📚 Year 7 CCEA Statistics: Formula & Theorems Quick Reference Handbook | Year 7 CCEA 统计：公式定理速查手册
This quick reference handbook is designed for Year 7 students following the CCEA curriculum. It brings together essential formulas, definitions and key theorems in statistics, helping you revise efficiently and build a solid foundation in data handling and probability. Each section presents the core concept with a clear explanation, and all formulas are highlighted for quick access.
本速查手册专为 CCEA 课程 Year 7 学生设计，汇集了统计学科中的基本公式、定义和关键定理，助你高效复习，为数据处理和概率打好坚实基础。每一节都以清晰的解释呈现核心概念，并突出所有公式，方便快速查阅。
1. Types of Data | 数据类型
Statistics begins with understanding data. Data can be qualitative (categorical) or quantitative (numerical). Qualitative data describes attributes like eye colour or favourite food. Quantitative data consists of numbers. Quantitative data can be further split into discrete data, which can only take specific separate values (e.g., number of siblings: 0, 1, 2, …), and continuous data, which can take any value in a given range (e.g., height, mass, time).
统计从理解数据开始。数据可以是定性（分类）或定量（数值）的。定性数据描述属性，如眼睛颜色或最喜欢的食物。定量数据由数字组成。定量数据可进一步分为离散数据，只能取特定的分离值（例如兄弟姐妹的数量：0、1、2……），以及连续数据，可以在给定范围内取任何值（例如身高、质量、时间）。
2. Tally Charts and Frequency Tables | 计数表与频数表
A tally chart uses marks to record data. Each group of five is shown as four vertical lines with a diagonal line crossing them (|||| with a strike through). The frequency is the total count for each category. A frequency table organises the tallied data into a clear summary, showing each category and its frequency.
计数表用划记符号记录数据。每组五个表示为四条竖线加一条对角线穿过（卌形计数）。频数是每个类别的计数总和。频数表将计数后的数据组织成清晰的摘要，显示每个类别及其频数。
From a frequency table you can also calculate relative frequency: Relative frequency = Frequency ÷ Total frequency. This tells you the proportion of the whole that the category represents.
通过频数表你还可以计算相对频率：相对频率 = 频数 ÷ 总频数。这告诉你该类别占整体的比例。
3. Bar Charts and Pictograms | 条形图与象形图
Bar charts represent data using rectangular bars of equal width. The height of each bar corresponds to the frequency of the category. Always start the frequency axis at 0, and leave gaps between bars because the categories are separate.
条形图使用等宽的矩形条表示数据。每个条形的高度对应该类别的频数。频数轴必须从0开始，并且条形之间要留有空隙，因为类别是彼此独立的。
Pictograms use pictures or symbols to represent data. A key tells you how many items each picture stands for. For example, one smiley face could represent 2 students. When a frequency is not a multiple of the symbol, part of a picture is drawn.
象形图用图片或符号表示数据。图例会告诉你每个图片代表多少个项目。例如，一个笑脸可能代表2个学生。当频数不是符号代表数量的整数倍时，需要画出部分图片。
4. Pie Charts | 饼图
Pie charts show data as slices of a circle, where each slice represents a category. The size of the angle of each slice is proportional to the frequency. The whole circle (360°) corresponds to the total frequency.
Angle = (Category Frequency ÷ Total Frequency) × 360°
饼图将数据显示为圆形的扇区，每个扇区代表一个类别。每个扇区的角度大小与频数成正比。整个圆（360°）对应总频数。计算扇区角度的公式如上。
For example, if a category has a frequency of 5 out of a total of 20, then the angle is (5 ÷ 20) × 360° = 90°. Always check that the angles sum to 360° before drawing.
例如，如果某个类别的频数是5，总频数是20，那么角度为 (5 ÷ 20) × 360° = 90°。绘制前务必检查所有角度之和是否等于360°。
5. Mean, Median and Mode | 平均数、中位数和众数
The mean (average) is calculated by adding up all the data values and then dividing by the number of values. If the data set has values x₁, x₂, …, xₙ, the mean is given by:
Mean = (Sum of all values) ÷ (Number of values) = (x₁ + x₂ + … + xₙ) / n
平均数（均值）的计算方法是把所有数据值加起来，然后除以值的个数。如果数据集有值 x₁, x₂, …, xₙ，均值公式如上。
The median is the middle value when the data are arranged in order. If there are n values:
中位数是数据按顺序排列后位于中间的值。如果有 n 个值：
• If n is odd, the median is the value at the (n+1)/2 position.
• If n is even, the median is the mean of the values at the n/2 and (n/2)+1 positions.
• 如果 n 是奇数，中位数是第 (n+1)/2 个位置的值。
• 如果 n 是偶数，中位数是第 n/2 和第 (n/2)+1 个位置的两个值的均值。
The mode is the value that appears most often in the data set. A set can have one mode, more than one mode (bimodal or multimodal), or no mode at all if all values occur equally often.
众数是数据集中出现次数最多的值。一组数据可能有一个众数，多个众数（双众数或多众数），或者如果所有值出现次数相同，则没有众数。
6. Range | 极差
The range measures the spread of the data. It is the difference between the largest and smallest values.
Range = Maximum value – Minimum value
极差衡量数据的分散程度，它是最大值与最小值的差。公式如上。
A larger range shows the data are more spread out; a smaller range indicates the data are closer together. Range is affected by outliers.
极差越大，说明数据越分散；极差越小，数据越集中。极差易受异常值的影响。
7. Line Graphs | 折线图
Line graphs are used to display data that change over time. Points are plotted and then connected with straight lines. The horizontal axis usually represents time, and the vertical axis represents the quantity being measured. They help us see trends and patterns.
折线图用于展示随时间变化的数据。在坐标平面上描点，然后用直线连接。横轴通常代表时间，纵轴代表被测量的量。折线图能帮助我们观察趋势和模式。
When reading a line graph, look for upward trends (increase), downward trends (decrease) and flat sections (no change). Always check the scales on both axes.
阅读折线图时，注意上升趋势（增加）、下降趋势（减少）和平坦部分（无变化）。务必检查两条轴的刻度。
8. Scatter Graphs and Correlation | 散点图与相关性
Scatter graphs show the relationship between two sets of data. Each pair of values is plotted as a point. By looking at the pattern, we can describe the correlation:
散点图展示两组数据之间的关系。每一对数值作为一个点绘制。通过观察图形的模式，我们可以描述相关性：
• Positive correlation: as one variable increases, the other also tends to increase (points slope upwards).
• Negative correlation: as one variable increases, the other tends to decrease (points slope downwards).
• No correlation: no clear pattern, the points are scattered randomly.
• 正相关：当一个变量增加时，另一个变量也倾向于增加（点群向上倾斜）。
• 负相关：当一个变量增加时，另一个变量倾向于减少（点群向下倾斜）。
• 无相关：没有明显的模式，点随机分散。
9. Data Collection and Questionnaires | 数据收集与问卷设计
Good data collection starts with a clear question. When designing a questionnaire or survey:
好的数据收集从一个明确的问题开始。在设计问卷或调查时：
• Use simple language and avoid leading questions.
• Include response options that are exhaustive and mutually exclusive.
• For closed questions, provide tick boxes with clear categories. For open questions, allow free text but recognise that it is harder to analyse.
• 使用简单的语言，避免引导性问题。
• 提供详尽且互斥的回答选项。
• 对于封闭式问题，给出有清晰类别的勾选框。对于开放式问题，允许自由填写，但要注意这类回答比较难分析。
A pilot survey tests the questionnaire on a small group first to spot any improvements needed. This helps ensure valid and reliable data.
试调查是先在小群体中测试问卷，以发现需要改进的地方。这有助于确保数据的有效性和可靠性。
10. Probability Basics | 概率基础
Probability is a measure of how likely an event is to happen. The probability of an event A is calculated as:
P(Event) = Number of favourable outcomes / Total number of possible outcomes
概率是衡量事件发生可能性的指标。事件 A 的概率计算公式如上。
The probability scale runs from 0 to 1. A probability of 0 means the event is impossible, and a probability of 1 means it is certain. Probabilities can be written as fractions, decimals or percentages. All outcomes must be equally likely for this formula to apply.
概率尺度的范围从0到1。概率为0表示事件不可能发生，概率为1表示事件必然发生。概率可以用分数、小数或百分比表示。使用该公式时，所有结果必须是等可能的。
11. Probability Scale and Language | 概率尺度与描述
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