📚 Year 7 CCEA Statistics: Formula & Theorem Quick Reference Guide | CCEA 七年级统计公式定理速查手册
This quick reference guide covers the essential formulas, theorems, and key concepts for Year 7 CCEA Statistics. It is designed to help students revise data types, averages, charts, probability, and data collection methods effectively.
本速查手册涵盖了 CCEA 七年级统计的重要公式、定理和关键概念，旨在帮助学生高效复习数据类型、平均数、图表、概率以及数据收集方法。
1. Types of Data | 数据类型
Data is information collected for analysis. It can be classified into categorical (qualitative) data, which describes qualities or categories, and numerical (quantitative) data, which involves numbers.
数据是为分析而收集的信息。它可以分为描述性质或类别的分类（定性）数据，以及涉及数字的数值（定量）数据。
Numerical data can be further split into discrete data, which can only take specific values (often counted), and continuous data, which can take any value within a range (measured).
数值数据可进一步分为只能取特定值的离散数据（通常可计数），以及可在某个范围内取任意值的连续数据（可测量）。
Examples: Eye colour is categorical. Number of siblings is discrete numerical. Height is continuous numerical.
例子：眼睛颜色是分类数据。兄弟姐妹个数是离散数值数据。身高是连续数值数据。
2. Frequency Tables | 频率表
A frequency table organises raw data by counting how often each value or category occurs. Tally marks are often used during data collection to make counting easier.
频率表通过统计每个值或类别出现的次数来整理原始数据。数据收集时常用划记符号以方便计数。
The frequency column simply records the total count for each item. A typical frequency table includes the categories, a tally column, and a frequency column.
频率列仅记录每个项目的总计数。一张典型的频率表包含类别、划记列和频率列。
For example, in a survey of favourite fruits: Apple |||| (freq. 4), Banana ||| (freq. 3), Orange || (freq. 2). This summary makes it easier to analyse data and draw charts.
例如，在一项关于最喜欢的水果的调查中：苹果 划记4次（频数4），香蕉 3次（频数3），橙子 2次（频数2）。这种总结使数据分析和图表绘制更容易。
3. Mode | 众数
The mode is the value that appears most frequently in a data set. It is the only average that can be used for categorical data.
众数是一组数据中出现频率最高的值。它是唯一可以用于分类数据的平均数。
If two values tie for most frequent, the data set is bimodal (two modes). If more than two tie, it is multimodal. If no value repeats, there is no mode.
如果两个值出现次数相同且最多，则该数据集为双峰分布（有两个众数）。如果超过两个值并列最多，则为多峰分布。如果没有重复值，则没有众数。
Example: In the set {red, blue, blue, green, blue, red}, mode is blue. In {2, 3, 4, 4, 5, 6, 6, 6, 7}, the mode is 6.
示例：在集合 {红, 蓝, 蓝, 绿, 蓝, 红} 中，众数是蓝色。在 {2, 3, 4, 4, 5, 6, 6, 6, 7} 中，众数为6。
4. Median | 中位数
The median is the middle value when all data values are arranged in order (usually smallest to largest). It splits the data into two equal halves.
中位数是将所有数据值按顺序（通常从小到大）排列后位于中间的值。它将数据分为两个相等的部分。
To find the median: arrange the n numbers in order. If n is odd, the median is the (n+1)/2 th value. If n is even, the median is the mean of the n/2 th and (n/2 + 1) th values.
求中位数的方法：将 n 个数按顺序排列。如果 n 为奇数，中位数为第 (n+1)/2 个值。如果 n 为偶数，中位数为第 n/2 个值和第 (n/2 + 1) 个值的平均数。
Example: Odd set {3, 5, 7, 9, 11} → median is 7. Even set {2, 4, 6, 8} → median = (4+6)/2 = 5.
示例：奇数数据集 {3, 5, 7, 9, 11} → 中位数为7。偶数数据集 {2, 4, 6, 8} → 中位数 = (4+6)/2 = 5。
The median is not affected by extremely large or small values (outliers), making it a useful average for skewed data.
中位数不受极大或极小值（异常值）的影响，因此对于偏斜数据是一种有用的平均数。
5. Mean | 平均数
The mean (also called the arithmetic average) is calculated by summing all data values and dividing by the number of values.
平均数（又称算术平均值）是将所有数据值相加，然后除以数据个数计算得出。
Mean = (x₁ + x₂ + … + xₙ) ÷ n
公式：平均数 = (x₁ + x₂ + … + xₙ) ÷ n，其中 n 是数据个数。
For example, the mean of {4, 8, 6, 10, 2} is (4+8+6+10+2) ÷ 5 = 30 ÷ 5 = 6.
例如，{4, 8, 6, 10, 2} 的平均数是 (4+8+6+10+2) ÷ 5 = 30 ÷ 5 = 6。
The mean uses every value in the data set, so it can be influenced by outliers. It is most appropriate when the data is fairly symmetric.
平均数使用了数据集中的每一个值，因此会受到异常值的影响。当数据大致对称时最适用。
6. Range | 极差
The range is a simple measure of spread or dispersion. It shows how far apart the smallest and largest values are.
极差是一种简单的离散程度度量。它展示了最小值与最大值相距多远。
Range = Largest value − Smallest value
极差 = 最大值 − 最小值
Example: Data set {12, 15, 11, 18, 14} → Range = 18 − 11 = 7. A small range indicates consistent data, while a large range shows more variation.
示例：数据集 {12, 15, 11, 18, 14} → 极差 = 18 − 11 = 7。极差小说明数据一致性好，极差大则说明变动较大。
Because the range only uses two values, it does not give information about how the other values are spread.
由于极差只使用了两个值，它无法反映其他数值的分布情况。
7. Bar Charts | 条形图
A bar chart displays categorical data with rectangular bars. The height (or length) of each bar is proportional to the frequency of the category it represents.
条形图用矩形条展示分类数据。每个条形的高度（或长度）与其所代表类别的频数成正比。
Bars should be of equal width and separated by gaps, because the categories are distinct. Both axes must be clearly labelled, and the chart should have a title.
条形应宽度相等并留有间隔，因为各类别互不相同。两个轴都必须清晰标注，图表应有标题。
The vertical axis must start at zero so that the bar lengths correctly represent the frequencies. Breaking this rule can make differences look exaggerated.
纵轴必须从零开始，这样条形的长度能正确表示频数。违反此规则可能会让差异显得过分夸大。
Bar charts can be drawn vertically or horizontally. They help compare frequencies at a glance.
条形图可以竖着画或横着画。它们有助于快速比较频数。
8. Pie Charts | 饼图
A pie chart represents
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