Year 7 Cambridge Statistics: Formulas & Theorems Quick Reference Handbook | Year 7 Cambridge 统计：公式定理速查手册

📚 Year 7 Cambridge Statistics: Formulas & Theorems Quick Reference Handbook | Year 7 Cambridge 统计：公式定理速查手册

Welcome to your handy quick-reference guide for Year 7 Cambridge Statistics. This handbook collects the essential formulas, definitions, and key concepts you need to interpret data, calculate averages, and draw clear statistical diagrams. Keep it by your side for homework and revision.

欢迎使用这份便捷的 Year 7 Cambridge 统计速查手册。本手册汇集了你需要解读数据、计算平均数以及绘制清晰统计图表所必备的公式、定义和关键概念。做作业和复习时放在身边随时查阅。

1. Key Statistical Terms | 关键统计术语

Statistics is the branch of mathematics that deals with collecting, organising, analysing, and presenting data. It helps us make sense of information from the world around us.

统计学是数学的一个分支，涉及数据的收集、整理、分析和展示。它帮助我们从周围世界的信息中获得见解。

Data can be qualitative (descriptive, e.g. eye colour, favourite subject) or quantitative (numerical, e.g. height, marks). Quantitative data may be discrete (countable, like number of siblings) or continuous (measurable, like mass or time).

数据可以是定性数据（描述性的，例如眼睛颜色、最喜欢的科目）或定量数据（数值的，例如身高、分数）。定量数据又可分为离散数据（可数的，如兄弟姐妹数量）和连续数据（可测量的，如质量或时间）。

A population is the entire group you want to study, while a sample is a smaller part of that group selected to represent the whole. Analysing a sample can give us reliable conclusions if it is chosen fairly.

总体是你想要研究的全部对象，而样本是从中选出的、能代表整体的较小部分。如果样本选取公平，分析样本可以得出可靠的结论。

A variable is any characteristic that can change or take different values, such as ‘age’ or ‘temperature’. Understanding variables helps us design surveys and experiments.

变量是任何可以变化或取不同值的特征，如“年龄”或“温度”。理解变量有助于我们设计调查和实验。

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2. Mean (Arithmetic Average) | 算术平均数

The mean is the most common measure of central tendency. It gives a single value that represents the centre of a data set.

平均数是最常用的集中趋势度量。它能用一个值代表数据集的中心。

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

平均数 =（所有数据值的总和）÷（数据值的个数）

To find the mean, add up all the numbers and then divide the total by how many numbers there are. It can also be written as Mean = Σx / n, where Σx is the sum of all values and n is the number of values.

求平均数，先把所有数相加，再将总和除以数的个数。它也可以写作平均数 = Σx / n，其中 Σx 是所有值的总和，n 是值的个数。

Example: Find the mean of 4, 8, 6, 5, 2. Sum = 4+8+6+5+2 = 25. Number of values = 5. Mean = 25 ÷ 5 = 5.

示例：求 4、8、6、5、2 的平均数。总和 = 4+8+6+5+2 = 25。值的个数 = 5。平均数 = 25 ÷ 5 = 5。

The mean can be affected by extreme values (outliers). If one number is much larger or smaller than the rest, the mean may not truly reflect the typical value.

平均数会受到极端值（异常值）的影响。如果有一个数比其他数大很多或小很多，平均数可能无法真实反映典型值。

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3. Median | 中位数

The median is the middle value in an ordered list of numbers. It splits the data into two equal halves.

中位数是按顺序排列的一列数中位于中间的值。它将数据分成相等的两半。

Steps to find the median: 1) Arrange all the data values from smallest to largest. 2) If the number of values (n) is odd, the median is the value at position (n+1)/2. 3) If n is even, the median is the mean of the two middle values.

求中位数的步骤：1) 将所有数据值从小到大排列。2) 如果值的个数 (n) 为奇数，中位数就是第 (n+1)/2 个位置上的值。3) 如果 n 为偶数，中位数是中间两个数的平均数。

Example (odd): Find the median of 7, 3, 9, 2, 12. Order: 2, 3, 7, 9, 12. n=5 (odd). Middle position = (5+1)/2 = 3rd value. Median = 7.

示例（奇数）：求 7、3、9、2、12 的中位数。排序：2, 3, 7, 9, 12。n=5（奇数）。中间位置 = (5+1)/2 = 第3个值。中位数 = 7。

Example (even): 4, 1, 6, 8, 2, 9. Order: 1, 2, 4, 6, 8, 9. n=6 (even). Middle two values are 4 and 6. Median = (4+6) ÷ 2 = 5.

示例（偶数）：4, 1, 6, 8, 2, 9。排序：1, 2, 4, 6, 8, 9。n=6（偶数）。中间两个值是 4 和 6。中位数 = (4+6) ÷ 2 = 5。

The median is not distorted by extreme values, which makes it useful when data has outliers, such as house prices or incomes.

中位数不受极端值的影响，因此在数据含有异常值（如房价或收入）时非常有用。

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4. Mode | 众数

The mode is the value that appears most frequently in a data set. There can be one mode (unimodal), two modes (bimodal) or more (multimodal). If all values appear only once, there is no mode.

众数是数据集中出现频率最高的值。可以有一个众数（单峰）、两个众数（双峰）或更多（多峰）。如果所有值都只出现一次，则没有众数。

To identify the mode, simply count how many times each value occurs. The one with the highest frequency is the mode.

要找出众数，只需数出每个值出现的次数。出现次数最多的那个就是众数。

Example: 5, 2, 7, 2, 9, 2, 5. The number 2 appears three times, which is more often than any other value, so mode = 2.

示例：5, 2, 7, 2, 9, 2, 5。数字 2 出现了三次，比其他任何值都多，所以众数 = 2。

The mode is the only measure of central tendency that can be used with qualitative data, such as ‘red’ being the most common car colour.

众数是唯一可用于定性数据的集中趋势度量，例如“红色”是最常见的汽车颜色。

Unlike the mean and median, the mode is not calculated through a formula but by observation, making it straightforward for categorical data.

与平均数和中位数不同，众数不是通过公式计算的，而是通过观察得出，因此对于类别数据来说很简单。

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5. Range | 极差

The range measures how spread out the data are. It is the simplest measure of dispersion.

极差衡量数据的分散程度。它是最简单的离散量度。

Range = Maximum value – Minimum value

极差 = 最大值 – 最小值

To find the range, identify the largest and smallest numbers in the data set and subtract the smallest from the largest.

要求极差，先找出数据集中最大的数和最小的数，然后用最大的数减去最小的数。

Example: The heights of five students in cm are 152, 149, 160, 155, 145. Maximum = 160, Minimum = 145. Range = 160 – 145 = 15 cm.

示例：五名学生的身高（厘米）为 152, 149, 160, 155, 145。最大值 = 160，最小值 = 145。极差 = 160 – 145 = 15 厘米。

A small range means the data values are close together, while a large range indicates they are more spread out. However, the range only depends on two values, so it may not give a full picture of spread on its own.

极差小意味着数据值比较集中，极差大则表明它们更加分散。不过，极差只依赖于两个值，因此单独使用可能无法全面反映分散情况。

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6. Comparing Data Sets Using Averages and Range | 用平均数和极差比较数据

When comparing two or more sets of data, always look at a measure of central tendency (mean or median) alongside the range. The mean tells you the typical value, while the range tells you about consistency.

比较两组或多组数据时，一定要同时参考集中趋势度量（平均数或中位数）和极差。平均数告诉你典型值，极差则说明一致性。

If Data Set A has a higher mean and a smaller range than Data Set B, then A’s values are generally higher and more consistent. If B has a similar mean but a much larger range, B’s values are more varied.

如果数据集 A 的平均数比 B 高、极差比 B 小，那么 A 的数值总体上更高且更一致。如果 B 的平均数相似但极差大得多，则 B 的数值变化更大。

Be careful: an outlier in a small data set can drag the mean up or down without affecting the median much. Always consider which average to use based on the data’s shape.

注意：小数据集中的异常值可能会拉高或拉低平均数，而对中位数的影响较小。应根据数据的分布形状，思考使用哪种平均数。

Example: Set X: 5, 5, 6, 6, 7; Mean = 5.8, Range = 2. Set Y: 1, 5, 6, 7, 20; Mean = 7.8, Range = 19. Although Set Y has a higher mean, its huge range warns us that the mean is inflated by the 20. The median of Y is 6, which may be a better comparison.

示例：数据集 X：5, 5, 6, 6, 7；平均数 = 5.8，极差 = 2。数据集 Y：1, 5, 6, 7, 20；平均数 = 7.8，极差 = 19。虽然 Y 的平均数更高，但它的极大极差提醒我们，平均数是受 20 的影响而虚高。Y 的中位数是 6，可能更适合比较。

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7. Tally Charts and Frequency Tables | 划记表与频数表

A tally chart is a quick way to record data as it is collected. Each vertical line represents one item, and every fifth line crosses the previous four to form a bundle of five, making counting easier.

划记表是一种在收集数据时快速记录的方法。每条竖线代表一个项目，每第五条线划过前面四条形成一个五线束，这样便于计数。

A frequency table summarises data by listing each distinct value or category alongside its frequency (the number of times it occurs). It may also include a tally column before the frequency.

频数表通过列出每个不同的值或类别及其频数（出现的次数）来汇总数据。它还可以在频数列之前增加一列划记。

Example of a frequency table for the number of pets students have:

Number of pets	Tally	Frequency
0	|||	3
1	~~||||~~	5
2	~~||||~~ ||	7
3	||	2

From a frequency table, you can quickly see the mode (2 pets) and calculate the total number of students by adding all frequencies (3+5+7+2 = 17). You can also compute the mean by multiplying each value by its frequency, summing those products, and dividing by the total frequency.

从频数表中，你可以快速找到众数（2 只宠物），并通过将所有频数相加（3+5+7+2 = 17）算出学生总数。你还可以通过将每个值乘以它的频数、求和、再除以总频数来计算平均数。

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

A bar chart displays categorical data using rectangular bars. The bars have equal width and are separated by gaps. The length (or height) of each bar represents the frequency or value for that category.

条形图用矩形条展示类别数据。条宽相等，条与条之间有间隔。每个条的长度（或高度）代表该类别的频数或数值。

Rules for a good bar chart: include a clear title, label both axes, use a consistent scale on the frequency axis, and start the scale at zero if possible. Bars can be drawn vertically or horizontally.

绘制规范条形图的要点：包含清晰的标题，标注两条坐标轴，在频数轴上使用一致的刻度，并尽可能从零开始。条形可以垂直或水平绘制。

A pictogram uses symbols or pictures to represent data. Each symbol represents a certain number of items, and a key must explain what one symbol stands for. Part of a symbol can show a fraction of the unit.

象形图用符号或图片表示数据。每个符号代表一定数量的项目，必须用图例说明一个符号代表什么。部分符号可以表示单位的一部分。

Example: If one smiley face represents 4 students, then 3 smiley faces represent 12 students. A half smiley face would represent 2 students.

示例：如果一个笑脸代表 4 个学生，那么 3 个笑脸就代表 12 个学生。半个笑脸代表 2 个学生。

When reading bar charts and pictograms, always check the scale and key to avoid misinterpreting the data. A bar that looks twice as tall does not mean twice the value if the scale is not linear or is truncated.

阅读条形图和象形图时，务必检查刻度和图例，以免误解数据。如果刻度不是线性的或被截断，一个看起来高两倍的条形并不意味着数值是两倍。

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9. Pie Charts | 饼图

A pie chart is a circular chart divided into sectors, where each sector represents a proportion of the total data. The entire circle represents the whole, or 360°. The size of each sector’s angle is proportional to the frequency of that category.

饼图是一种划分成扇区的圆形图表，每个扇区代表总数据中的一个比例。整个圆代表整体或 360°。每个扇区的角度大小与该类别的频数成比例。

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

扇区角度 =（类别频数 ÷ 总频数）× 360°

To draw a pie chart: 1) Calculate the total frequency. 2) For each category, find the angle using the formula. 3) Use a protractor to measure and draw each sector. 4) Label each sector or provide a key, and give the chart a title.

绘制饼图步骤：1) 计算总频数。2) 对每个类别用公式求出角度。3) 使用量角器测量并画出每个扇区。4) 标注每个扇区或提供图例，并给图表加上标题。

Example: In a survey of 30 students’ favourite fruits, 12 chose apple, 9 chose banana, 6 chose orange, 3 chose grape. Total = 30. Apple angle = (12/30) × 360° = 144°. Banana = (9/30)×360° = 108°. Orange = (6/30)×360° = 72°. Grape = (3/30)×360° = 36°. Check that the angles sum to 360°.

示例：在一项对 30 名学生最喜欢水果的调查中，12 人选苹果，9 人选香蕉，6 人选橙子，3 人选葡萄。总数 = 30。苹果角度 = (12/30) × 360° = 144°。香蕉 = (9/30)×360° = 108°。橙子 = (6/30)×360° = 72°。葡萄 = (3/30)×360° = 36°。检查角度之和为 360°。

Pie charts are excellent for showing proportions, but they become difficult to read if there are too many categories or very small slices. In such cases, a bar chart might be clearer.

饼图非常适合展示比例，但如果类别过多或扇区过小，阅读起来就困难。这种情况下，条形图可能更清晰。

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10. Line Graphs and Scatter Graphs | 折线图与散点图

A line graph is used to display data that changes over time. Points are plotted and then connected with straight line segments. The horizontal axis usually represents time, and the vertical axis represents the measured quantity.

折线图用于展示随时间变化的数据。先描点，然后用直线段连接。水平轴通常表示时间，竖直轴表示测量的量。

Example: A line graph can show how the temperature in a classroom changes every hour from 8:00 to 15:00. By looking at the line’s slope, you can see when the temperature rose quickly or stayed constant.

示例：折线图可以显示教室温度从 8:00 到 15:00 每小时的变化情况。通过观察线条的斜率，可以看出温度何时快速升高或保持不变。

A scatter graph (or scatter plot) is used to see if there is a relationship (correlation) between two sets of numerical data. Each pair of numbers is plotted as a point on a coordinate grid.

散点图（散点图）用于判断两组数值数据之间是否存在关系（相关性）。每对数值作为一个点画在坐标网格上。

If the points roughly form an upward pattern, there is a positive correlation (as one variable increases, the other tends to increase). A downward pattern indicates negative correlation (as one increases, the other tends to decrease). If the points show no clear pattern, the correlation is zero or very weak.

如果点大致呈上升趋势，则为正相关（一个变量增加，另一个也倾向于增加）。下降趋势表示负相关（一个增加，另一个倾向于减少）。如果点没有明显规律，则相关性为零或很弱。

In Year 7, you may simply describe the relationship as ‘positive’, ‘negative’ or ‘no correlation’ and note how strong it looks. A line of best fit is not usually required until later years.

在 Year 7 阶段，你只需将关系描述为“正相关”、“负相关”或“无相关”，并说明看起来的强弱程度。通常到更高年级才要求画出最佳拟合线。

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11. Interpreting Statistical Diagrams | 解读统计图表

Being able to draw diagrams is only half the story. Interpreting them correctly is an essential skill. Always begin by reading the title, axis labels, scale, and legend (key). Ask yourself what the chart is trying to show.

会画图表只是一半的功夫，能够正确解读是一项关键技能。一定要先看标题、坐标轴标签、刻度和图例。问自己，这张图表想说明什么。

Look at the range of data and identify any noticeable peaks or trends. Compare categories or time periods carefully, and use numbers from the graph to support your statements, rather than just saying ‘a lot’ or ‘a little’.

观察数据的变化范围，识别明显的峰值或趋势。仔细比较不同类别或时间段，并用图中的具体数字来支持你的陈述，而不仅仅说“很多”或“很少”。

Be alert to misleading diagrams. A bar chart

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