📚 Year 8 Edexcel Statistics: Formula & Theorem Quick Reference Handbook | Year 8 爱德思统计：公式定理速查手册
This quick reference guide brings together the key formulas, definitions, and theorems you need for Year 8 Edexcel Statistics. Keep it handy for homework and revision to quickly look up averages, probability, and how to interpret charts.
本速查手册汇集了 Year 8 爱德思统计课程所需的关键公式、定义和定理。可随时用于作业和复习，快速查阅平均数、概率以及如何解读图表。
1. Mean, Median, Mode and Range | 平均数、中位数、众数和极差
The three measures of central tendency describe the ‘centre’ of a data set, while the range describes how spread out the data are.
三个集中趋势量度描述数据集的“中心”，而极差描述数据的离散程度。
The mean is the numerical average. Add all data values and divide by the number of values.
Mean = Σx / n
where Σx is the sum of all data values and n is the number of values.
平均数（均值）是数值平均值。将所有数据值相加，再除以数据的个数。
平均数 = Σx / n
其中 Σx 是所有数据值的总和，n 是数据的总个数。
The median is the middle value when the data are arranged in ascending order. If there are an even number of values, take the mean of the two middle values.
中位数是按升序排列后位于中间的数据值。如果数据个数为偶数，则取中间两个值的平均数。
The mode is the value that occurs most frequently. A data set can have one mode, more than one mode (bimodal), or no mode at all.
众数是出现次数最多的数据值。一个数据集可能有一个众数、一个以上的众数（双众数），也可能没有众数。
The range is a measure of spread. It is the difference between the largest and smallest values.
Range = Maximum value − Minimum value
极差是衡量离散程度的量，等于最大值与最小值之差。
极差 = 最大值 − 最小值
2. Calculating the Mean from a Frequency Table | 从频率表计算平均数
When data are grouped in a frequency table, each value must be weighted by its frequency. Multiply each data value (x) by its frequency (f), sum these products, then divide by the total frequency.
当数据以频率表呈现时，每个值必须乘以其频数加权。将每个数据值 (x) 乘以其频率 (f)，求出这些乘积的总和，再除以总频率。
Mean = Σ(f x) / Σf
where f is the frequency of each value, x is the data value, and Σf is the total number of data items.
其中 f 是每个值的频数，x 是数据值，Σf 是数据的总个数。
平均数 = Σ(f × x) / Σf
3. Finding the Median from a List and Stem-and-Leaf Diagram | 从列表和茎叶图求中位数
To find the median from an ordered list, count the number of values, n. The position of the median is (n + 1) / 2. If this position gives a decimal, the median is the number halfway between the two middle values.
从有序列表中求中位数时，先计数数据个数 n。中位数的位置是 (n + 1) / 2。如果这个位置是小数，中位数是位于中间两个值正中间的数。
In a stem-and-leaf diagram, each number is split into a stem (leading digit(s)) and a leaf (final digit). A key must be given, e.g., 2 | 5 means 25. Count the total leaves, then locate the middle leaf using the position rule.
在茎叶图中，每个数被分成茎（前导数字）和叶（最后一位数字）。必须给出键，例如 2 | 5 代表 25。数出总叶数，然后使用位置规则找到中间的叶。
Example: For data stems: 1 | 2 5, 2 | 0 3 4, with key 1 | 2 = 12. The ordered list is 12, 15, 20, 23, 24. n = 5, median position = (5+1)/2 = 3rd value, which is 20.
示例：数据茎：1 | 2 5, 2 | 0 3 4，键为 1 | 2 = 12。有序列表为 12, 15, 20, 23, 24。n=5，中位数位置= (5+1)/2=第3个值，即20。
4. Probability Scale and Basic Probability | 概率尺度和基本概率
Probability measures how likely an event is to happen. It always lies between 0 and 1: 0 means impossible, 1 means certain. Probabilities can be written as fractions, decimals, or percentages.
概率衡量一个事件发生的可能性大小。它总是在 0 到 1 之间：0 表示不可能，1 表示一定发生。概率可以用分数、小数或百分数表示。
P(Event) = Number of favourable outcomes / Total number of equally likely outcomes
provided all outcomes are equally likely.
P(事件) = 有利结果数 / 所有等可能结果的总数
前提是所有结果等可能发生。
The probability of an event not occurring is: P(not A) = 1 − P(A).
某事件不发生的概率为：P(非A) = 1 − P(A)。
5. Experimental Probability and Relative Frequency | 实验概率与相对频率
When an experiment is repeated, the relative frequency of an event can be used to estimate its probability.
Relative frequency = Number of times the event occurs / Total number of trials
当重复进行实验时，事件的相对频率可用于估计其概率。
相对频率 = 事件发生的次数 / 总的试验次数
As the number of trials increases, the relative frequency tends to get closer to the theoretical probability. This is sometimes called the law of large numbers.
随着试验次数的增加，相对频率往往会越来越接近理论概率。这有时被称为大数定律。
6. Sample Space Diagrams | 样本空间图
A sample space diagram lists all possible outcomes of a combined event. It can be shown as a list, a table, or a two-way grid. For example, when rolling a fair dice and tossing a fair coin, there are 12 equally likely outcomes.
样本空间图列出一个组合事件所有可能的结果。它可以用列表、表格或双向网格表示。例如，同时掷一个均匀骰子和抛一枚均匀硬币，共有 12 种等可能结果。
| Dice/Coin | Head (H) | Tail (T) |
|---|---|---|
| 1 | (H,1) | (T,1) |
| 2 | (H,2) | (T,2) |
| … | … | … |
| 6 | (H,6) | (T,6) |
Using the sample space, you can directly count favourable outcomes. For instance, P(H and even number) = number of outcomes with H and an even number (3) / 12 = 1/4.
利用样本空间，可以直接数出有利结果的个数。例如，P(正面且偶数) = 正面且偶数的结果数(3) / 12 = 1/4。
7. Two-way Tables | 双向表
Two-way tables organise data for two categorical variables. The row and column totals help to calculate probabilities directly from the table.
双向表用于整理两个类别变量的数据。行和列的合计数有助于直接从表中计算概率。
Example table: 30 students, gender and left/right-handed.
| Left-handed | Right-handed | Total | |
|---|---|---|---|
| Boys | 2 | 13 | 15 |
| Girls | 3 | 12 | 15 |
| Total | 5 | 25 | 30 |
P(boy and left-handed) = 2/30 = 1/15.
P(男生且左撇子) = 2/30 = 1/15。
8. Bar Charts and Pictograms | 条形图和象形图
Bar charts display frequency with equal-width bars, where the height (for vertical bars) or length (for horizontal bars) corresponds to the frequency. The bars must not touch, and both axes should be clearly labelled.
条形图用等宽的条形表示频率，条形的高度（垂直条形图）或长度（水平条形图）对应频率。条形之间不能接触，且两轴应清晰标注。
Pictograms use symbols to represent a fixed number of items. Always check the key to interpret the frequency correctly. For example, one whole picture may represent 5 books; a half picture represents 2.5 books (or round appropriately as per the context).
象形图使用符号代表一定数量的项目。务必查看健以正确解读频率。例如，一个完整图形可代表5本书；半个图形代表2.5本书（或根据上下文适当取整）。
9. Pie Charts and Calculating Angles | 饼图和计算角度
A pie chart displays proportions by dividing a circle into sectors. The angle of each sector represents the frequency for that category. To construct a pie chart, first find the total frequency, then compute each sector angle.
饼图通过将圆分成若干个扇形来显示比例。每个扇形的角度代表该类别的频率。要绘制饼图，先求出总频率，然后计算每个扇形的角度。
Sector angle = (Frequency of category / Total frequency) × 360°
Always check that the sum of all sector angles equals 360°.
扇形角度 = (类别频率 / 总频率) × 360°
始终要验证所有扇形角度之和等于360°。
10. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph shows the relationship between two sets of numerical data. Each point represents a pair of values (x, y). Correlation describes the pattern:
- Positive correlation: as x increases, y tends to increase.
- Negative correlation: as x increases, y tends to decrease.
- No correlation: there is no clear pattern.
散点图显示两组数值数据之间的关系。每个点代表一对值 (x, y)。相关性描述模式：
- 正相关：x 增大时，y 也趋于增大。
- 负相关：x 增大时，y 趋于减小。
- 无相关：没有明显模式。
A line of best fit (trend line) can be drawn to show the general direction of the data. It should have roughly equal numbers of points above and below the line, and outliers can be identified as points far from this line.
可以画一条最佳拟合线（趋势线）来显示数据的大致方向。该线上下两侧的点数应大致相等，远离该线的点可被识别为异常值。
11. Misleading Graphs and Data Interpretation | 误导性图表与数据解读
Always read the axes labels, scales, and titles carefully. Graphs can be misleading in several ways:
- The vertical axis may not start at zero, making differences appear larger.
- Uneven scales or missing intervals can distort the picture.
- 3D effects and perspective can make bars look longer or shorter than they really are.
- In pictograms, using images of different sizes (instead of repeating the same symbol) can exaggerate differences.
务必仔细阅读轴标签、刻度和标题。图表可能通过
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