📚 Year 8 CAIE Statistics: Formula & Theorem Quick Reference Handbook | Year 8 CAIE 统计：公式定理速查手册
This quick reference handbook brings together all the essential formulas and theorems you need for Year 8 CAIE Statistics. Keep it handy when you tackle averages, spread, probability and frequency tables – each rule is explained with clear steps and examples so you can revise efficiently and avoid mixing up concepts.
这本速查手册汇总了 Year 8 CAIE 统计所需的所有核心公式与定理。在学习平均数、离散程度、概率和频率表时，随时翻阅这本手册，每个规则都配有清晰的步骤和实例，帮助你高效复习，避免混淆概念。
1. Mean (Average) | 平均数
The mean is the sum of all data values divided by the number of values. It gives the balancing point of the data set.
平均数等于所有数据值的总和除以数据的个数，它代表数据的平衡中心点。
Mean = (Sum of all values) ÷ (Number of values) or x̄ = Σx / n
Here Σx means the sum of every observation, and n is the total count. To calculate the mean, first add all the numbers together, then divide the total by how many numbers there are. For the data set 8, 12, 15, 21, 24, the sum is 80 and there are 5 values, so the mean is 80 ÷ 5 = 16.
其中 Σx 表示所有观测值的总和，n 表示数据的总个数。计算平均数时，先将所有数字相加得到总和，再用总和除以数据个数。对于数据集 8, 12, 15, 21, 24，总和为 80，共有 5 个数值，因此平均数为 80 ÷ 5 = 16。
2. Median | 中位数
The median is the middle value when the data are arranged in order. It splits the data into two equal halves and is not affected by extreme values.
中位数是将数据按大小排序后位于中间位置的数值。它将数据分成两等份，并且不受极端值的影响。
Position of median = (n + 1) ÷ 2
If n is odd, the median is the value exactly at that position. If n is even, the median is the mean of the two values around that position. For example, with six ordered scores 4, 7, 9, 12, 15, 18, the median position is (6+1)÷2 = 3.5, so take the average of the 3rd and 4th values: (9+12)÷2 = 10.5.
如果 n 为奇数，中位数就是该位置上的数值；如果 n 为偶数，中位数就是该位置两侧两个数值的平均数。例如，六个有序分数 4, 7, 9, 12, 15, 18，中位数的位置为 (6+1)÷2 = 3.5，因此取第 3 和第 4 个值的平均数：(9+12)÷2 = 10.5。
3. Mode | 众数
The mode is the value that occurs most frequently in a data set. A set may have one mode, more than one mode (bimodal or multimodal), or no mode at all if all values appear equally often.
众数是数据集中出现频率最高的数值。一组数据可以有一个众数、多个众数（双众数或多众数），如果所有值出现次数相同也可能没有众数。
To find the mode, list the frequency of each distinct value and pick the one with the highest frequency. In the list 5, 8, 5, 12, 8, 5, 7, the value 5 appears three times, more than any other, so the mode is 5.
要找出众数，列出每个不同数值的频数，选择频数最高的那个。在数列 5, 8, 5, 12, 8, 5, 7 中，5 出现了三次，多于其他任何数，因此众数是 5。
4. Range | 极差
The range measures the spread of the data. It is the difference between the largest and smallest values. Although easy to calculate, it can be heavily influenced by outliers.
极差衡量数据的分散程度，等于最大值与最小值之差。它计算简单，但容易受异常值的影响。
Range = Maximum value – Minimum value
For the temperatures (in °C) 22, 25, 19, 28, 24, the maximum is 28 and the minimum is 19, so the range is 28 – 19 = 9°C. A larger range indicates more variability in the data.
对于温度（°C）22, 25, 19, 28, 24，最大值为 28，最小值为 19，因此极差为 28 – 19 = 9°C。极差越大，说明数据的变异性越大。
5. Mean from a Frequency Table | 从频率表中计算平均数
When data are summarized in a frequency table, the mean is calculated by multiplying each value by its frequency, summing those products, and dividing by the total frequency.
当数据以频率表的形式呈现时，计算平均数需要将每个值乘以其频数，求出这些乘积之和，再除以总频数。
Mean = (Σ f × x) ÷ (Σ f)
Here f is the frequency and x is the data value. For a table showing score 2 (frequency 5), score 3 (frequency 8), score 4 (frequency 7), Σf = 5+8+7 = 20, Σfx = 2×5 + 3×8 + 4×7 = 10 + 24 + 28 = 62. The mean is 62 ÷ 20 = 3.1.
这里 f 代表频数，x 代表数据值。例如，一张表格显示分数 2（频数 5）、分数 3（频数 8）、分数 4（频数 7），总频数 Σf = 5+8+7 = 20，乘积和 Σfx = 2×5 + 3×8 + 4×7 = 10 + 24 + 28 = 62。平均数为 62 ÷ 20 = 3.1。
6. Probability Scale and Basic Terms | 概率尺度与基本术语
Probability measures how likely an event is to happen, expressed as a number between 0 and 1 (or 0% and 100%). A probability of 0 means the event is impossible; a probability of 1 means it is certain.
概率用来衡量某个事件发生的可能性大小，用 0 到 1（或 0% 到 100%）之间的数字表示。概率为 0 表示事件不可能发生，概率为 1 表示事件必然发生。
An outcome is a possible result of an experiment. An event is a set of one or more outcomes. The sample space is the list of all possible outcomes. For a fair coin toss, the sample space is {Head, Tail}. Equally likely outcomes have the same chance of occurring.
结果是一次试验可能出现的状况。事件由一个或多个结果组成。样本空间是所有可能出现的结果的集合。抛一枚均匀硬币时，样本空间为 {正面, 反面}。等可能结果是指每个结果发生的可能性相同。
7. Theoretical Probability | 理论概率
Theoretical probability tells us what we expect to happen in the long run, based on the nature of the experiment, without actually carrying out trials. It relies on equally likely outcomes.
理论概率基于试验本身的特征，告诉我们在长期运行中预期会发生什么，而不需要实际进行多次试验。它依赖于等可能结果。
P(Event) = Number of favourable outcomes / Total number of possible outcomes
If a regular six-sided die is rolled, the probability of rolling an even number is P(even) = 3/6 = 1/2, because there are 3 favourable outcomes (2, 4, 6) out of 6 equally likely outcomes.
如果抛掷一个标准的六面骰子，掷出偶数的概率为 P(偶数) = 3/6 = 1/2，因为在 6 个等可能结果中有 3 个有利结果（2, 4, 6）。
8. Experimental Probability | 实验概率
Experimental probability is based on actual trials or observations. It is also called relative frequency and helps us estimate probability when outcomes are not equally likely or when theoretical probability cannot be determined easily.
实验概率基于实际的试验或观察数据，也称为相对频率。当结果不等可能或难以确定理论概率时，它可以帮助我们估算概率。
Experimental Probability = Number of times the event occurs / Total number of trials
If a drawing pin is dropped 200 times and lands point down 136 times, the experimental probability of landing point down is 136/200 = 0.68. As the number of trials increases, experimental probability usually gets closer to the theoretical probability (if one exists).
如果一枚图钉被抛落 200 次，其中有 136 次钉尖朝下，那么钉尖朝下的实验概率为 136/200 = 0.68。随着试验次数的增加，实验概率通常会越来越接近理论概率（如果存在的话）。
9. Complementary Events | 互补事件
The complement of an event A, written as ‘not A’, consists of all outcomes in the sample space that are not in A. The probabilities of A and its complement always add up to 1.
事件 A 的互补事件，记作“非 A”，包含样本空间中所有不属于 A 的结果。事件 A 与其互补事件的概率之和总是等于 1。
P(not A) = 1 – P(A)
This rule is extremely useful when it is easier to find the probability of the complement. If the probability of winning a game is 0.35, then the probability of not winning is 1 – 0.35 = 0.65. Complementary events are always mutually exclusive, but not all mutually exclusive events are complements.
当互补事件的概率更容易计算时，这条规则特别有用。如果赢下一场比赛的概率是 0.35，那么没有赢的概率就是 1 – 0.35 = 0.65。互补事件一定是互斥的，但并非所有互斥事件都是互补关系。
10. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. In a single roll of a die, getting a 2 and getting a 5 are mutually exclusive – both outcomes cannot occur simultaneously.
如果两个事件不能同时发生，则它们互斥。掷一次骰子时，得到 2 点和得到 5 点是互斥事件——这两个结果不可能同时出现。
For mutually exclusive events A and B: P(A or B) = P(A) + P(B)
This addition rule works only when A and B have no overlap. For the spinner with colours red, blue, green and yellow, each equally likely, the probability of getting red or blue is 1/4 + 1/4 = 1/2. Never add probabilities unless you are sure the events cannot occur together.
这条加法规则仅在 A 和 B 没有重叠时成立。对于一个等分为红、蓝、绿、黄四色的转盘，每种颜色的概率相等，则得到红色或蓝色的概率为 1/4 + 1/4 = 1/2。除非确定事件不可能同时发生，否则切勿直接相加概率。
11. Expected Frequency | 期望频数
Expected frequency estimates how many times an event will occur in a given number of trials. It uses theoretical probability (or a reliable experimental probability) and the number of trials.
期望频数用于估计在给定的试验次数中，某个事件预计会发生的次数。计算时使用理论概率（或可靠的实验概率）与试验次数。
Expected Frequency = Probability of event × Number of trials
If the probability that a biased coin lands heads is 0.4 and we toss it 250 times, we expect heads to appear 0.4 × 250 = 100 times. The expected frequency may not be a whole number, and it may not match the actual observed frequency exactly in a real experiment.
如果一枚偏心的硬币抛出正面的概率是 0.4，我们抛掷 250 次，那么正面预计会出现 0.4 × 250 = 100 次。期望频数不一定是整数，而且在实际实验中，它可能并不完全等于观察到的频数。
12. Quick Summary Table | 公式速览表
This table gathers all the key formulas at a glance, perfect for last-minute revision before a test.
这张表格将所有关键公式集中展示，非常适合考前快速回顾。
| Concept | Formula / Rule | 概念 |
|---|---|---|
| Mean | Mean = (Σx) ÷ n | 平均数 |
| Median position | (n + 1) ÷ 2 | 中位数位置 |
| Range | Max – Min | 极差 |
| Mean from frequency table | (Σ f × x) ÷ (Σ f) | 频率表平均数 |
| Theoretical probability | Favourable / Total outcomes | 理论概率 |
| Experimental probability | Occurrences / Total trials | 实验概率 |
| Complementary events | P(not A) = 1 – P(A) | 互补事件 |
| Mutually exclusive (or rule) | P(A or B) = P(A) + P(B) | 互斥事件（或规则） |
| Expected frequency | Probability × Number of trials | 期望频数 |
Keep this reference close while working on past paper questions – it will help you build speed and confidence. Remember that understanding when and why to apply each formula is just as important as memorising it.
在练习历年真题时，保持这本手册在手边，能帮助你提高解题速度和信心。记住，理解每一条公式的适用场景和原理，与记住公式本身同样重要。
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