📚 Year 7 WJEC Statistics: Formula & Theorem Quick Reference Handbook | Year 7 WJEC 统计：公式定理速查手册
This handbook provides a concise and structured reference for all the essential formulas, definitions, and theorems covered in the Year 7 WJEC Statistics curriculum. Whether you are revising for a test or tackling a data analysis project, having these foundational tools at your fingertips will help you work efficiently and accurately. Each section pairs an English explanation with its Chinese equivalent, ensuring bilingual learners can master the terminology and concepts with confidence.
本手册为Year 7 WJEC统计课程中所有必备公式、定义和定理提供了简洁、结构化的参考。无论你是在为考试复习，还是在进行数据分析项目，手边准备好这些基础工具都将帮助你高效、准确地完成学习任务。每个小节都配对了英文解释和对应的中文说明，确保双语学习者能够自信地掌握相关术语和概念。
1. Mean | 平均数
The mean is the sum of all data values divided by the number of values. It is often called the average and is the most commonly used measure of central tendency. To calculate the mean, use the formula below.
平均数是指所有数据值的总和除以数据的个数。它常被直接称为“平均值”，是最常用的集中趋势度量。要计算平均数，请使用以下公式。
Mean = Sum of all data values ÷ Number of data values
For example, if a student scores 7, 8, 9, 6, and 10 in five tests, the mean score is (7+8+9+6+10) ÷ 5 = 40 ÷ 5 = 8. The mean can be affected by extremely high or low values, which are called outliers. When outliers are present, the median may give a better idea of the typical value.
例如，如果一名学生在五次测验中得了7分、8分、9分、6分和10分，那么平均分为 (7+8+9+6+10) ÷ 5 = 40 ÷ 5 = 8。平均数可能会受到极高或极低数值的影响，这些数值被称为异常值。当存在异常值时，中位数可能更能体现典型数值。
2. Median | 中位数
The median is the middle value when the data are arranged in order from smallest to largest. If there is an odd number of data values, the median is the exact middle value. If there is an even number of data values, the median is the mean of the two middle values.
中位数是将数据按从小到大的顺序排列后，位于中间位置的数值。如果数据个数是奇数，中位数就是正中间的那个值。如果数据个数是偶数，中位数则是中间两个数值的平均数。
For the data set 3, 7, 9, 12, 15, the median is 9 because it is the third value when ordered. For the data set 4, 8, 10, 14, the two middle numbers are 8 and 10, so the median is (8+10) ÷ 2 = 9. The median is not affected by outliers, which makes it useful when analysing data such as house prices or incomes.
对于数据集3, 7, 9, 12, 15，中位数为9，因为排序后它是第三个数值。对于数据集4, 8, 10, 14，中间的两个数是8和10，因此中位数为 (8+10) ÷ 2 = 9。中位数不受异常值的影响，这使得它在分析房价或收入等数据时非常有用。
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, more than one mode, or no mode at all if all values occur with the same frequency. The mode is particularly useful for categorical data, where we want to know the most common category.
众数是数据集中出现频率最高的值。一组数据可能有一个众数、多个众数，或者如果所有数值出现的频率都相同，则没有众数。众数对于分类数据特别有用，因为我们可以通过它了解最常见的类别。
For the shoe sizes 4, 5, 5, 6, 7, 5, 8, the mode is 5 because it occurs three times. If a survey asks students to name their favourite colour and ‘blue’ gets the most votes, then blue is the mode. In Year 7 statistics, mode is often used alongside mean and median to give a complete picture of the data.
对于鞋码数据4, 5, 5, 6, 7, 5, 8，众数是5，因为它出现了三次。如果一项调查要求学生说出他们最喜欢的颜色，并且“蓝色”得票最多，那么蓝色就是众数。在Year 7统计中，众数通常与平均数和中位数一起使用，以便全面了解数据特征。
4. Range | 极差
The range is a simple measure of spread that tells us how spread out the data values are. It is calculated by subtracting the smallest value from the largest value. A larger range indicates greater variability in the data set.
极差是一种简单的离散程度度量，它告诉我们数据值的分布范围有多大。其计算方法是用最大值减去最小值。极差越大，表明数据集中的变异性越大。
Range = Largest value − Smallest value
For the temperatures recorded over a week: 12°C, 15°C, 14°C, 18°C, 11°C, 17°C, 13°C, the largest value is 18 and the smallest is 11. The range is 18 − 11 = 7°C. While the range is very quick to calculate, it only uses two values and does not give any information about the data in between.
对于一周内记录的温度：12°C, 15°C, 14°C, 18°C, 11°C, 17°C, 13°C，最大值为18，最小值为11。极差为 18 − 11 = 7°C。虽然极差计算起来非常快，但它只使用了两个数值，并不能提供中间数据的任何信息。
5. Frequency Tables | 频数表
A frequency table organises raw data into a clear, structured format by listing each distinct value alongside how many times it occurs. The total frequency should equal the number of data items collected. Frequency tables make it easier to spot patterns and calculate statistics such as the mode and mean.
频数表通过列出每个不同的数值及其出现的次数，将原始数据整理成清晰、结构化的格式。总频数应等于所收集数据项的个数。频数表使我们更容易发现数据模式，并计算众数和平均数等统计量。
When calculating the mean from a frequency table, we multiply each value by its frequency, sum these products, and then divide by the total frequency. The formula is:
当用频数表计算平均数时，我们将每个数值乘以其频数，求出这些乘积的总和，然后除以总频数。公式如下：
Mean = Σ(value × frequency) ÷ Total frequency
For a table showing the number of pets: 0 pets (frequency 3), 1 pet (frequency 7), 2 pets (frequency 5), the total is 3+7+5=15. The sum of products is (0×3)+(1×7)+(2×5)=0+7+10=17, so the mean is 17÷15 ≈ 1.13 pets per student.
对于显示宠物数量的表格：0只宠物（频数3），1只宠物（频数7），2只宠物（频数5），总数为3+7+5=15。乘积之和为(0×3)+(1×7)+(2×5)=0+7+10=17，因此平均数为 17÷15 ≈ 1.13 只宠物每名学生。
6. Bar Charts | 柱状图
Bar charts are used to represent data visually, with the height or length of each bar showing the frequency or value for each category. The bars must be of equal width and should be separated by gaps, because the data are categorical or discrete. Labelling axes clearly is essential for correct interpretation.
柱状图用于直观地表示数据，每个柱子的高度或长度显示每个类别的频数或数值。柱子必须等宽，并且柱子之间应有空隙，因为这些数据是分类数据或离散数据。清晰地标注坐标轴对于正确解读图表至关重要。
In Year 7, students learn to draw bar charts using a pencil and ruler, ensuring that all bars are proportional to the frequencies. The vertical axis usually starts at zero, and the scale should be chosen so that the tallest bar fits well within the grid. A well-constructed bar chart allows comparison of categories at a glance.
在Year 7，学生学习使用铅笔和尺子绘制柱状图，确保所有柱子的高度与频数成比例。纵轴通常从零开始，并且应选择合适的刻度，使最高的柱子能很好地容纳在网格内。一个制作精良的柱状图可以让人一目了然地比较各个类别。
7. Pictograms | 象形图
A pictogram uses small pictures or symbols to represent data. Each symbol stands for a certain number of units, and part of a symbol can be used to represent a fraction of that unit. Pictograms are visually appealing and make data easy to understand, but they require a clear key to show what each symbol represents.
象形图使用小图片或符号来表示数据。每个符号代表一定数量的单位，而部分符号可用于表示该单位的一部分。象形图在视觉上很吸引人，使数据易于理解，但它们需要一个清晰的图例来说明每个符号所代表的内容。
When constructing a pictogram, the symbol should be easy to draw and closely related to the topic. For example, one book icon could represent 5 books read. If a student read 12 books, you would draw two whole book icons and a fraction of a third icon. Accuracy and proportionality are key skills tested in the WJEC exam.
在绘制象形图时，符号应易于画出并与主题密切相关。例如，一个书本图标可以代表读了5本书。如果一名学生读了12本书，你就要画出两个完整的书本图标和第三个图标的一部分。准确性和比例性是WJEC考试中考查的关键技能。
8. Pie Charts | 饼状图
Pie charts display data as sectors of a circle, where each sector’s angle is proportional to the frequency it represents. The whole circle represents the total data set, and the size of each slice shows the relative size of each category. The formula for calculating the angle of a sector is shown below.
饼状图将数据显示为圆的扇形区域，每个扇形的角度与其所代表的频数成比例。整个圆代表整个数据集，每一块的大小则显示了每个类别的相对大小。计算扇形角度的公式如下。
Sector angle = (Category frequency ÷ Total frequency) × 360°
For example, if a survey of 30 students shows that 12 prefer football, the sector angle for football is (12 ÷ 30) × 360° = 144°. Students must practise using a protractor to draw sectors accurately and should label each sector or provide a legend. Pie charts are excellent for showing proportions but cannot show exact frequencies as clearly as a bar chart.
例如，如果对30名学生的调查显示12人更喜欢足球，那么足球对应的扇形角度为 (12 ÷ 30) × 360° = 144°。学生必须练习使用量角器准确绘制扇形，并应为每个扇形添加标签或提供图例。饼状图非常适合展示比例，但不如柱状图那样能清晰地显示确切的频数。
9. Probability Scale and Language | 概率尺度与语言
Probability measures how likely an event is to happen, and it is expressed as a number between 0 and 1. A probability of 0 means the event is impossible, while a probability of 1 means it is certain. In Year 7, students also use words such as impossible, unlikely, even chance, likely, and certain to describe probability.
概率衡量一个事件发生的可能性，用一个介于0和1之间的数字表示。概率为0表示事件不可能发生，而概率为1表示事件必然发生。在Year 7，学生也使用诸如不可能、不太可能、均等机会、很可能以及必然等词语来描述概率。
The probability scale is often drawn as a horizontal line from 0 to 1, with events placed along it according to their chance of occurring. An event with a probability of ½ or 0.5 is said to have an even chance. This conceptual understanding forms the foundation for later calculation of theoretical probability using fractions.
概率尺度通常画成一条从0到1的水平线，根据事件发生的机会大小，将各种事件标在线上的相应位置。概率为½或0.5的事件被称为具有均等机会。这种概念性的理解，为日后使用分数计算理论概率奠定了基础。
10. Calculating Theoretical Probability | 计算理论概率
Theoretical probability is calculated when all outcomes are equally likely. It is the number of favourable outcomes divided by the total number of possible outcomes. The formula must be memorised and applied in a variety of contexts, including dice, spinners, and cards.
当所有结果出现的可能性都相同时，就可以计算理论概率。它等于有利结果的数量除以所有可能结果的总数。这个公式必须记住，并能应用于骰子、转盘和扑克牌等多种情境中。
Probability = Number of favourable outcomes ÷ Total number of possible outcomes
If a fair six-sided die is rolled, the probability of rolling a 3 is 1 ÷ 6, because there is one favourable outcome and six possible outcomes. The probability of rolling an even number is 3 ÷ 6 = ½, because there are three even numbers (2, 4, 6). Probability is often written as a fraction in its simplest form, a decimal, or a percentage.
如果掷一个公平的六面骰子，掷出3点的概率是 1 ÷ 6，因为有利结果只有一个，而可能的结果有六个。掷出偶数的概率是 3 ÷ 6 = ½，因为有3个偶数（2, 4, 6）。概率通常以最简分数、小数或百分比的形式表示。
11. Mutually Exclusive Events and Sum of Probabilities | 互斥事件与概率之和
Mutually exclusive events are events that cannot happen at the same time. For example, when rolling a die, getting a 2 and getting a 5 are mutually exclusive. For any set of mutually exclusive events that cover all possible outcomes, the sum of their probabilities is always 1.
互斥事件是指不可能同时发生的事件。例如，掷一个骰子时，掷出2点和掷出5点就是互斥的。对于任何一组涵盖了所有可能结果的互斥事件，其概率之和总是1。
This fundamental theorem is extremely useful for finding the probability that an event does not happen. The probability of an event not occurring is 1 minus the probability that it does occur. So if the probability of rain tomorrow is 0.3, the probability it does not rain is 1 − 0.3 = 0.7.
这条基本定理在求某个事件不发生的概率时极其有用。一个事件不发生的概率等于1减去该事件发生的概率。因此，如果明天下雨的概率是0.3，那么不下雨的概率就是 1 − 0.3 = 0.7。
12. Two-Way Tables and Sample Spaces | 双向表与样本空间
Two-way tables organise data about two categorical variables and help us find frequencies and probabilities. The rows represent one variable and the columns represent the other. The totals in the margins are very helpful for checking calculations and for finding probabilities.
双向表用于整理两个分类变量的数据，帮助我们找到频数和概率。表格的行代表一个变量，列代表另一个变量。表格边缘的“合计”栏对检查计算和求出概率非常有帮助。
A sample space is a list or diagram of all possible outcomes of an experiment. For two events combined, a sample space diagram or two-way table can be drawn. For example, if two coins are flipped, the sample space is {HH, HT, TH, TT}. Using this, the probability of at least one head is 3 out of 4.
样本空间是列出或图示某个实验所有可能结果的方式。对于两个事件的组合，可以画出样本空间示意图或双向表。例如，如果抛两枚硬币，样本空间为{HH, HT, TH, TT}。据此，至少出现一次正面的概率为4种结果中的3种。
13. Stem-and-Leaf Diagrams | 茎叶图
A stem-and-leaf diagram is a way of displaying numerical data that keeps the original values visible while showing the shape of the distribution. The ‘stem’ represents the leading digit or digits, and the ‘leaf’ represents the final digit. An ordered stem-and-leaf diagram makes it easy to find the median, mode, and range.
茎叶图是一种显示数值数据的方式，它既能保持原始数据的可见性，又能展示分布的形状。“茎”代表前导数字，“叶”代表最后一位数字。一个有序的茎叶图可以方便地找出中位数、众数和极差。
For the test scores 23, 25, 31, 32, 32, 45 and 47, the stems would be 2, 3 and 4 (representing the tens). The leaf for stem 2 would be 3 and 5, producing 23 and 25. A key must always be included, for example ‘2|3 means 23’. Stem-and-leaf diagrams are a WJEC favourite for testing data handling skills.
对于测验分数23, 25, 31, 32, 32, 45和47，茎为2、3和4（代表十位数）。茎为2的叶是3和5，即代表23和25。图中必须始终包含图例说明，例如“2|3表示23”。茎叶图是WJEC考试中测验数据处理技能的常用题型。
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