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

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

This quick reference handbook is designed for Year 7 students following the CAIE curriculum. It brings together all the key formulas, definitions, and statistical methods you need for handling data, calculating averages, drawing charts, and understanding probability. Use it to revise efficiently and to check your work during practice. The explanations are given in both English and Chinese to support bilingual learners.

这份速查手册专为学习CAIE课程的七年级学生编写。手册汇总了数据处理、平均值计算、图表绘制以及概率理解所需的所有关键公式、定义和统计方法。你可以用它高效复习，也可以在练习时核对思路。所有解释均提供中英双语，以帮助双语学习者更好地掌握内容。

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

Statistical investigation always begins with data. Data can be collected first-hand by the researcher, which is called primary data. When you use data that someone else has already gathered, such as from a book or website, it is secondary data. Understanding the type of data you are working with is crucial because it determines which graph to use and which average is appropriate.

统计调查总是从数据开始。由研究者亲自收集的数据称为一手数据。当你使用他人已经收集好的数据，例如来自书籍或网站的数据时，就是二手数据。理解你所处理的数据类型至关重要，因为它决定了使用哪种图表以及哪种平均值最为合适。

Data is often described as qualitative or quantitative. Qualitative data is non-numerical; it describes qualities, categories or labels, such as eye colour, favourite sport or type of pet. Quantitative data is numerical and can be further divided into discrete and continuous data. Discrete data can only take certain values (usually whole numbers), like the number of students in a class or the score on a dice. Continuous data can take any value within a range, like height, mass or time.

数据通常被描述为定性数据或定量数据。定性数据是非数值型的，它描述性质、类别或标签，例如眼睛的颜色、最喜爱的运动或宠物的种类。定量数据是数值型的，可进一步分为离散数据和连续数据。离散数据只能取特定的值（通常是整数），比如班级学生人数或骰子的点数。连续数据可以在一个范围内取任何值，例如身高、质量或时间。

2. Frequency Tables | 频率表

A frequency table organises raw data by showing how many times each value or category occurs. The frequency is simply the count of a particular item. Tally marks are often used when collecting data to keep an accurate running count. You group the tallies in fives, with the fifth stroke crossing the previous four, to make counting easy.

频率表通过显示每个数值或类别出现的次数来整理原始数据。频率就是某一特定项的计数。在收集数据时，通常使用画记符号来保持准确的实时计数。一般以五个为一组标记，第五笔划过前四笔，这样便于清点。

A typical frequency table has two columns: the data value or category, and its frequency. If the data is numeric, you can also add a third column for a running total called the cumulative frequency, but for Year 7 the focus is on reading and completing simple frequency tables.

典型的频率表有两列：数据值或类别及其频率。如果数据是数值型的，还可以增加第三列用来记录累计频数，但在七年级阶段，重点是读懂和填写简单的频率表。

3. Bar Charts | 条形图

Bar charts are used to display categorical or discrete data. Each category is represented by a bar, and the height of the bar shows its frequency. All bars have the same width and are separated by equal gaps to emphasise that the categories are distinct, not continuous.

条形图用于显示分类数据或离散数据。每个类别用一根条形表示，条形的高度表示其频率。所有条形的宽度相同，并且条形之间留有相等的间隙，以强调这些类别是独立的，并非连续。

When drawing a bar chart, always label both axes clearly: the horizontal axis shows the categories, and the vertical axis shows the frequency. Choose a suitable scale that allows all frequencies to fit and makes the chart easy to read. The bars can be drawn vertically or horizontally; a horizontal bar chart is often used when category names are long.

绘制条形图时，务必清楚地标注两个坐标轴：横轴显示类别，纵轴显示频率。选择合适的刻度，使所有频率都能容纳，且图表易于阅读。条形可以垂直或水平绘制；当类别名称较长时，常使用水平条形图。

4. Pie Charts | 饼图

A pie chart displays data as slices of a circle, where each slice represents a category. The size of each slice is proportional to its frequency. The whole circle represents the total frequency of all categories. To draw a pie chart accurately, you need to calculate the angle for each sector using the key formula.

饼图以圆形切片的形式显示数据，每个切片代表一个类别。每个切片的大小与其频率成正比。整个圆代表所有类别的总频率。要准确绘制饼图，你需要使用关键公式计算每个扇形的角度。

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

Use a protractor to measure the calculated angles from a starting radius. Always label each sector with its category name and, if helpful, write the frequency or percentage inside or next to the slice.

使用量角器从起始半径开始量出计算好的角度。务必在每个扇形上标注其类别名称，如果对理解有帮助，还可以在切片内部或旁边标注频率或百分比。

5. Mean | 平均数

The mean is the most commonly used measure of average. It is found by adding up all the data values and then dividing by the number of values. The mean can be a decimal even if all the data values are whole numbers. It is sometimes called the arithmetic mean or simply the average.

平均数是最常用的平均度量。计算方法是把所有数据值加起来，然后除以数值的个数。即使所有数据值都是整数，平均数也可能是一个小数。它有时被称为算术平均数或简称为平均值。

Mean = Sum of all values ÷ Number of values

Mean = (∑ x) / n

The symbol ∑ (sigma) means ‘sum of’, x represents each individual value, and n is the total number of values. The mean is sensitive to extremely high or low values; a single outlier can pull the mean in its direction.

符号 ∑（西格玛）表示“求和”，x 代表每个单独的数值，n 是数值的总个数。平均数对极高或极低的数值很敏感；一个异常值就能把平均数拉向自己的方向。

6. Median | 中位数

The median is the middle value when the data is arranged in order from smallest to largest. It splits the data set into two halves, each containing the same number of values. The median is a useful average when the data contains extreme values, because it is not affected by them.

中位数是将数据从小到大排列后位于中间的那个数值。它将数据集分成数量相等的两部分。当数据包含极端值时，中位数是一个很有用的平均数，因为它不受极端值的影响。

To find the median: first, order the data. If the number of values (n) is odd, the median is the value at position (n + 1) ÷ 2. For example, with 7 values, the median is the 4th value. If n is even, there is no single middle value, so you find the mean of the two middle values, i.e. the values at positions n/2 and (n/2) + 1.

寻找中位数的方法：首先将数据排序。如果数值的个数 n 为奇数，中位数就是位于 (n + 1) ÷ 2 位置上的值。例如有7个数值，中位数就是第4个值。如果 n 为偶数，就没有单一的中间值，此时应取两个中间值的平均数，即位于 n/2 和 (n/2) + 1 位置上的两个值。

7. Mode and Range | 众数与极差

The mode, or modal value, is the value that appears most frequently in a data set. A set of data can have one mode (unimodal), two modes (bimodal), or more. If no value repeats, the data set has no mode. The mode is especially useful for qualitative or categorical data, where you cannot calculate a mean or median.

众数或模态值，是数据集中出现频率最高的值。一组数据可以有一个众数（单峰），两个众数（双峰），甚至更多。如果没有重复值，则数据集没有众数。众数对于定性数据或分类数据特别有用，因为这种情况下无法计算平均数或中位数。

The range is a simple measure of how spread out the data is. It tells you the difference between the greatest value and the smallest value.

极差是衡量数据分散程度的一个简单指标。它告诉我们最大值与最小值之间的差距。

Range = Highest value − Lowest value

A larger range means the data is more spread out, while a smaller range indicates the values are closer together. The range is easy to calculate but can be heavily influenced by a single outlier.

极差越大表示数据越分散，极差越小则表示数据值越集中。极差容易计算，但也很容易受到单个异常值的影响。

8. Stem-and-Leaf Diagrams | 茎叶图

A stem-and-leaf diagram is a way of ordering numerical data and showing its shape while keeping each original value visible. The ‘stem’ represents the leading digit(s) and the ‘leaf’ represents the final digit. For example, in the number 37, the stem is 3 and the leaf is 7. A key must always be included to show what the stem and leaf represent.

茎叶图是一种将数值数据排序并显示其分布形状的方法，同时还能保留每个原始值。“茎”代表前一位或几位数字，“叶”代表最后一位数字。例如，在数字37中，茎是3，叶是7。必须始终包含一个说明，以明示茎和叶代表的意义。

To construct a stem-and-leaf diagram, first write the stems in a vertical column from smallest to largest. Then add each leaf next to its stem, also arranged in ascending order. Once the diagram is drawn, you can easily find the mode, median, and range from it. The median can be identified by counting through the leaves until you reach the middle value.

要构建茎叶图，先把茎从小到大写成一竖列。然后在每个茎旁边添加对应的叶，叶也按升序排列。绘制完成后，你可以轻松地从图中找出众数、中位数和极差。通过逐个数叶子直到到达中间位置，就可以找出中位数。

9. Probability Basics | 概率基础

Probability measures how likely an event is to happen. It is always a number between 0 and 1, where 0 means the event is impossible, and 1 means it is certain. Probability can be written as a fraction, a decimal, or a percentage.

概率度量一个事件发生的可能性大小。它总是介于0和1之间的一个数，0表示事件不可能发生，1表示事件必然发生。概率可以用分数、小数或百分数来表示。

When all outcomes are equally likely, the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes.

当所有的结果都是等可能时，一个事件的概率由有利结果的数量与所有可能结果的总数之比给出。

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

The probability of an event NOT happening is equal to 1 minus the probability that it does happen. This is known as the complement rule.

一个事件不发生的概率等于1减去它发生的概率。这被称为互补规则。

10. Experimental Probability | 实验概率

Sometimes we cannot list all the outcomes or assume they are equally likely. In these cases we carry out an experiment or survey and use the results to estimate probability. This estimate is called the relative frequency or experimental probability.

有时我们无法列出所有结果或假设它们是等可能的。在这些情况下，我们进行实验或调查，并利用结果来估计概率。这种估计值称为相对频数或实验概率。

Relative frequency = Frequency of the event ÷ Total number of trials

As the number of trials increases, the relative frequency tends to get closer to the true theoretical probability. This idea is called the law of large numbers. For Year 7, it is enough to know that more trials usually give a better estimate.

随着试验次数的增加，相对频数往往会接近真正的理论概率。这个思想叫做大数定律。对于七年级来说，知道更多试验通常能给出更好的估计就足够了。

11. Choosing the Right Average | 选择适当的平均数

The mean, median and mode each describe the centre of a data set, but they are useful in different situations. The mean uses all the data values, so it is best when there are no extreme outliers and the data is fairly symmetric. The median is the preferred average when the data is skewed or contains outliers, because it ignores the actual magnitude of extreme values. The mode is the only average that can be used with non-numerical data; it is also helpful when you want to know the most common or popular choice.

平均数、中位数和众数都能描述数据集的中心，但它们适用于不同的情况。平均数用到了所有的数据值，因此最适合没有极端异常值且数据大致对称的情况。当数据偏斜或包含异常值时，中位数是首选的集中趋势度量，因为它不受极端值实际大小的影响。众数是唯一可用于非数值型数据的平均度量；当你想知道最常见或最受欢迎的选择时，众数也很有用。

Always think about the context of the data before choosing which average to use. For example, when reporting the typical price of houses in an area, the median is often used because a few very expensive properties would distort the mean.

在选择使用哪个平均数之前，一定要考虑数据的背景意义。例如，在报告一个地区的典型房价时，通常使用中位数，因为少数非常昂贵的房产会使平均数发生扭曲。

12. Quick Reference Formulas | 公式速查汇总

The table below brings together all the essential formulas and measures covered in this handbook. Keep it handy when you are solving statistics problems.

下表汇总了本手册涵盖的所有基本公式和度量指标。在解答统计问题时，请将它放在手边参考。

Statistical Measure	Formula (Symbols)	Formula (Words)
Mean	Mean = (∑ x) / n	Sum of values divided by number of values
Median	Middle value (or mean of two middle values)	Order data, find central position(s)
Mode	Most frequent value	Value with highest frequency
Range	Range = Max − Min	Highest value minus lowest value
Pie chart sector angle	Sector angle = (Frequency / Total) × 360°	Frequency divided by total, times 360 degrees
Probability	P(E) = Favourable outcomes / Total outcomes	Number of successful outcomes divided by total possible outcomes
Relative frequency	Relative frequency = Event freq. / Total trials	Frequency of event divided by number of trials

Memorise these formulas and practise applying them to different types of questions. A solid grasp of these basics will make Year 7 statistics much easier and prepare you for more advanced topics in the future.

记住这些公式，并练习将它们应用到不同类型的题目中。扎实掌握这些基础知识，将使七年级统计学习变得轻松许多，并为未来更进阶的主题做好准备。

Published by TutorHao | Statistics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）