概率尺度：
0 ≤ P(event) ≤ 1
For equally likely outcomes, the probability of an event is:
对于等可能的结果，某事件的概率为：
P(event) = 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
Example: Rolling a fair six-sided die, the probability of rolling an even number is 3/6 = 1/2 = 0.5.
示例：投掷一枚均匀的六面骰子，掷出偶数的概率是 3/6 = 1/2 = 0.5。
8. Experimental Probability and Expected Frequency | 实验概率与期望次数
When we cannot assume equally likely outcomes, we estimate probability using data from experiments or surveys. This is called experimental probability or relative frequency.
当我们无法假设结果等可能时，我们会通过实验或调查的数据来估计概率。这被称为实验概率或相对频率。
Experimental Probability = Number of times the event occurs ÷ Total number of trials
实验概率 = 事件发生的次数 ÷ 试验总次数
As the number of trials increases, the experimental probability tends to get closer to the theoretical probability (the Law of Large Numbers).
随着试验次数的增加，实验概率会趋向于接近理论概率（大数定律）。
We can also use theoretical probability to predict how many times an event might occur in a given number of trials, called the expected frequency.
我们还可以利用理论概率来预测在给定试验次数中某事件可能发生的次数，这称为期望次数。
Expected Frequency = Probability of event × Number of trials
期望次数 = 事件发生的概率 × 试验次数
Example: If a coin is flipped 200 times, the expected frequency of heads is 0.5 × 200 = 100.
示例：如果抛硬币 200 次，出现正面的期望次数为 0.5 × 200 = 100。
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