📚 Year 9 CIE Statistics: Formula & Theorem Quick Reference | 九年级 CIE 统计：公式定理速查手册
This quick reference guide compiles all the essential formulae and theorems for the Year 9 Cambridge IGCSE (or Lower Secondary) Statistics syllabus. Mastering these concepts will help you analyse data, calculate probabilities, and interpret statistical graphs with confidence. Keep this guide handy for homework, revision, and exam preparation.
本速查手册汇编了九年级剑桥 IGCSE（或初中阶段）统计课程中的所有核心公式与定理。掌握这些概念将帮助你自信地分析数据、计算概率并解读统计图表。随时参考本手册，助你完成作业、复习和备考。
1. Mean, Median, Mode | 平均数、中位数、众数
The mean is the arithmetic average, found by adding all data values together and dividing by the number of values. The median is the middle value when the data are arranged in ascending or descending order; if there is an even number of values, take the mean of the two central numbers. The mode is the value that occurs most frequently in the data set.
平均数（均值）是算术平均值，将全部数据值相加后除以数值的个数。中位数是将数据按升序或降序排列后的中间值；若数值个数为偶数，则取最中间两个数的平均数。众数是数据集中出现频率最高的数值。
Mean = Σx / n
+3+5+7) / 5 = 4; median (ordered: 2, 3, 3, 5, 7) is 3; mode is 3 because it appears twice.
对于数据集 2, 3, 3, 5, 7：平均数 = (2+3+3+5+7) / 5 = 4；中位数（排序后：2, 3, 3, 5, 7）为 3；众数为 3，因为它出现两次。
2. Range and Interquartile Range (IQR) | 极差与四分位距
The range is a measure of spread: Range = maximum value – excluding the overall median if n is odd, but conventions vary slightly; use your syllabus' The upper quartile Q₃ is the median of the upper half. The interquartile range is IQR = Q₃ – Q₁. It describes the spread of the middle 50% of values, making it resistant to outliers.
极差衡量数据的分散程度：极差 = 最大值 – 最小值。四分位数将有序数据分成四等份。下四分位数 Q₁ 是数据下半部分的中位数（如果 n 为奇数，通常排除总中位数，但方法可能略有不同；以课程大纲为准）。上四分位数 Q₃ 是上半部分的中位数。四分位距 IQR = Q₃ – Q₁，用于描述中间 50% 数据的分布范围，因而能抵抗异常值的影响。
Range = Max – Min
IQR = Q₃ – Q₁
For 1, 3, 4, 7, 8, 9, 12: Q₁ = 3.5, Q₃ = 9, so IQR = 5.5; range = 12 – 1 = 11.
对于 1, 3, 4, 7, 8, 9, 12：Q₁ = 3.5，Q₃ = 9，因此 IQR = 5.5；极差 = 12 – 1 = 11。
3. Frequency Tables and Estimated Mean | 频率表与估算平均数
When data are presented in a frequency table, the mean is calculated by multiplying each value (x) by its frequency (f), summing these products, and dividing by the total frequency. For grouped data, we use the midpoint of each class interval as x. This gives an estimated mean because we assume all values within an interval are at the midpoint. The modal class is the interval with the highest frequency.
当数据以频率表呈现时，平均数的计算方法是：将每个数值 x 乘以其频数 f，求和，再除以总频数。对于分组数据，我们取每个组区间的组中值作为 x。这样得到的是一个估算平均数，因为我们假设组内所有值都位于组中值。众数所在组是指频数最高的区间。
Mean = Σ(f × x) / Σf
Estimated Mean = Σ(f × midpoint) / Σf
curs 3 times, value 5 occurs 4 times. Mean = (2×3 + 5×4) / (3+4) = (6+20)/7 = 26/7 ≈ 3.71.
未分组示例：数值 2 出现 3 次，数值 5 出现 4 次。平均数 = (2×3 + 5×4) / (3+4) = (6+20)/7 ≈ 3.71。
4. Bar Charts, Histograms and Frequency Polygons | 条形图、直方图与频率多边形
A bar chart displays categorical data with spaces between the bars; the height of each bar represents the frequency or count of that category. A histogram is used for continuous grouped data; bars touch to reflect the continuous scale, and the area of each bar is proportional to its frequency. If class widths are unequal, we plot frequency density y polygon can be drawn by connecting the midpoints of the tops of histogram bars (or from a frequency table) and is often used to compare distributions.
条形图用于展示分类数据，条形之间有间隙；每个条形的高度代表该类别出现的频率或计数。直方图用于连续分组数据；条形紧贴以体现连续刻度，每个条形的面积与其频率成正比。若组距不等，则需绘制 频率密度 = 频率 / 组距。频率多边形可以通过连接直方图顶部中点（或直接从频率表中构建）来绘制，常用于比较分布。
Frequency Density = Frequency ÷ Class Width
When constructing a histogram from unequal class widths, always calculate frequency density and use it for the vertical axis. For equal class widths, frequency can be plotted directly on the vertical axis.
当组距不等绘制直方图时，必须计算频率密度并将其作为纵轴。如果组距相等，则可直接将频率标在纵轴上。
5. Pie Charts | 饼图
total angle in a circle is 360°. To find the angle for any category, use: sector angle = (category frequency / total frequency) × 360°. Pie charts are excellent for showing relative proportions but become unclear if there are too many small categories.
饼图以圆的扇区来表示数据，每个扇区的角度与其所代表类别的频率成正比。圆的总角度为 360°。计算任一扇区角度的公式为：扇区角度 = (类别频率 / 总频率) × 360°。饼图非常适合显示相对比例，但如果存在过多微小类别，图表会变得难以解读。
Sector Angle = (f / Σf) × 360°
tudents choose Mathematics as a favourite subject, the sector angle = (12/40)×360° = 108°.
例如：若 40 名学生中有 12 人选择数学作为最喜爱的科目，则扇区角度 = (12/40)×360° = 108°。
6. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph displays the relationship between two variables by plotting points on a coordinate plane. If the points tend to rise together, there is positive correlation. If one variable increases while the other decreases, it shows negative correlation. If no pattern is visible, there is no correlation. The strength of correlation can be described as strong, moderate, or weak. A line of best fit (trend line) can be drawn to model the relationship. Outliers are points that deviate markedly from the overall pattern.
散点图通过在坐标平面上描点来展示两个变量之间的关系。若点呈共同上升趋势，则是 正相关；若一个变量增加而另一个减少，则是 负相关；若无规律可循，则是 无相关。相关性的强弱可用强、中等或弱来描述。可以绘制一条最佳拟合线（趋势线）来模拟这种关系。异常值是指明显偏离整体模式的点。
Remember, correlation does not imply causation – a strong link between two variables might be caused by a third hidden factor.
请记住，相关性并不意味着因果关系——两个变量之间的强关联可能是由第三个隐藏因素导致的。
7. Probability Basics | 概率基础
ble outcomes. Probabilities lie between 0 (impossible) and 1 (certain). The sum of probabilities of all mutually exclusive outcomes in a sample space is 1. The complement of event A, written A', means 'not A', and P(A') = 1 – ased on trials: Relative frequency = number of successful trials / total number of trials.
概率衡量事件发生的可能性。对于等可能结果：P(事件) = 有利结果的数量 / 所有可能结果的总数。概率值介于 0（不可能）和 1（必然）之间。样本空间中所有互斥结果的概率之和为 1。事件 A 的互补事件记为 A'，表示“非 A”，且 P(A') = 1 – P(A)。实验概率（相对频率）基于试验得出：相对频率 = 成功试验次数 / 总试验次数。
P(A) = n(A) / n(S)
P(A') = 1 – P(A)
ou roll a die 60 times and get a 3 ten times, the experimental probability is 10/60 = 1/6.
对于一枚均匀六面骰子，P(掷出 3 点) = 1/6；对于一枚硬币，P(正面) = 0.5。若掷骰子 60 次中出现 3 点 10 次，则实验概率为 10/60 = 1/6。
8. Sample Space Diagrams | 样本空间图
A sample space is the set of all possible outcomes of an experiment. For a single event, listing outcomes is straightforward. For two combined events, a two-way table or a sample space diagram helps list all possible pairs of outcomes systematically. Once the sample space is obtained, probabilities can be found by counting favourable outcomes.
样本空间是一个试验所有可能结果的集合。对于单一事件，列出结果很简单。对于两个组合事件，双向表或样本空间图有助于系统地列出所有可能的结果对。获得样本空间后，便可通过计数有利结果来求概率。
1/6 because six combinations give a total of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
示例：掷两个公平骰子时，样本空间包含 36 个等可能的有序对。点数和为 7 的概率为 6/36 = 1/6，因为有六种组合得到 7：(1,6)、(2,5)、(3,4)、(4,3)、(5,2)、(6,1)。
9. Tree Diagrams | 树形图
Tree diagrams are used for multistage events, especially when outcomes are not equally likely at each stage or when events are sequential. Branches represent possible outcomes, with probabilities written along each branch. To find the probability of a particular combined outcome, multiply the probabilities along the branches that lead to that outcome. If more than one path leads to the same final event, add the probabilities of those paths. The total of probabilities on all complete paths must equal 1.
树形图用于多阶段事件，尤其当各阶段结果并非等可能或事件是顺序发生时。每个分支代表一种可能结果，并在分支旁标出概率。要求出某个特定组合结果的概率，将通向该结果的路径上的概率相乘。若有多条路径通向同一个最终事件，则将这些路径的概率相加。所有完整路径的概率总和必须等于 1。
thout replacement. Construct tree diagram to find P(both red) = (3/5)×(2/4) = 6/20 = 3/10.
示例：袋中有 3 个红球和 2 个蓝球，不放回地抽取两次。构建树形图可求得 P(两个都是红球) = (3/5)×(2/4) = 6/20 = 3/10。
10. Sets and Venn Diagrams | 集合与韦恩图
The rectangle represents the universal set. Circles represent subsets. The intersection A ∩ B contains elements in both A and B. The union A ∪ . For probability involving two events, the addition rule helps avoid double counting: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). If A and B are mutually exclusive (cannot happen together), then P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
韦恩图用来表示集合及其关系。矩形代表全集，圆代表子集。交集 A ∩ B 包含同时在 A 和 B 中的元素。并集 A ∪ B 包含在 A 或 B 或两者之中的元素。对于涉及两个事件的概率，加法法则可避免重复计算：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若 A 与 B 互斥（不能同时发生），则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
When using a Venn diagram with frequencies, place the intersection first, then fill remaining parts, ensuring all regions sum to the total in the universal set.
使用频率绘制韦恩图时，先填写交集部分，再填其他部分，确保所有区域之和等于全集中元素总数。
11. Stem-and-Leaf Diagrams | 茎叶图
A stem-and-leaf diagram organises numerical data by separating each value into a stem (the leading digit(s)) and a leaf (the final significant digit). This preserves the original data while showing the distribution shape. The stems are listed in vertical order, and leaves are written next to their stems, usually in ascending order. A key must be provided to show the place value. Back-to-back stem-and-leaf diagrams can compare two related data sets by sharing a central stem.
茎叶图通过将每个数值分成“茎”（前导数字）和“叶”（最后一个有效数字）来整理数值型数据。这种方法既能保留原始数据，又能展示分布形态。茎按顺序纵向排列，叶片写在对应茎的旁边，通常按升序排列。必须提供图例以说明数位值。背靠背茎叶图可以共用中间茎来比较两个相关的数据集。
Example: data 12, 15, 16, 21, 23 → stem 1 | 2 5 6, stem 2 | 1 3. Key: 1 | 2 means 12. The median is easily located by counting leaves from the centre.
示例：数据 12, 15, 16, 21, 23 → 茎 1 | 叶 2 5 6，茎 2 | 叶 1 3。图例：1 | 2 表示 12。通过从中心数叶片可以轻松定位中位数。
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