KS3 CCEA Statistics: Formula & Theorem Quick Reference | KS3 CCEA 统计：公式定理速查手册

📚 KS3 CCEA Statistics: Formula & Theorem Quick Reference | KS3 CCEA 统计：公式定理速查手册

This quick reference guide brings together all the essential formulas, definitions and rules you need for Key Stage 3 Statistics under the CCEA curriculum. Keep it handy as a revision tool so you can work through averages, chance, diagrams and data summaries with confidence.

这份速查手册汇集了 CCEA 课程关键阶段 3 统计所需的所有核心公式、定义和规则。将它作为复习工具放在手边，你就能自信地处理平均数、概率、图表与数据汇总问题。

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1. Mean | 算术平均数

The mean is the most common measure of average. It takes every data value into account and is sometimes called the ‘arithmetic mean’.

平均数是使用最广泛的集中量数，它考虑每一个数据值，有时也被称作“算术平均数”。

To calculate the mean, add up all the data values and then divide by the number of values.

计算平均数时，先将所有数据值相加，然后除以数据值的个数。

The formula:

x̄ = Σx / n

公式为：

x̄ = Σx / n

Here, Σx stands for the sum of all the individual data values, and n is the total number of values. The symbol x̄ is read as ‘x-bar’.

其中，Σx 代表所有单个数据值的总和，n 是数据的总个数。符号 x̄ 读作“x 拔”。

Example: For the data set 4, 7, 9, 6, 4, the sum is 4+7+9+6+4 = 30, and there are n = 5 values. The mean is 30 ÷ 5 = 6.

示例：对于数据集 4,7,9,6,4，总和为 4+7+9+6+4 = 30，共有 n = 5 个值。平均数为 30 ÷ 5 = 6。

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2. Median | 中位数

The median is the middle value when the data are arranged in order from smallest to largest. It splits the data into two equal halves.

中位数是将数据从小到大排序后正中间的那个值，它将数据分成数量相等的两部分。

When the number of data values n is odd, the median is the value at position (n+1) / 2.

当数据个数 n 为奇数时，中位数是第 (n+1)/2 个位置上的值。

When n is even, the median is the mean of the two middle values, which are at positions n/2 and (n/2)+1.

当 n 为偶数时，中位数是中间两个值的平均数，这两个值分别位于第 n/2 和 第 (n/2)+1 个位置。

Example (odd): Find the median of 11, 5, 8, 3, 9. First order the data: 3, 5, 8, 9, 11. n = 5 (odd). Position (5+1)/2 = 3rd value, which is 8.

示例（奇数）：求 11,5,8,3,9 的中位数。先排序：3,5,8,9,11。n = 5（奇数）。位置 (5+1)/2 = 3，第 3 个值是 8。

Example (even): Find the median of 20, 14, 18, 12. Order: 12, 14, 18, 20. n = 4 (even). Two middle values: 2nd = 14 and 3rd = 18. Median = (14+18)/2 = 16.

示例（偶数）：求 20,14,18,12 的中位数。排序：12,14,18,20。n = 4（偶数）。中间两个值：第 2 个 14、第 3 个 18。中位数 = (14+18)/2 = 16。

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3. Mode | 众数

The mode is the value that appears most often. A data set can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all if all values occur with the same frequency.

众数是出现次数最多的值。一组数据可以有一个众数（单峰）、多个众数（双峰或多峰），如果所有值出现次数相同则没有众数。

You find the mode simply by counting how many times each value occurs; the value with the highest frequency is the mode.

寻找众数只需数出每个值出现了多少次，出现频率最高的那个值就是众数。

Example: In the list 2, 3, 5, 3, 7, 3, 9, the value 3 occurs three times, more than any other value, so the mode is 3.

示例：在数列 2,3,5,3,7,3,9 中，数值 3 出现了三次，比其他任何值都多，因此众数是 3。

The mode is the only average that can be used with non‑numerical data, for example finding the most common colour of car.

众数是唯一可用于非数值数据的平均数，例如找出最常见的汽车颜色。

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4. Range | 极差

The range measures how spread out the data are. It is the difference between the largest value and the smallest value.

极差衡量的是数据的分散程度，它是最大值与最小值的差。

Formula:

Range = Highest value – Lowest value

公式：

极差 = 最大值 – 最小值

A small range means the data are tightly clustered; a large range shows they are widely spread.

极差小说明数据比较集中，极差大表明数据分布较广。

Example: In the data set 34, 42, 28, 51, 37, the highest value is 51 and the lowest is 28. Range = 51 – 28 = 23.

示例：在数据集 34,42,28,51,37 中，最大值是 51，最小值是 28。极差 = 51 – 28 = 23。

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5. Mean from a Frequency Table | 根据频数表求平均数

When data are presented in a frequency table, the mean is calculated using the formula:

x̄ = Σfx / Σf

当数据以频数表的形式给出时，平均数的计算公式为：

x̄ = Σfx / Σf

where f stands for frequency and x is the data value. First add an extra column for f × x, then sum those products (Σfx) and sum the frequencies (Σf).

其中 f 代表频数，x 是数据值。先增加一列计算 f × x，然后求出这些乘积的总和 Σfx 以及频数的总和 Σf。

Score (x)	Frequency (f)	f × x
2	5	10
3	8	24
4	7	28
Total	Σf = 20	Σfx = 62

In this example, mean = 62 / 20 = 3.1.

在此例中，平均数 = 62 / 20 = 3.1。

If the data are grouped, use the midpoint of each class interval as x. The formula remains the same.

如果数据是分组的，则用每个组区间的中点值作为 x，公式不变。

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6. Probability Basics | 概率基础

Probability is a measure of how likely an event is to happen. It always lies between 0 (impossible) and 1 (certain).

概率是对事件发生可能性的一种度量，其值总是在 0（不可能）与 1（必然）之间。

The basic probability formula for equally likely outcomes is:

P(A) = Number of favourable outcomes / Total number of possible outcomes

对于等可能结果，基本的概率公式为：

P(A) = 有利结果的数量 / 所有可能结果的总数

Example: A fair six‑sided dice is rolled. The probability of rolling a 4 is P(4) = 1/6.

示例：掷一个均匀的六面骰子，掷出 4 的概率为 P(4) = 1/6。

Probability can also be written as a fraction, decimal or percentage. So P(4) = 1/6 ≈ 0.167 = 16.7%.

概率也可以用分数、小数或百分数表示，因此 P(4) = 1/6 ≈ 0.167 = 16.7%。

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7. Sample Spaces | 样本空间

A sample space is the set of all possible outcomes of an experiment. Listing the sample space systematically helps you find every outcome without missing any.

样本空间是一次实验所有可能结果的集合。系统性地列出样本空间有助于你找出每一个结果，不遗漏。

You can list outcomes in a simple list, a two‑way table, or a tree diagram. For example, when flipping a coin twice, the sample space is {HH, HT, TH, TT}.

可以用简单列表、双向表格或树形图列出结果。例如，抛掷一枚硬币两次，样本空间为 {HH, HT, TH, TT}。

Once the sample space is known, the probability of an event is the number of outcomes in the event divided by the total number of outcomes in the sample space.

一旦知道了样本空间，事件的概率就等于该事件包含的结果数量除以样本空间的总结果数。

Example: From the sample space for two coin flips, P(exactly one head) = 2/4 = 1/2.

示例：从两次抛硬币的样本空间中，P(恰好一个正面) = 2/4 = 1/2。

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8. Probability of an Event Not Happening | 对立事件的概率

For any event A, the event ‘not A’ (written A’) consists of all outcomes that are not in A. The probabilities of A and A’ add up to 1.

对于任何事件 A，它的对立事件“非 A”（写作 A’）由所有不属于 A 的结果组成。A 与 A’ 的概率之和为 1。

Formula:

P(A’) = 1 – P(A)

公式：

P(A’) = 1 – P(A)

This rule is especially useful when it is easier to calculate the probability of the complement than the event itself.

当计算对立事件的概率比计算事件本身更容易时，这条规则特别有用。

Example: The probability that a football team wins is 0.4. The probability that it does not win is P(does not win) = 1 – 0.4 = 0.6.

示例：一支足球队获胜的概率是 0.4，那么它不胜的概率为 P(不胜) = 1 – 0.4 = 0.6。

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9. Expected Frequency | 期望频数

The expected frequency tells you how many times you would expect an event to happen if you repeat an experiment a set number of times.

期望频数告诉你，如果将实验重复固定次数，预期事件会发生多少次。

Formula:

Expected frequency = P(A) × Number of trials

公式：

期望频数 = P(A) × 试验次数

Example: A biased coin lands on heads with probability 0.3. If the coin is flipped 200 times, the expected number of heads is 0.3 × 200 = 60.

示例：一枚不均匀的硬币出现正面的概率为 0.3。如果抛掷这枚硬币 200 次，期望出现的正面次数为 0.3 × 200 = 60。

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10. Bar Charts & Pie Charts | 条形图与饼图

Bar charts and pie charts are used to display categorical data. In a bar chart, the height or length of each bar represents the frequency (or value) for that category.

条形图和饼图用来展示类别数据。在条形图中，每个条形的高度或长度表示该类别的频数（或数值）。

In a pie chart, each category is shown as a slice of a circle, where the angle of the slice is proportional to its frequency: angle = (frequency / total) × 360°.

在饼图中，每个类别显示为圆形的一个扇形，其圆心角与频数成正比：圆心角 = (频数 / 总数) × 360°。

When constructing a bar chart, make sure the bars are equally wide and separated by equal gaps, and label both axes clearly.

绘制条形图时，要保证所有条形宽度相等、间距均匀，并清晰地标注两根坐标轴。

For pie charts, use a protractor to measure the angles and always include a key or labels for the sectors.

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