📚 Year 8 CAIE Statistics: Quick Reference Handbook of Formulas and Theorems | 8年级CAIE统计：公式定理速查手册
This quick reference handbook covers all essential formulas and theorems for Year 8 CAIE Statistics, including measures of central tendency, data representation, probability, and more. Use it for revision, homework, or exam preparation.
本速查手册涵盖8年级CAIE统计所有关键公式与定理，包括集中趋势的度量、数据表示、概率等。可用于复习、作业或备考。
1. Mean | 平均数
The mean is the average of a set of numbers. It is calculated by adding all the values together and dividing by the number of values.
平均数是一组数据的平均值，计算方法是将所有数值相加，再除以数值的个数。
Mean = x̄ = Σx / n
Where Σx is the sum of all data values, and n is the number of values. The symbol x̄ (pronounced 'x-bar') represents the sample mean.
其中 Σx 是所有数据值的总和，n 是数值的个数。符号 x̄（读作 “x-bar”）代表样本平均数。
: Find the mean of 5, 8, 12, 15. Sum = 5 + 8 + 12 + 15 = 40, n = 4, so Mean = 40 ÷ 4 = 10.
示例：求 5、8、12、15 的平均数。总和 = 5 + 8 + 12 + 15 = 40，n = 4，因此平均数 = 40 ÷ 4 = 10。
2. Median | 中位数
The median is the middle value in an ordered data set. Arrange the numbers from smallest to largest. The position of the median is found using (n + 1) ÷ 2, where n is the number of values.
中位数是有序数据集的中间值。将数字从小到大排列。中位数的位置用 (n + 1) ÷ 2 求得，其中 n 是数值的个数。
Median position = (n + 1) ÷ 2
If the position is a whole number, the median is the value at that position. If it is a decimal (e.g., 5.5), the median is the mean of the two surrounding values (the 5th and 6th values).
如果该位置是整数，中位数就是该位置上的值。如果是小数（如 5.5），中位数则是两个相邻值（第 5 个和第 6 个值）的平均数。
, median = 4. Even case: 2, 5, 8, 10 → position = (4+1)÷2 = 2.5, median = (5+8) ÷ 2 = 6.5.
示例（奇数 n）：数据 4, 1, 7, 3, 9 → 排序后 1, 3, 4, 7, 9；n = 5，位置 = (5+1)÷2 = 3，中位数 = 4。偶数情况：2, 5, 8, 10 → 位置 = (4+1)÷2 = 2.5，中位数 = (5+8) ÷ 2 = 6.5。
3. Mode and Range | 众数和极差
The mode is the value that appears most often. A data set can have no mode, one mode (unimodal), or multiple modes (bimodal, multimodal). The mode is useful for categorical data.
众数是出现次数最多的值。一组数据可以没有众数、有一个众数（单峰）或多个众数（双峰、多峰）。众数对分类数据尤其有用。
The range measures the spread of the data. It is found by subtracting the smallest value from the largest value.
极差衡量数据的离散程度，计算方法是用最大值减去最小值。
Range = Maximum – Minimum
or the data set 3, 7, 7, 9, 10, the mode is 7 (appears twice) and the range is 10 − 3 = 7.
示例：对于数据集 3, 7, 7, 9, 10，众数为 7（出现两次），极差为 10 − 3 = 7。
4. Frequency Tables | 频数表
A frequency table organises data by showing each distinct value (or group) together with its frequency — how many times it occurs. It is a powerful tool for finding averages and the spread of data.
频数表通过列出每个不同值（或组）及其频数（出现的次数）来整理数据。它是求取平均数与数据分布的有力工具。
Mean from a frequency table: x̄ = Σ(f × x) ÷ Σf
Multiply each value (x) by its frequency (f), add all these products, then divide by the total frequency Σf.
将每个值 (x) 乘以其频数 (f)，把所有这些乘积相加，再除以总频数 Σf。
| Score (x) | Frequency (f) | f × x |
|---|---|---|
| 2 | 3 | 6 |
| 3 | 5 | 15 |
| 4 | 2 | 8 |
| Total | Σf = 10 | Σfx = 29 |
Then mean = 29 ÷ 10 = 2.9.
因此平均数 = 29 ÷ 10 = 2.9。
To find the median from a frequency table, add a cumulative frequency column. The median position is (Σf + 1) ÷ 2. Locate the smallest value whose cumulative frequency is at least this position.
要从频数表求中位数，可添加累计频数列。中位数位置为 (Σf + 1) ÷ 2。找到累计频数至少达到此位置的最小值即可。
sition = (10+1)÷2 = 5.5. The cumulative frequency reaches 8 by the score 3, so median = 3.
在上例中，累计频数为 3, 8, 10；中位数位置 = (
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