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

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

This handbook brings together the key formulas, definitions and statistical theorems you will meet in the KS3 Cambridge Statistics course. Use it for quick checks, last-minute revision and to build your confidence with numerical and graphical methods.

本手册汇集了 KS3 剑桥统计课程中你会遇到的关键公式、定义和统计定理。既可用于快速查阅、考前速览，也能帮助你建立对数值和图形方法的信心。

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1. Mean | 平均数

The mean (also called the arithmetic average) is the most common measure of central tendency. To find the mean, add all the data values together and divide by the total number of values.

Mean = (x₁ + x₂ + … + xₙ) / n

In the formula, x₁, x₂, …, xₙ represent each individual data value, and n is the number of data items. You can also think of the mean as the ‘fair share’ when the total is spread evenly among all the items.

平均数是最常用的集中趋势度量。计算方法是将所有数据值相加，再除以数据的总个数。在上面的公式里，x₁, x₂, …, xₙ 代表每一个数据值，n 代表数据的总个数。你也可以把平均数想象成将总量平均分配给每一个个体之后的“公平份额”。

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2. Median | 中位数

The median is the middle value of an ordered data set. It splits the data into two equal halves and is not affected by extremely high or low values.

中位数是一组排序数据中位于正中间的值，它把数据分成数量相等的两部分，并且不受极端高值或低值的影响。

To locate the median position, use the formula:

Median position = (n + 1) / 2

If n is odd, the median is the value at that position. If n is even, the median is the mean (average) of the two middle values at positions n/2 and (n/2)+1.

中位数的位置可以通过公式 (n+1)/2 找到。如果数据总数 n 是奇数，中位数就是该位置上的数值；如果 n 是偶数，中位数是位于第 n/2 和 (n/2)+1 位置的两个中间数值的平均数。

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3. Mode | 众数

The mode is the value that appears most often in a data set. A set of data 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.

众数是数据集中出现次数最多的数值。一组数据可能只有一个众数，也可能有多个众数（双众数或多众数），如果所有数值出现次数相同，则没有众数。

Unlike the mean and median, the mode can be used with non-numerical (categorical) data, such as favourite colours or types of pet.

和平均数、中位数不同，众数可以用于非数值（分类）数据，比如最受欢迎的颜色或宠物种类。

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4. Range | 极差

The range is a simple measure of spread. It tells you how far the data extends from the smallest value to the largest value.

极差是一种简单的离散程度度量，它告诉我们数据从最小值到最大值的分布范围。

Range = Maximum value – Minimum value

A larger range means the data are more spread out; a smaller range means the values are closer together. Always identify the largest and smallest numbers before subtracting.

极差越大，说明数据越分散；极差越小，说明数值越集中。计算时一定要先找出最大值和最小值，再做减法。

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5. Frequency Tables | 频数表

A frequency table organises raw data by showing each distinct value (or group) and how many times it occurs (frequency). You can calculate the mean directly from a frequency table using the weighted formula:

Mean = Σ(f × x) / Σf

where f is the frequency of each value and x is the data value itself. Σf gives the total number of data items. If the table uses grouped data (class intervals), use the midpoint of each interval as x.

频数表用每个不同数值（或分组）及其出现次数整理原始数据。你可以直接从频数表计算平均数，使用加权公式：Mean = Σ(f × x) / Σf，其中 f 表示每个数值的频数，x 表示该数值，Σf 表示数据总个数。如果表格用的是分组数据，就用每一组的组中值作为 x。

• Multiply each value by its frequency: f × x
• Add all these products to get Σ(f × x)
• Add all frequencies to get Σf
• Divide the two sums

• 每个数值乘以它的频数：f × x
• 把所有乘积相加得到 Σ(f × x)
• 把所有频数相加得到 Σf
• 将两个总和相除

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6. Bar Charts and Pictograms | 条形图与象形图

Bar charts represent categorical or discrete data using rectangular bars. The length (or height) of each bar is proportional to the frequency it represents. Bars are drawn with equal width and equal gaps between them.

条形图用长方形条块表示分类数据或离散数据，条块的长度（或高度）与对应的频数成正比。条形之间宽度相等，并且间隔相同。

A pictogram uses pictures or symbols to represent data. Each symbol stands for a fixed number of items. You must include a key (e.g. one heart = 5 people) so that readers can interpret the chart correctly.

象形图用图形或符号来表示数据，每个符号代表固定数量的单位。必须配有图例（例如一个爱心代表5人），以便读者正确理解图表。

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7. Pie Charts | 饼图

A pie chart is a circular diagram divided into sectors. Each sector’s angle is proportional to the frequency of the category it represents. The key relationship that links frequency to angle is:

Sector angle = (Category frequency / Total frequency) × 360°

饼图是将圆形分割成若干扇形的统计图，每个扇形的圆心角与该类别频数所占的比例成正比。连接频数与角度的核心公式是：扇形角度 = (类别频数 ÷ 总频数) × 360°。

Follow these steps to construct a pie chart:

• Find the total frequency.
• For each category, calculate angle = (frequency / total) × 360°.
• Draw a circle and use a protractor to measure and draw each sector.
• Label each sector or provide a clear legend.

绘制饼图的步骤：

• 求出总频数。
• 对每一个类别，计算角度 = (频数 / 总数) × 360°。
• 画一个圆，用量角器量出并绘制每个扇形。
• 为每个扇形标注名称或添加清晰的图例。

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8. Probability Basics | 概率基础

Probability measures how likely an event is to occur. It is always a number between 0 (impossible) and 1 (certain). The basic probability formula is:

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

概率用来衡量某个事件发生的可能性，取值范围总是在 0（不可能发生）到 1（必然发生）之间。基本的概率公式为：P(事件) = 有利结果的数量 / 所有可能结果的总数。

For any event A, the probability that A does not happen is written P(not A) = 1 − P(A). This is called the complement rule.

对于任意事件 A，A 不发生的概率记为 P(非 A) = 1 − P(A)，这叫做补集规则。

If all outcomes are equally likely, you can list the sample space and use the formula directly. Probabilities can be expressed as fractions, decimals or percentages.

如果所有可能出现的结果可能性相等，你可以列出样本空间，然后直接使用公式。概率可以用分数、小数或百分数表示。

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9. Sample Space | 样本空间

The sample space is the set of all possible outcomes of an experiment. Listing the sample space systematically helps you count outcomes accurately. For two events, you can use a table or a tree diagram.

样本空间是指一次试验所有可能结果的集合，系统性地列出样本空间可以帮助你准确计数。涉及两个事件时，可以使用表格或树状图。

When two independent events occur one after the other, the total number of outcomes equals the product of the number of outcomes for each event.

Total outcomes = m × n

where m and n are the numbers of outcomes for the first and second events respectively.

当两个独立事件接连发生时，最终结果的总数等于两个事件各自结果数的乘积：总结果数 = m × n，其中 m 和 n 分别是第一个事件和第二个事件的结果数。

For example, rolling a die and flipping a coin gives 6 × 2 = 12 equally likely outcomes.

比如，掷一枚骰子并抛一枚硬币，就有 6 × 2 = 12 种等可能的结果。

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10. Stem-and-Leaf Diagrams | 茎叶图

A stem-and-leaf diagram is a way of displaying numerical data while keeping every original value visible. Each number is split into a ‘stem’ (often the tens digit) and a ‘leaf’ (the units digit). The leaves are ordered from smallest to largest.

茎叶图既能展示数据分布，又能保留每一个原始数据。每个数值被分成“茎”（通常是十位数）和“叶”（个位数），并将叶从小到大排列。

To read the median from a stem-and-leaf plot, count to the middle value using the ordered leaves. If the total number of leaves is n, the median position is (n+1)/2, just as before.

要从茎叶图中找到中位数，数到排列有序的叶的中间位置即可。如果叶的总数是 n，中位数的位置仍然是 (n+1)/2，和之前一样。

Stem-and-leaf diagrams also make it easy to spot the mode (the leaf that appears most often) and to compare two data sets by drawing back-to-back plots.

茎叶图还可以方便地找到众数（出现次数最多的叶），并通过背靠背茎叶图比较两组数据。

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11. Scatter Graphs and Correlation | 散点图与相关性

A scatter graph (or scatter plot) shows the relationship between two sets of numerical data. Each point on the graph corresponds to one paired observation (x, y).

散点图展示两组数值数据之间的关系，图上每一个点对应一对观测值 (x, y)。

From the pattern of points, you can describe the correlation:

• Positive correlation: as x increases, y tends to increase.
• Negative correlation: as x increases, y tends to decrease.
• No correlation: there is no obvious pattern.

根据点的分布模式可以描述相关性：

• 正相关：x 增大时 y 也倾向于增大。
• 负相关：x 增大时 y 倾向于减小。
• 无相关：看不出明显的变化模式。

Correlation does not imply causation. Even if two variables show a strong correlation, it does not prove that one causes the other. Always consider other factors.

相关不代表因果关系。即使两个变量表现出很强的相关性，也不能证明一个导致了另一个，要始终考虑其它可能的因素。

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12. Key Statistical Vocabulary | 关键统计词汇

Understanding technical terms is essential for answering exam questions precisely. The table below lists some important KS3 statistics vocabulary with their definitions.

准确理解专业术语对于在考试中准确作答至关重要。下表列出了一些重要的 KS3 统计词汇及其定义。

English Term	中文术语	Definition / 定义
Population	总体	The entire group being studied.
Sample	样本	A subset of the population used to represent the whole.
Discrete data	离散数据	Data that can only take specific, separate values (e.g. number of students).
Continuous data	连续数据	Data that can take any value within a range (e.g. height, time).
Primary data	一手数据	Data you collect yourself for a specific purpose.
Secondary data	二手数据	Data that someone else has already collected.
Outlier	异常值	A value that lies far away from the rest of the data.
Hypothesis	假设	A testable statement about what you expect to find.

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