Year 7 Cambridge Statistics: Formula & Theorem Quick Reference Handbook | 七年级剑桥统计：公式定理速查手册

📚 Year 7 Cambridge Statistics: Formula & Theorem Quick Reference Handbook | 七年级剑桥统计：公式定理速查手册

Welcome to your quick reference guide for Year 7 Cambridge Statistics. This handbook summarises all the essential formulas, definitions, and theorems you need to confidently tackle data handling questions. Keep it handy when solving exercises or revising for assessments.

欢迎使用七年级剑桥统计速查手册。本手册汇总了所有你需要熟练掌握的核心公式、定义和定理，帮助你自信地解答数据处理问题。在做练习或备考复习时，随时查阅。

1. Mean | 平均数

The mean is the average of a set of numbers. To calculate it, add up all the values and then divide by the total number of values. The mean is sometimes called the arithmetic mean and is very sensitive to extremely high or low values (outliers).

平均数是一组数据的平均值。计算方法是把所有数值加起来，再除以数值的总个数。平均数有时也称为算术平均数，对极大或极小值（异常值）非常敏感。

Mean = (Sum of all values) ÷ (Number of values)

平均数 = (所有数值之和) ÷ (数值个数)

Example: Find the mean of 4, 8, 6, 5, 3. First, 4+8+6+5+3 = 26. Then divide 26 by 5. Mean = 5.2.

示例：求 4、8、6、5、3 的平均数。先算 4+8+6+5+3=26，然后 26÷5，平均数为 5.2。

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2. Median | 中位数

The median is the middle value when a data set is ordered from smallest to largest. If there is an odd number of values, the median is the single middle number. If there is an even number of values, the median is the mean of the two middle numbers.

中位数是将数据从小到大排列后位于中间位置的数值。如果数据个数为奇数，中位数就是正中间的那个数；如果数据个数为偶数，则取中间两个数的平均数作为中位数。

Steps to find the median: (1) Arrange the numbers in ascending order. (2) Count how many values there are, n. (3) If n is odd, median = the ((n+1)/2)th value. If n is even, median = mean of the (n/2)th and ((n/2)+1)th values.

求中位数的步骤：(1) 将数字从小到大排列。(2) 数出有多少个值，记为 n。(3) 若 n 为奇数，中位数 = 第 (n+1)/2 个值。若 n 为偶数，中位数 = 第 n/2 个值与第 (n/2)+1 个值的平均数。

Example odd: 3, 5, 7, 8, 9 → n=5, median = 3rd value, which is 7.

奇数示例：3, 5, 7, 8, 9 → n=5，中位数是第 3 个值，即 7。

Example even: 2, 4, 6, 8 → n=4, middle two are 4 and 6. Mean = (4+6)÷2 = 5. Median = 5.

偶数示例：2, 4, 6, 8 → n=4，中间两个数是 4 和 6，平均数为 (4+6)÷2=5。中位数 = 5。

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3. Mode | 众数

The mode is the value that appears most frequently in a data set. A set of data can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode if all values occur equally often. The mode is the only average that can be used with non-numerical data, such as colours or categories.

众数是一组数据中出现次数最多的值。一组数据可以有一个众数（单峰）、多个众数（双峰或多峰），或者如果没有重复则没有众数。众数是唯一可以用于非数值数据（如颜色或类别）的平均数。

Example: In the list 2, 3, 3, 5, 6, 3, 7, the number 3 appears three times, more than any other, so mode = 3.

示例：在数列 2, 3, 3, 5, 6, 3, 7 中，数字 3 出现了三次，次数最多，因此众数 = 3。

For data grouped in a frequency table, the mode is the value with the highest frequency.

对于频率表中的分组数据，众数是频率最高的那个值。

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4. Range | 范围（极差）

The range measures the spread of a data set. It is the difference between the largest and smallest values. A larger range indicates more variability, while a smaller range suggests the data are more consistent. Range is extremely simple but can be influenced heavily by a single outlier.

范围（极差）衡量一组数据的分散程度。它是最大值与最小值之差。范围越大，表示数据变化越大；范围越小，表示数据越集中。范围计算非常简单，但极易受单个异常值的影响。

Range = Largest value − Smallest value

范围 = 最大值 − 最小值

Example: Data set: 12, 7, 22, 5, 16. Largest = 22, smallest = 5. Range = 22 − 5 = 17.

示例：数据集：12, 7, 22, 5, 16。最大值=22，最小值=5。范围 = 22 − 5 = 17。

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5. Frequency Tables | 频率表

A frequency table organises data by recording how often each value (or category) occurs. The ‘frequency’ column shows the count. The sum of all frequencies must equal the total number of data items. Frequency tables make it easier to find the mode, calculate the mean of large data sets, and construct graphs.

频率表通过记录每个值（或类别）出现的次数来整理数据。“频率”一栏显示计数。所有频率之和必须等于数据的总个数。频率表有助于更容易地找到众数、计算大数据集的平均数以及绘制图表。

To find the mean from a frequency table: multiply each value by its frequency, sum these products, then divide by the total frequency.

从频率表求平均数的方法：将每个值乘以其频率，把这些乘积相加，再除以总频率。

Mean from frequency table = Σ(value × frequency) ÷ Σ(frequency)

频率表平均数 = Σ(数据值 × 频率) ÷ Σ(频率)

Example: Value 2 occurs 4 times, value 5 occurs 6 times. Sum = 2×4 + 5×6 = 8+30=38. Total frequency = 10. Mean = 3.8.

示例：数值 2 出现 4 次，数值 5 出现 6 次。总和 = 2×4 + 5×6 = 8+30=38，总频率=10，平均数=3.8。

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6. Bar Charts | 条形图

A bar chart uses rectangular bars to represent data. The height (or length) of each bar is proportional to the frequency or value it represents. Bars must be of equal width and separated by equal gaps. Bar charts are ideal for comparing distinct categories or discrete data.

条形图用矩形条来表示数据。每个条的高度（或长度）与其所代表的频率或数值成比例。条形必须等宽，且条与条之间的间隔相等。条形图非常适合比较不同的类别或离散数据。

When drawing a bar chart: label the axes (categories on horizontal, frequency on vertical), choose a suitable scale, and ensure all gaps are uniform. A bar-line chart is a variation where lines are drawn instead of thick bars; the same rules apply.

绘制条形图时：标注坐标轴（横轴为类别，纵轴为频率），选择合适的刻度，并确保间隔一致。条形折线图是一种变体，用线条代替粗条；规则相同。

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7. Pictograms | 象形图

A pictogram uses symbols or pictures to represent a fixed number of items. Each picture stands for a certain value, and part-symbols can represent fractions of that value. A key must be included to explain what one symbol represents. Pictograms make data visually attractive but can be less precise.

象形图使用符号或图片来表示固定数量的项目。每个图片代表某个值，部分符号可以表示这个值的分数。必须包含图例，说明每个符号代表多少。象形图使数据在视觉上更具吸引力，但可能不够精确。

Example: If one apple icon represents 2 students, then 3 and a half apples represent 7 students. Always check the key before interpreting.

示例：如果一个苹果图标代表 2 名学生，那么 3 个半苹果代表 7 名学生。解读时务必先查看图例。

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8. Pie Charts | 饼图

A pie chart represents data as slices of a circle. The whole circle (360°) corresponds to the total frequency. Each slice angle is proportional to the frequency of the category. Pie charts show proportions clearly and are best used when there are not too many categories.

饼图用圆形的扇形来表示数据。整个圆（360°）对应总频率。每个扇形的角度与该类别的频率成比例。饼图能清楚地显示比例，最适合类别不太多的情况。

Formula to find the angle for a sector:

计算扇形角度的公式：

Angle = (Frequency of category ÷ Total frequency) × 360°

角度 = (该类别频率 ÷ 总频率) × 360°

After calculating each angle, draw the circle, measure angles with a protractor, and label each sector. Sum of all angles must be 360°.

计算完每个角度后，画圆，用量角器测量角度，并给每个扇形贴上标签。所有角度之和必须为 360°。

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9. Data Collection and Types | 数据收集与类型

Data can be collected through surveys, observations, experiments, or existing records. It is essential to distinguish between qualitative (categorical) data and quantitative (numerical) data. Quantitative data can be discrete (countable, e.g. number of pets) or continuous (measurable, e.g. height).

数据可以通过调查、观察、实验或现有记录收集。区分定性（分类）数据和定量（数值）数据非常重要。定量数据可以是离散的（可数的，如宠物数量）或连续的（可测量的，如身高）。

Primary data is collected directly by the researcher for a specific purpose. Secondary data is data taken from existing sources, like websites, books, or databases. Both must be evaluated for reliability and relevance.

一手数据是研究者为特定目的直接收集的。二手数据是从现有来源（如网站、书籍或数据库）获取的。两者都需要评估其可靠性和相关性。

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10. Interpreting Graphs and Charts | 图形与图表的解读

Reading data displays accurately is a key skill. Always check title, axis labels, scale, and key. Look for trends, highest and lowest values, and comparisons. Ask: What does this graph actually show? Misleading scales or missing labels can distort the message.

准确地读取数据展示是一项关键技能。务必检查标题、坐标轴标签、刻度和图例。寻找趋势、最大值和最小值，并进行比较。问问自己：这张图表到底说明了什么？误导性的刻度或缺失的标签可能歪曲信息。

When describing a bar chart or pictogram, state which category is the most or least frequent, and note any interesting differences. For a pie chart, discuss the largest and smallest proportions. Use the language of comparison – ‘more than’, ‘twice as many’, ‘half of’, etc.

描述条形图或象形图时，指出哪个类别最频繁或最少，并注意任何有趣的差异。对于饼图，讨论最大和最小的比例。使用比较性语言——“多于”、“两倍”、“一半”等。

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11. Common Pitfalls and Quick Tips | 常见误区与速查提示

Common mistakes include: forgetting to order data before finding the median; confusing mode with frequency; using a scale that does not start from zero in a bar chart (which can mislead); and miscalculating the mean from a frequency table by dividing by the number of distinct values instead of total frequency.

常见错误包括：求中位数前忘记将数据排序；混淆众数与频率；条形图采用不从零开始的刻度（可能误导）；以及从频率表求平均数时，错用不同值的个数去除，而不是除以总频率。

Quick checks: mean should generally lie between the smallest and largest values. Sum of frequencies in a table must equal the stated total. Pie chart angles must add to 360°. Always include a key in a pictogram. When comparing two data sets, use mean and range together to describe central tendency and spread.

速查：平均数通常应位于最小值和最大值之间。表中频率之和必须等于给定的总数。饼图角度相加必须等于 360°。象形图一定要有图例。比较两组数据时，同时使用平均数和范围来描述集中趋势和分散程度。

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12. Summary of Key Formulas | 关键公式汇总

Here is a compact reminder of the essential formulas covered in this handbook. Memorise them and practise applying them to different questions.

以下是对本手册所涵盖基本公式的精简汇总。请熟记并练习将其应用在不同题目中。

Concept	Formula / Method	中文公式/方法
Mean	Sum ÷ Count	总和÷个数
Median	Middle number (or mean of two middle)	中间的数（或两个中间数的平均数）
Mode	Most frequent value	出现次数最多的值
Range	Max − Min	最大值−最小值
Mean from frequency table	Σ(value × frequency) ÷ Σ(frequency)	Σ(值×频率) ÷ Σ(频率)
Pie chart angle	(Category frequency ÷ Total) × 360°	(类别频率÷总数) × 360°

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