📚 Year 9 WJEC Statistics: Quick Reference Formula & Theorems Handbook | WJEC Year 9 统计公式定理速查手册
This quick reference handbook draws together all the essential formulas, rules and statistical theorems you will meet in the Year 9 WJEC Statistics curriculum. Use it to check definitions, reinforce key methods and avoid slip-ups in assessments. Each formula is presented with a brief explanation so that you can apply it confidently to data handling, probability and diagrammatic work.
这本速查手册汇总了 WJEC Year 9 统计课程中所有核心公式、定理和规则。你可以用它来核对定义、巩固关键方法、避免考试失误。每个公式都配有简要解释，帮助你自信地处理数据、概率和统计图表问题。
1. Measures of Central Tendency | 集中趋势的度量
The mean (arithmetic average) is found by adding all data values and dividing by the number of values. For ungrouped data: Mean, x̄ = Σx / n, where Σx is the sum of values and n is the sample size.
平均数（算术平均值）通过将所有数据值相加再除以数据个数求得。对于未分组数据：平均数 x̄ = Σx / n，其中 Σx 是数值总和，n 是样本容量。
The median is the middle value when data are arranged in order of size. For n data points, the position of the median is (n+1)/2. If there is an even number of data, take the average of the two central values.
中位数是将数据按大小排序后的中间值。对于 n 个数据点，中位数的位置是 (n+1)/2。若数据个数为偶数，则取中间两个数的平均值。
The mode is the value that appears most frequently in a data set. A set of data can have one mode, more than one mode (bimodal or multimodal) or no mode at all.
众数是数据集中出现次数最多的值。一组数据可以有一个众数、多个众数（双众数或多众数），也可以没有众数。
The mode is the only measure of central tendency that can be used for non-numerical (categorical) data, such as colours or favourite subjects.
众数是唯一可以用于非数值（类别）数据的集中趋势度量，例如颜色或最喜欢的科目。
2. Measures of Spread | 离散程度的度量
The range is the simplest measure of spread. It is the difference between the largest and smallest values: Range = Maximum value – Minimum value. Because it uses only two data points, it can be heavily affected by outliers.
极差是最简单的离散度量。它是最大值与最小值之差：极差 = 最大值 – 最小值。由于只用到了两个数据点，极差受离群值影响很大。
The interquartile range (IQR) measures the spread of the middle 50% of data. It is given by IQR = Upper quartile (Q₃) – Lower quartile (Q₁). The IQR is more robust against outliers than the range.
四分位数间距（IQR）衡量中间 50% 数据的离散程度，公式为 IQR = 上四分位数 (Q₃) – 下四分位数 (Q₁)。与极差相比，IQR 对离群值更具稳健性。
To find the position of the lower quartile Q₁, use (n+1)/4. For the upper quartile Q₃, use 3(n+1)/4. When the positions are not whole numbers, use interpolation or take a simple average of the surrounding data points, following the method taught in your course.
下四分位数 Q₁ 的位置公式为 (n+1)/4。上四分位数 Q₃ 的位置公式为 3(n+1)/4。当位置不是整数时，需要按照课堂所教的方法进行插值或取相邻两个数据的平均值。
3. Frequency Tables and the Estimated Mean | 频数表与估算均值
When data are summarised in a frequency table, the mean is estimated by working with midpoints of class intervals. The formula becomes: Estimated mean = Σ(f × x) / Σf, where f is the frequency of each class and x is the class midpoint.
当数据以频数表形式概括时，需要用组中值来估算均值。公式为：估算均值 = Σ(f × x) / Σf，其中 f 是每组的频数，x 是该组的组中值。
The class midpoint is found by adding the lower and upper boundaries of the interval and dividing by two. For open-ended intervals (e.g. ’50 and above’), you must make an assumption about the upper boundary before finding a midpoint.
组中值等于组下限加上组上限再除以二。对于开口组（例如“50 及以上”），必须先对上限作出假设才能计算组中值。
Finding the median or quartiles from a grouped frequency table requires cumulative frequency. The cumulative frequency for a class is the running total of frequencies up to and including that class. The median, Q₁ and Q₃ positions are found using the same position formulas on the cumulative frequency scale.
从分组频数表求中位数或四分位数需要用到累积频数。一个组的累积频数是之前各组频数加上当前组频数的累计值。中位数、Q₁ 和 Q₃ 的位置可以在累积频数尺度上利用相同的位置公式求得。
4. Stem and Leaf Diagrams & Box Plots | 茎叶图与箱线图
A stem and leaf diagram sorts data into stems (the leading digit(s)) and leaves (the final digit). It preserves the original data values while showing their distribution. A key must always be provided, e.g. ‘3 | 5 means 35’.
茎叶图将数据按茎（前导数字）和叶（最后一位数字）排列，既保留了原始数值，又展示了数据的分布。必须附上图例说明，例如“3 | 5 表示 35”。
An ordered stem and leaf diagram arranges the leaves in ascending order to the right of each stem. This makes it easy to read off the median, quartiles and range directly.
有序茎叶图将每个茎右边的叶按升序排列，这样可以直接从图上读出中位数、四分位数和极差。
A box plot (box-and-whisker diagram) uses the five-number summary: minimum, Q₁, median, Q₃ and maximum. The box spans from Q₁ to Q₃ with the median marked inside. The whiskers extend to the minimum and maximum, unless outliers are shown separately. The IQR is the length of the box.
箱线图（盒须图）使用五数概括：最小值、Q₁、中位数、Q₃ 和最大值。箱体从 Q₁ 延伸到 Q₃，中位数标在箱内。须线延伸至最小值和最大值，除非离群值单独标出。箱体的长度就是 IQR。
5. Probability Basics | 概率基础
Probability measures the chance of an event happening. It is always a number between 0 and 1 (or between 0% and 100%).
概率衡量事件发生的可能性，始终是一个介于 0 和 1 之间的数（或 0% 至 100% 之间）。
The probability of an event A is calculated as: P(A) = number of favourable outcomes / total number of possible outcomes, provided all outcomes are equally likely.
事件 A 的概率计算公式为： P(A) = 有利结果数 / 所有可能结果的总数，前提是所有结果等可能发生。
The probability that event A does not happen is called the complement of A, written as P(A’). The relationship is: P(A’) = 1 – P(A).
事件 A 不发生的概率叫做 A 的补集，记作 P(A′)。关系式为： P(A′) = 1 – P(A)。
Two events are mutually exclusive if they cannot happen at the same time. For mutually exclusive events A and B: P(A or B) = P(A) + P(B). This is the ‘or rule’ for mutually exclusive events.
如果两个事件不可能同时发生，则它们互斥。对于互斥事件 A 和 B： P(A 或 B) = P(A) + P(B)。这就是互斥事件的“或规则”。
6. Tree Diagrams and the Multiplication Rule | 树形图与乘法法则
Tree diagrams show all the possible outcomes of a sequence of events. Probabilities are written on the branches. The sum of probabilities on all branches from a single point must equal 1.
树形图展示一系列事件所有可能的结果。概率标在树的分支上。从同一点出发的所有分支的概率之和必须等于 1。
The probability of a combination of independent events is found by multiplying the probabilities along the relevant branches: P(A and B) = P(A) × P(B given A). When events are independent, P(B given A) simplifies to P(B).
独立事件组合的概率等于相应路径上各分支概率的乘积： P(A 和 B) = P(A) × P(给定 A 时 B 的概率)。当事件独立时，P(给定 A 时 B 的概率) 简化为 P(B)。
To find the probability of an event that can occur in more than one way, add the probabilities of all the paths that lead to that event. This is the ‘and/or’ reasoning on tree diagrams.
若一个事件可通过多条路径发生，则将该事件所有路径的概率相加。这就是树形图上的“且/或”逻辑。
7. Scatter Graphs and Correlation | 散点图与相关
A scatter graph shows the relationship between two sets of numerical data (bivariate data). Each plotted point represents a pair of corresponding values.
散点图展示两组数值数据（二元数据）之间的关系。每一个点代表一对对应的取值。
Correlation describes the direction and strength of the relationship. Positive correlation means as one variable increases, the other tends to increase. Negative correlation means as one variable increases, the other tends to decrease. No correlation indicates no clear pattern.
相关描述变量间关系的方向和强度。正相关意味着一个变量增大时，另一个变量也趋于增大。负相关意味着一个变量增大时，另一个变量趋于减小。无相关表示没有明显规律。
A line of best fit (trend line) can be drawn through the points to model the relationship. It should pass as close as possible to all points, with roughly equal numbers of points above and below the line. The equation of the line of best fit is usually in the form y = mx + c, and it can be used to estimate unknown values.
最佳拟合线（趋势线）可以穿过散点来建模这种关系。该直线应尽可能靠近所有点，且直线两侧的点数大致相等。最佳拟合线的方程通常形如 y = mx + c，可用于估算未知数值。
8. Pie Charts | 饼图
A pie chart shows proportions of a whole. The angle for each sector is calculated using the formula: Angle = (Category frequency / Total frequency) × 360°. Angles are measured using a protractor.
饼图用于显示各部分占整体的比例。每个扇形的角度按下式计算：角度 = (类别频数 / 总频数) × 360°。角度用量角器量取。
When constructing a pie chart, always label each sector clearly with its category and include either the frequency or the percentage. Charts should have a title.
绘制饼图时，务必清楚地标注每个扇形的类别，并附上频数或百分比。图表应配有标题。
9. Bar Charts and Histograms | 条形图与直方图
Bar charts display categorical or discrete data. Each category has a separate bar, and the height of the bar represents the frequency or value. Gaps between bars emphasise that the data are separate categories.
条形图用于显示类别数据或离散数据。每个类别都有独立的条形，条形的高度表示频数或数值。条形之间的间隔强调数据属于不同的类别。
Histograms are used for continuous data grouped into class intervals. In a histogram, the area of each bar is proportional to the frequency. When class intervals are equal, height is proportional to frequency. When intervals are unequal, use frequency density: Frequency density = Frequency / Class width. Then the bar height is frequency density.
直方图用于分组连续数据。在直方图中，每个条形的面积与频数成正比。当组距相等时，高度与频数成正比。当组距不等时，需要使用频率密度：频率密度 = 频数 / 组距。此时条形的高度就是频率密度。
10. Sampling Methods | 抽样方法
A sample is a subset of a population selected to represent the whole. To avoid bias, a sample should be chosen randomly so that every member of the population has an equal chance of being included.
样本是从总体中选出的一个子集，用以代表整个总体。为了避免偏差，样本应通过随机方式选取，使总体中每个成员都有均等的机会被抽中。
Simple random sampling can be achieved by using random number tables, lottery methods or computer-generated random selections. A sample frame (a list of the whole population) is required.
简单随机抽样可以通过随机数表、抽签法或计算机生成的随机数来实现。实施时需要一份抽样框（整个总体的名单）。
Stratified sampling divides the population into distinct groups (strata) that share a common characteristic. The sample size from each stratum is calculated as: (Stratum size / Population size) × Overall sample size. Within each stratum, members are chosen randomly. This guarantees representation of all key groups.
分层抽样将总体按共同特征划分为不同的层。每层应抽取的样本量计算公式为：(层的大小 / 总体大小) × 总样本量。各层内部再随机选择成员。这确保了所有关键群体都有代表。
11. Misleading Graphs and Critical Interpretation | 误导性图表与批判性解读
Statistical diagrams can be misleading if they use a non-zero starting point on the vertical axis, change the scale partway along an axis, or use distorted pictograms where the area rather than the length of the symbol is altered.
如果统计图表的纵轴不是从零开始、轴上中途改变刻度，或者使用面积被扭曲的象形图（改变的是图符面积而非长度），那么图表就可能产生误导。
Always check the scales, labels and source of a graph before drawing conclusions. A missing title, unlabelled axes or an exaggerated vertical scale can create a false impression of difference.
在得出结论之前，一定要检查图表的刻度、标注和数据来源。缺少标题、坐标轴没有标注或纵轴比例过度放大都可能营造出差异的假象。
12. Key Formulas Summary | 关键公式汇总
| Concept / 概念 | Formula / 公式 | Notes / 备注 |
|---|---|---|
| Mean (ungrouped data) / 均值（未分组） | x̄ = Σx / n | Σx = sum of values, n = count |
| Mean (grouped data) / 均值（分组） | x̄ ≈ Σ(f × x) / Σf | x = class midpoint |
| Median position / 中位数位置 | (n+1) / 2 | Order data first |
| Range / 极差 | Max – Min | Affected by outliers |
| Interquartile range (IQR) / 四分位数间距 | Q₃ – Q₁ | Q₁ position: (n+1)/4; Q₃: 3(n+1)/4 |
| Probability / 概率 | P(A) = favourable / total | 0 ≤ P(A) ≤ 1 |
| Complement rule / 补集规则 | P(A’) = 1 – P(A) | A’ means ‘not A’ |
| Addition rule (mutually exclusive) / 加法法则（互斥） | P(A or B) = P(A) + P(B) | Only if A and B cannot occur together |
| Multiplication rule (tree diagrams) / 乘法法则（树形图） | P(A and B) = P(A) × P(B|A) | Multiply along branches |
| Pie chart angle / 饼图角度 | Angle = (frequency / total) × 360° | Use a protractor |
| Frequency density / 频率密度 | Frequency density = Frequency / Class width | Used in histograms with unequal intervals |
| Stratified sampling number / 分层抽样人数 | (Stratum size / Population) × sample size | Ensures proportional representation |
Published by TutorHao | Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply