Year 8 AQA Statistics: Quick Reference Formula & Theorem Handbook | Year 8 AQA 统计学公式定理速查手册

📚 Year 8 AQA Statistics: Quick Reference Formula & Theorem Handbook | Year 8 AQA 统计学公式定理速查手册

This compact handbook gathers the essential formulas, theorems and key ideas you will meet in Year 8 AQA Statistics. Use it to revise efficiently, check definitions and remind yourself how to work with data and probability. Every section gives you the English explanation followed by a matching Chinese version so you can master the concepts in both languages.

这本小手册汇集了 Year 8 AQA 统计学中必备的公式、定理和核心思想。你可以用它来高效复习、核对定义并回顾如何处理数据和概率。每个部分都先提供英文解释，随后给出对应的中文版本，帮助你用双语掌握这些概念。

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1. Types of Data | 数据类型

Data can be split into two broad types: qualitative and quantitative. Qualitative data (also called categorical data) describes qualities or categories, such as eye colour, favourite subject or car brand. Quantitative data measures quantities with numbers, like height, test marks or the number of pets.

数据可分为两大类：定性数据和定量数据。定性数据（也称分类数据）描述的是性质或类别，比如眼睛颜色、最喜欢的科目或汽车品牌。定量数据用数字来计量，如身高、考试分数或宠物数量。

Quantitative data is further divided into discrete and continuous. Discrete data can only take certain separate values (usually whole numbers) – for example, the number of students in a class. Continuous data can take any value within a range and is measured, such as time (3.52 seconds) or mass (1.6 kg).

定量数据又分为离散数据和连续数据。离散数据只能取某些分开的值（通常是整数），例如一个班的学生人数。连续数据可以在一个范围内取任意值，是测量得到的，比如时间（3.52 秒）或质量（1.6 kg）。

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2. Mean, Median and Mode | 平均数：均值、中位数与众数

The three most common averages are the mean, median and mode. Each one summarises a data set with a single typical value but is calculated differently and suits different situations.

最常见的三种平均数是均值、中位数和众数。它们各自用一个典型值概括整个数据集，但计算方式不同，适用的场合也不同。

The mean (often just called the average) is found by adding up all the data values and dividing by the number of values.

均值（通常直接称作平均数）是把所有数据值相加，然后除以数据的个数得到。

Mean = Σx ÷ n

平均数 = Σx ÷ n

where Σx means the sum of all the data values and n is the number of values.

其中 Σx 代表所有数据值的总和，n 是数据的个数。

The median is the middle value when the data is arranged in order. For an odd number of values, it is the value at position (n+1)÷2. If there is an even number of values, the median is the mean of the two middle values.

中位数是将数据从小到大排列后位于中间的值。如果数据个数为奇数，中位数在 (n+1)÷2 的位置；如果数据个数为偶数，中位数则是中间两个值的平均数。

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

众数（众数值）是出现次数最多的值。一个数据集可以有一个众数、多个众数（双峰、多峰），也可以没有众数（当所有值出现次数相同时）。

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3. Range | 极差

The range is a simple measure of spread (how spread out the data is). It only uses the largest and smallest values.

极差是一个简单的离散程度（数据的分散程度）指标，只用到最大值和最小值。

Range = Largest value − Smallest value

极差 = 最大值 − 最小值

A larger range tells you the data is more spread out; a smaller range means the values are more clustered together. The range is easy to calculate but can be heavily affected by a single extremely high or low value (an outlier).

极差越大，说明数据越分散；极差越小，数据越集中。极差计算简便，但很容易受到个别极高或极低数值（离群值）的影响。

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4. Calculating Averages from Frequency Tables | 从频率表计算平均数

When data is given in a frequency table, each value (x) appears with a certain frequency (f). To find the mean, you multiply each value by its frequency, add up those products and then divide by the total frequency.

当数据以频率表形式给出时，每个数据值 (x) 都对应一个频数 (f)。要求平均数，需要把每个值乘以它的频数，将所有这些乘积相加，再除以总频数。

Mean = Σ(f × x) ÷ Σf

平均数 = Σ(f × x) ÷ Σf

If the data is grouped into class intervals, you cannot see the exact original values. In this case, use the midpoint of each class interval as your x. The formula remains the same.

如果数据被归入组段中，你无法看到确切的原始数值。这时就用每个组段的组中点作为 x，计算公式不变。

Example: A frequency table shows scores 1 (frequency 3), 2 (frequency 5), 3 (frequency 2). Σf = 10. Σ(f × x) = (1×3) + (2×5) + (3×2) = 3+10+6 = 19. Mean = 19 ÷ 10 = 1.9.

例子：一个频率表显示得分 1（频数 3）、2（频数 5）、3（频数 2），Σf = 10。Σ(f × x) = (1×3)+(2×5)+(3×2)=3+10+6=19。平均数 = 19 ÷ 10 = 1.9。

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5. Statistical Diagrams: Bar Charts, Pie Charts & Scatter Graphs | 统计图：条形图、饼图与散点图

A bar chart uses bars of equal width to show the frequency of categories. The height of each bar represents the frequency. Bar charts are for categorical or discrete data, and the bars do not touch each other.

条形图用等宽的条形来表示各类别的频数。每个条形的高度代表频数。条形图适用于分类数据或离散数据，且条形之间不接触。

In a pie chart, a circle is divided

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