📚 Year 8 Cambridge Statistics: Formula & Theorem Quick Reference Guide | 剑桥8年级统计：公式定理速查手册
This quick reference guide is designed for Year 8 Cambridge students. It brings together all the essential formulas, theorems and key points you need for statistics. Use it for fast revision and to boost your confidence when working with data and probability.
这份速查手册专为剑桥8年级学生打造。它汇总了你需要掌握的所有统计核心公式、定理及要点，帮助你快速复习、增强处理数据和概率问题的信心。
1. Types of Data | 数据类型
Data can be qualitative (categorical) or quantitative (numerical). Quantitative data is further split into discrete data (countable, e.g. number of books) and continuous data (measurable, e.g. height in cm).
数据可以是定性（分类）数据或定量（数值）数据。定量数据又分为离散数据（可数，如书本数量）和连续数据（可测量，如以厘米为单位的身高）。
Recognising the data type helps you choose the right chart and the right way to calculate averages.
识别数据类型有助于你选择正确的图表以及正确的平均数计算方法。
2. Frequency Tables and Diagrams | 频率表与统计图
A frequency table organises raw data using tally marks and a frequency column. It shows how many times each value or group occurs.
频率表使用计数符号和频数列整理原始数据，显示每个值或组出现的次数。
he frequency. For a pie chart, each sector angle is: (frequency ÷ total frequency) × 360°.
条形图以等宽长方形展示数据，每个长方形的高度代表频数。饼图中，每个扇形的圆心角计算公式为：(频数 ÷ 总频数) × 360°。
Line graphs are often used to show trends over time.
折线图常用于显示随时间变化的趋势。
3. Mean, Median and Mode | 平均数、中位数与众数
The mean is the sum of all data values divided by the number of values. The median is the middle value when the data is arranged in order. The mode is the value that appears most frequently.
平均数是所有数据值的总和除以数据的个数。中位数是将数据按顺序排列后位于中间的值。众数是出现次数最多的值。
Mean (x̄) = (∑x) ÷ n
Median position = (n + 1) ÷ 2 (for ordered data)
For an odd number of values, the median is the single middle value. For an even number, it is the mean of the two middle numbers.
如果数据个数为奇数，中位数就是正中间的那个值。如果数据个数为偶数，中位数则是中间两个数的平均数。
The mode can be used for numerical or categorical data. A data set can have one mode, more than one mode (bimodal) or no mode at all.
众数既可用于数值数据也可用于分类数据。一组数据可以有一个众数、多个众数（双众数）或者没有众数。
4. Mean from a Frequency Table | 从频数表求平均数
When working with a frequency table, multiply each data value by its frequency, add all these products, then divide by the sum of the frequencies.
使用频数表时，将每个数据值乘以其频数，把所有乘积相加，再除以频数总和。
Mean = (∑(value × frequency)) ÷ ∑frequency
× frequency) ÷ total frequency. This is an estimate because the exact values are unknown.
对于分组数据（组距分组），先找出每组的组中点。估算平均数 = ∑(组中点 × 频数) ÷ 总频数。这是一个估计值，因为原始精确值未知。
5. Range | 极差
The range measures how spread out the data is. It is found by subtracting the smallest value from the largest value.
极差衡量数据的分散程度，由最大值减去最小值得到。
Range = Maximum value – Minimum value
A larger range means greater variability. The range is sensitive to extreme values (outliers).
极差越大表示数据变动性越大。极差容易受极端值（异常值）影响。
6. Stem-and-Leaf Diagrams | 茎叶图
A stem-and-leaf diagram shows the shape of the distribution while keeping all the original data values. The 'stem' is the leading part of each number and the 'leaf' is the final digit. You must include a key explaining the place value.
茎叶图既能显示分布形态，又能保留所有原始数据。'茎'是每个数字的前导部分，'叶'是最后一位数字。你必须附上图例说明数位。
To draw one: order leaves from smallest to largest. To find the median, count from the smallest leaf inwards until you reach the middle.
绘制步骤：将叶子从小到大排列。寻找中位数时，从最小的叶子开始向中间数，直到到达中间位置。
7. Basic Probability | 基础概率
Probability is a measure of how likely an event is to happen. It lies on a scale from 0 (impossible) to 1 (certain).
概率是事件发生可能性的度量，其刻度介于0（不可能）和1（必然发生）之间。
P(Event) = Number of favourable outcomes ÷ Total number of equally likely outcomes
Probabilities can be written as fractions, decimals or percentages.
概率可以写成分数、小数或百分数。
ith a fair six-sided die, P(rolling a 2) = 1/6, and P(rolling an even number) = 3/6 = 1/2.
例如，掷一枚公平的六面骰子，P(掷出2) = 1/6，P(掷出偶数) = 3/6 = 1/2。
8. Sample Spaces and Listing Outcomes | 样本空间与结果列举
A sample space is the set of all possible outcomes of an experiment. It can be listed as a set or shown in a two-way table.
样本空间是一个实验所有可能结果的集合。你可以用集合的形式列出，也可以用双向表展示。
When two events happen together (e.g., tossing a coin and rolling a die), systematic listing avoids missing outcomes. The outcomes for a coin and a die: (H,1), (H,2), …, (T,6).
当两个事件同时发生（如掷硬币和骰子），系统列举可以避免遗漏结果。硬币与骰子的结果有：(H,1), (H,2), …, (T,6).
Using a sample space diagram helps you count total outcomes and identify favourable outcomes for probability calculations.
使用样本空间图有助于计数总结果，并找出概率计算中的有利结果。
9. Mutually Exclusive Events and Sum of Probabilities | 互斥事件与概率之和
th occur at the same time. For mutually exclusive events A and B: P(A or B) = P(A) + P(B).
如果两个事件不能同时发生，则它们是互斥的。对于互斥事件A和B：P(A 或 B) = P(A) + P(B)。
The sum of the probabilities of all mutually exclusive outcomes in an experiment is always 1.
一个实验中所有互斥结果的概率之和总是1。
This also means P(not A) = 1 – P(A), which is useful for finding the probability of a complement.
这也意味着 P(非A) = 1 – P(A)，此关系在求补集概率时非常有用。
10. Experimental Probability | 实验概率
Experimental probability, also called relative frequency, is based on actual trials or observations.
实验概率，也称相对频率，是基于实际试验或观察得到的。
Experimental probability = Number of times the event occurs ÷ Total number of trials
As the number of trials increases, the experimental probability usually gets closer to the theoretical probability.
随着试验次数的增加，实验概率通常会越来越接近理论概率。
n lands point up 37 times out of 100 drops, the experimental probability is 37/100 = 0.37.
例如，一枚图钉落地100次中有37次针尖朝上，实验概率就是37/100 = 0.37。
11. Common Mistakes and Tips | 常见错误与提示
Always order the data before finding the median; forgetting to order is a common error.
求中位数前务必先排序；忘记排序是常见错误。
When reading a stem-and-leaf diagram, remember to use the key to get the correct values.
解读茎叶图时，记住通过图例还原正确的数值。
For grouped frequency tables, use midpoints, not the class boundaries, to estimate the mean.
使用分组频数表估算平均数时，要用组中点，而非组界限。
Check that the probabilities you find for all possible outcomes add up to 1. If they do not, your counting may be incomplete.
检查你找出的所有可能结果的概率和是否为1。如果不等于1，说明你的计数可能不完整。
Label your charts and diagrams clearly; missing labels can lose marks even when the drawing is correct.
为图表配上清晰的标签；即使绘图正确，缺少标签仍可能丢分。
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