📚 KS3 CAIE Statistics: Quick Reference Formula and Theorem Handbook | KS3 CAIE 统计：公式定理速查手册
This handbook provides a concise reference of key formulas and theorems for KS3 CAIE Statistics. Designed for quick revision, each concept is explained in clear, student-friendly language, with both English and Chinese explanations to support bilingual learners. Keep this guide handy for homework, tests, and end‑of‑year examinations.
本手册为 KS3 CAIE 统计课程的关键公式和定理提供简明参考，旨在帮助快速复习。每个概念均以清晰易懂的学生语言解释，并配有中英双语说明，以支持双语学习者。在作业、测验和年终考试时随时查阅本指南。
1. Mean | 平均数
The mean (arithmetic average) is a measure of central tendency. It is found by adding all the data values together and then dividing by the number of values.
平均数（算术平均值）是一种集中趋势的度量。计算方法是：将所有数据值相加，然后除以数值的个数。
Mean = Σx ÷ n
其中 Σx 表示所有数据值的总和，n 是数据值的个数。
Example: Find the mean of 3, 7, 8, 2, 5.
示例：求 3, 7, 8, 2, 5 的平均数。
Sum = 3 + 7 + 8 + 2 + 5 = 25, n = 5, so Mean = 25 ÷ 5 = 5.
总和 = 25，个数 = 5，因此平均数 = 25 ÷ 5 = 5。
2. Median | 中位数
The median is the middle value when the data are arranged in order of size. If there are two middle values (even number of data), the median is the mean of those two values.
中位数是将数据按大小顺序排列后位于中间位置的数值。如果有两个中间值（数据个数为偶数），中位数是这两个数值的平均数。
To find the median: order the data; count the number of values, n. If n is odd, the median is the (n+1)/2-th value. If n is even, take the average of the n/2-th and (n/2 + 1)-th values.
求中位数的方法：将数据排序；统计数据个数 n。如果 n 是奇数，中位数是第 (n+1)/2 个值。如果 n 是偶数，取第 n/2 个和第 (n/2 + 1) 个值的平均数。
Example (odd): data 4, 1, 7, 3, 9 → ordered: 1, 3, 4, 7, 9. n=5, median is the 3rd value, which is 4.
示例（奇数个）：数据 4, 1, 7, 3, 9 → 排序后：1, 3, 4, 7, 9。n=5，中位数是第3个值，即 4。
Example (even): data 12, 5, 8, 15, 10, 20 → ordered: 5, 8, 10, 12, 15, 20. n=6, median = (10 + 12) ÷ 2 = 11.
示例（偶数个）：数据 12, 5, 8, 15, 10, 20 → 排序后：5, 8, 10, 12, 15, 20。n=6，中位数 = (10+12) ÷ 2 = 11。
3. Mode | 众数
The mode is the value that appears most often in a data set. A set of data may have one mode, more than one mode (bimodal or multimodal), or no mode at all if no value repeats.
众数是数据集中出现次数最多的数值。一组数据可能有一个众数、多个众数（双峰或多峰），或者如果没有重复值则没有众数。
Example: 5, 2, 5, 3, 5, 8 → mode is 5.
示例：5, 2, 5, 3, 5, 8 → 众数是 5。
Example of bimodal: 1, 2, 3, 2, 4, 3 → modes are 2 and 3.
双众数示例：1, 2, 3, 2, 4, 3 → 众数是 2 和 3。
Note: For grouped data, the modal class is the class interval with the highest frequency.
注意：对于分组数据，众数所在的组称为众数组，即频数最高的组距。
4. Range | 极差
The range is a measure of spread. It is the difference between the largest and the smallest values in the data set.
极差是一种离散程度的度量。它是数据中最大值与最小值的差。
Range = Maximum value − Minimum value
极差 = 最大值 − 最小值
Example: Data 23, 45, 12, 67, 34. Maximum = 67, minimum = 12, range = 67 − 12 = 55.
示例：数据 23, 45, 12, 67, 34。最大值 = 67，最小值 = 12，极差 = 67 − 12 = 55。
A larger range indicates greater variability; a smaller range means the data are more clustered.
极差越大表示变异性越大；极差越小意味着数据越集中。
5. Quartiles and Interquartile Range | 四分位数和四分位距
Quartiles divide an ordered data set into four equal parts. The first quartile (Q₁) is the median of the lower half of the data; the second quartile (Q₂) is the median of the whole data; the third quartile (Q₃) is the median of the upper half.
四分位数将排序后的数据分成四等份。第一四分位数 (Q₁) 是数据下半部分的中位数；第二四分位数 (Q₂) 是整个数据的中位数；第三四分位数 (Q₃) 是上半部分的中位数。
Interquartile range (IQR) = Q₃ − Q₁
四分位距 (IQR) = Q₃ − Q₁
The IQR measures the spread of the middle 50% of the data and is not affected by extreme values.
四分位距衡量中间 50% 数据的分散程度，且不受极端值的影响。
To find quartiles: order the data. Locate the median (Q₂). Then find the median of the values before Q₂ (this gives Q₁) and the median of the values after Q₂ (this gives Q₃). If the number of data points is odd, exclude the median when forming the halves.
求四分位数的方法：将数据排序。找到中位数 (Q₂)。然后找出在 Q₂ 之前的那部分数据的中位数（得到 Q₁）和在 Q₂ 之后的那部分数据的中位数（得到 Q₃）。如果数据个数为奇数，划分两半时不包括中位数。
6. Frequency Tables and Mean from a Frequency Table | 频数表及由频数表求平均数
A frequency table lists distinct data values or groups alongside the number of times each occurs (frequency). Tally marks are often used to record frequencies.
频数表列出不同的数据值或组别，以及每个值出现的次数（频数）。划记符号常用于记录频数。
For discrete data in a frequency table, the mean is calculated using:
对于频数表中的离散数据，计算平均数使用下式：
Mean = Σ(f × x) ÷ Σf
where x represents each data value and f its frequency.
其中 x 代表每个数据值，f 代表该值的频数。
Example:
示例：
| Value (x) | Frequency (f) | f × x |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 5 | 10 |
| 3 | 2 | 6 |
| 4 | 1 | 4 |
| Total | Σf = 11 | Σ(f×x) = 23 |
Mean = 23 ÷ 11 ≈ 2.09
平均数 = 23 ÷ 11 ≈ 2.09
If data are grouped into class intervals, use the midpoint of each interval as x, and the result is an estimate of the mean.
如果数据被分成组距，则用每组的组中值作为 x，这样求出的平均数是估计值。
7. Bar Charts, Pictograms and Pie Charts | 条形图、象形图和饼图
Bar chart: uses bars of equal width to represent frequencies for different categories. The height of each bar corresponds to the frequency. Bars should not touch for discrete data.
条形图：用等宽的条形表示不同类别的频数。每个条形的高度代表频数。对于离散数据，条形之间不应接触。
Pictogram: uses pictures or symbols to represent frequencies. A key must show the value of one symbol (e.g., 1 picture = 2 students).
象形图：用图片或符号表示频数。必须用一个图例说明一个符号代表的数量（例如，1个图形代表2名学生）。
Pie chart: displays data as sectors of a circle. The angle of each sector is proportional to the frequency.
饼图：将数据表示为圆的扇形区域。每个扇形的角度与频数成正比。
Sector angle = (Frequency ÷ Total frequency) × 360°
扇形角度 = (频数 ÷ 总频数) × 360°
Example: if 15 out of 30 students prefer football, the pie chart sector angle = (15 ÷ 30) × 360° = 180°.
示例：如果 30 名学生中有 15 名偏爱足球，则饼图的扇形角度 = (15 ÷ 30) × 360° = 180°。
8. Line Graphs and Time Series | 折线图与时间序列
A line graph is used to display data that change over a continuous scale, often over time. Points are plotted and connected by straight lines to show trends.
折线图用于显示随连续尺度（通常是时间）变化的数据。在图上标出数据点并用直线连接，以展示趋势。
Time series graphs are line graphs where the horizontal axis always represents time. They help identify patterns such as increasing, decreasing or seasonal trends.
时间序列图是一种折线图，横轴始终代表时间。它们有助于识别增长、下降或季节性等模式。
When reading time series, look for overall trend (upward or downward) and any regular fluctuations.
解读时间序列时，注意整体趋势（上升或下降）以及任何有规律的波动。
Always label both axes and give the graph a title.
始终给两个坐标轴加注标签，并给图表加上标题。
9. Scatter Graphs and Correlation | 散点图与相关
A scatter graph displays pairs of numerical data. Each point represents two values for one item (e.g., height and weight).
散点图展示成对的数值数据。每个点表示同一个对象的两个值（例如身高和体重）。
Correlation describes the relationship between the two variables:
相关描述两个变量之间的关系：
- Positive correlation: as one variable increases, the other also tends to increase.
- 正相关：一个变量增加，另一个也趋于增加。
- Negative correlation: as one variable increases, the other tends to decrease.
- 负相关：一个变量增加，另一个趋于减少。
- No correlation: no clear pattern between the variables.
- 无相关：变量之间没有明显的模式。
The strength of correlation can be described as strong (points close to a line) or weak (points widely scattered).
相关的强弱程度可以描述为强相关（点紧密围绕一条直线）或弱相关（点分布散乱）。
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