📚 Year 9 CCEA Statistics: Formula & Theorem Quick Reference Handbook | Year 9 CCEA 统计：公式定理速查手册
This quick reference handbook brings together all the essential formulas, theorems and key concepts you need for Year 9 CCEA Statistics. Whether you are preparing for a class test or an end‑of‑year examination, this guide will help you quickly recall how to describe data, calculate averages, measure spread and work with probability. Each section pairs an English explanation with a Chinese translation, making it easy to follow and to revise from at any time.
本速查手册汇集了 Year 9 CCEA 统计课程所需的核心公式、定理和关键概念。无论你是在准备课堂测验还是年终考试，这本指南都能帮你快速回顾如何描述数据、计算平均数、度量离散程度以及处理概率问题。每个小节都配有中英文对照讲解，便于随时学习和复习。
1. Types of Data | 数据类型
Data can be either qualitative (describing qualities) or quantitative (measuring quantities). Quantitative data is split into discrete data, which can only take certain values (e.g. shoe size), and continuous data, which can take any value in a range (e.g. height).
数据可以分为定性数据（描述属性）和定量数据（衡量数量）。定量数据又分为离散数据（只能取特定数值，如鞋号）和连续数据（在一定范围内可取任意数值，如身高）。
2. Organising Data in Frequency Tables | 用频数表整理数据
A frequency table lists each data value together with the number of times it appears (its frequency). The total frequency is the sum of all the individual frequencies. For grouped data, data values are put into class intervals, and the class frequency tells you how many data points fall into that interval.
频数表列出每一个数据值及其出现的次数（即频数）。总频数等于所有单个频数之和。对于分组数据，将数据值归入组距内，组频数告诉你落入该区间的数据点数。
- Total frequency = sum of frequencies: Σ f
- 总频数 = 所有频数之和：Σ f
When choosing class intervals, make sure they are equal in width and cover the full range of the data without overlapping.
选择组距时，要确保各区间宽度相等，覆盖数据的全部范围且不重叠。
3. Bar Charts, Pie Charts and Pictograms | 条形图、饼图与象形图
Bar charts display categorical data using rectangular bars whose heights (or lengths) represent frequency or value. The bars are separated by small gaps to show that the categories are distinct. Pie charts use sectors of a circle to show proportions, where each sector angle is calculated as:
条形图用长方形的高（或长）来表示类别数据的频数或数值，长条之间有间隙以区分不同类别。饼图用扇区展示比例，每个扇区角度计算公式为：
Sector angle = (Frequency / Total frequency) × 360°
扇区角度 = （频数 / 总频数）× 360°
Pictograms use symbols to represent a number of items; a key tells you what one symbol stands for.
象形图用图标代表若干数量，图例标明每个图标所代表的数量。
4. Mean, Median and Mode | 平均数、中位数与众数
The mean is the sum of all data values divided by the number of values. The median is the middle value when the data are arranged in order. The mode is the value that appears most often. These three measures give a summary of the centre of a data set.
平均数是指所有数据值的总和除以数据个数。中位数是将数据按顺序排列后位于中间位置的数值。众数是出现次数最多的数值。这三个量概括了数据集的中心位置。
Formulas and steps:
公式与步骤：
Mean = (Sum of data values) / (Number of values) = Σx / n
平均数 = （数据值总和）/ （数据个数）= Σx / n
To find the median:
- Put n numbers in ascending order.
- If n is odd, median = value at position (n + 1)/2.
- If n is even, median = mean of the two middle values at positions n/2 and (n/2)+1.
确定中位数：
- 将 n 个数从小到大排列。
- 如果 n 为奇数，中位数 = 第 (n + 1)/2 个位置上的值。
- 如果 n 为偶数，中位数 = 第 n/2 和 (n/2)+1 位置上两个数的平均数。
The mode is simply the value with the highest frequency. A set may have one mode, more than one mode (bimodal or multimodal) or no mode at all.
众数就是频数最高的数值。一组数据可能有一个众数、多个众数（双众数或多众数）或完全没有众数。
5. Range and Interquartile Range (IQR) | 极差与四分位距
Spread tells you how spread out the data values are. The simplest measure of spread is the range. The interquartile range uses the lower quartile Q₁ and upper quartile Q₃ to ignore extreme values.
离散程度告诉我们数据值分散的程度。最简单的度量是极差。四分位距则使用下四分位数 Q₁ 和上四分位数 Q₃，可以忽略极端值的影响。
Range = Maximum value – Minimum value
极差 = 最大值 – 最小值
To find Q₁ and Q₃:
- First find the median (Q₂).
- Q₁ is the median of the lower half of the data (not including Q₂ if n is odd).
- Q₃ is the median of the upper half of the data.
求 Q₁ 和 Q₃：
- 先求出中位数（Q₂）。
- Q₁ 是数据下半部分的中位数（若 n 为奇数，不包括 Q₂）。
- Q₃ 是数据上半部分的中位数。
IQR = Q₃ – Q₁
IQR = Q₃ – Q₁
A small range or IQR means the data are tightly packed together; a large one means they are widely spread.
极差或四分位距小，表示数据集中紧凑；数值大则表示数据分散较开。
6. Stem‑and‑Leaf Diagrams | 茎叶图
A stem‑and‑leaf diagram keeps all the original data values while showing their distribution. Each value is split into a stem (the first digit or digits) and a leaf (the last digit). A key shows how to read the diagram, e.g. ‘3 | 4 means 34’.
茎叶图既能保留所有原始数据值，又能展示分布情况。每个数值被分成茎（前一位或多位数字）和叶（最后一位数字）。图例说明如何读图，如“3 | 4 代表 34”。
Ordered stem‑and‑leaf diagrams arrange the leaves in each row from smallest to largest, making it easy to spot the median, mode and range.
在有序茎叶图中，每一行的叶按从小到大的顺序排列，这样可以很容易地找出中位数、众数和极差。
7. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph plots two related sets of data as points on a coordinate grid. The pattern of points shows the relationship, or correlation, between the two variables.
散点图将两组相关数据作为点绘制在坐标网格上。点的分布形态显示出两个变量之间的关系，即相关性。
- Positive correlation: as one variable increases, the other tends to increase.
- Negative correlation: as one variable increases, the other tends to decrease.
- No correlation: no clear pattern.
- 正相关：一个变量增加，另一个也倾向于增加。
- 负相关：一个变量增加，另一个倾向于减少。
- 无相关：没有明显规律。
Correlation does not imply causation; a strong correlation only tells us that the two variables are associated, not that one causes the other.
相关性不意味着因果关系；强相关只告诉我们两个变量有关联，并不表示一个变量导致另一个变量发生变化。
A line of best fit can be drawn to model the trend, helping to make predictions. It should pass as close to as many points as possible, with roughly equal numbers of points on each side.
可以画一条最佳拟合线来模拟整体趋势，辅助做出预测。这条线应尽可能穿过更多的点，且两侧点数大致相等。
8. Introduction to Probability | 概率基础
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal or percentage.
概率衡量事件发生的可能性大小，取值范围从 0（不可能）到 1（一定发生）。概率可以用分数、小数或百分数表示。
Probability of an event = Number of favourable outcomes / Total number of equally likely outcomes
事件的概率 = 有利结果数 / 所有等可能结果的总数
The probability that an event does NOT happen is:
事件不发生的概率为：
P(not A) = 1 – P(A)
P(非 A) = 1 – P(A)
All possible outcomes added together give a total probability of 1. This is the ‘probability scale’ rule.
所有可能的结果的概率之和为 1。这就是“概率尺度”规则。
9. Experimental Probability and Expected Frequency | 实验概率与期望频数
Experimental probability is based on actual trials or experiments rather than on theoretical calculation:
实验概率基于实际的试验或实验，而非理论计算：
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
If an experiment is repeated a large number of times, the experimental probability gets closer to the theoretical probability (the law of large numbers).
如果实验重复足够多次，实验概率会逐渐接近理论概率（大数定律）。
Expected frequency allows you to predict how many times an event might occur:
期望频数可用于预测事件可能发生的次数：
Expected frequency = P(event) × Number of trials
期望频数 = P(事件) × 试验次数
This formula is useful for planning experiments and checking whether outcomes match predictions.
这一公式在设计实验和检验结果是否符合预测时十分有用。
10. Sample Space Diagrams | 样本空间图
A sample space is the set of all possible outcomes of an experiment. Sample space diagrams help you list outcomes systematically so that you can calculate probabilities correctly. For two independent events, a two‑way table or a grid often works well.
样本空间是一次试验所有可能出现的结果的集合。样本空间图帮助你系统性地列出结果，从而准确计算概率。对于两个独立事件，常用双向表或网格来表示。
For example, when rolling two fair six‑sided dice, the sample space has 6 × 6 = 36 equally likely ordered pairs. The probability of getting a sum of 7 can then be found by counting the pairs (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6/36 = 1/6.
例如，同时掷两个正常的六面骰子，样本空间含有 6 × 6 = 36 个等可能的有序数对。要得到点数之和为 7 的概率，可数出符合的数对：(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6/36 = 1/6。
11. Averages from Simple Frequency Tables | 从简单频数表求平均数
When data are shown in a frequency table, the mean is calculated by multiplying each value by its frequency, summing the products, and then dividing by the total frequency.
当数据以频数表呈现时，计算平均数的方法是：将每个数值乘以其频数，将乘积求和，再除以总频数。
Mean = Σ(f × x) / Σf
平均数 = Σ(f × x) / Σf
Where f is the frequency of each value x. The same method also works for grouped data using the midpoint of each class interval as an estimated value.
其中 f 是数值 x 的频数。对于分组数据，可使用各组的组中值作为估计值，用同样方法计算平均数的估算值。
Use Σf to find the total number of data points. The median and mode can also be located by examining the frequency column: the modal class is the class with the highest frequency, and the median lies at the position where the cumulative frequency first exceeds n/2.
利用 Σf 可得出数据点的总数。通过观察频数列，也可以定位中位数和众数：众数类别是频数最高的类别，中位数则位于累计频数首次超过 n/2 的位置。
12. Key Statistical Words and Notation | 常用统计词汇与符号
Below is a quick reference table for the notation and terms you are likely to meet in Year 9 CCEA Statistics. Understanding these will help you read questions accurately and present your answers clearly.
下表是 Year 9 CCEA 统计中常见的符号和术语速查，掌握它们有助于准确理解题意并清晰作答。
| Symbol / Term | Meaning | 中文含义 |
| n | Number of data values | 数据值的个数 |
| Σ | Sum of | 求和 |
| x | A data value | 一个数据值 |
| f | Frequency | 频数 |
| Q₁, Q₂, Q₃ | Lower quartile, median, upper quartile | 下四分位数、中位数、上四分位数 |
| IQR | Interquartile range = Q₃ – Q₁ | 四分位距 = Q₃ – Q₁ |
| P(A) | Probability of event A | 事件 A 的概率 |
| ~ | Approximately equal | 约等于 |
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