📚 Year 9 CCEA Statistics: Quick Reference Formula and Theorem Handbook | Year 9 CCEA 统计：公式定理速查手册
This quick reference handbook covers the essential formulas and theorems you will meet in Year 9 CCEA Statistics. Understanding these building blocks will help you describe data, calculate probabilities, and interpret statistical information confidently. Each entry is presented with a clear statement, an explanation in simple English, and the matching Chinese translation so you can study bilingually.
本速查手册涵盖了 Year 9 CCEA 统计中你会遇到的核心公式和定理。掌握这些基础模块将帮助你描述数据、计算概率并自信地解读统计信息。每个条目都配有清晰的陈述、简单的英文解释和对应的中文翻译，便于你双语学习。
1. The Mean (Arithmetic Average) | 算术平均数
The mean is the sum of all data values divided by the number of values. It gives a measure of central tendency that uses every piece of data.
平均数是所有数据值的总和除以数据的个数。它是一种使用了每一个数据点的集中趋势度量。
Mean = (Sum of all values) ÷ (Number of values) or x̄ = Σxᵢ / n
For example, for the set 3, 5, 8, 8, 11, the sum is 35 and there are 5 values, so the mean is 35 ÷ 5 = 7.
例如，对于数据集 3, 5, 8, 8, 11，总和为 35，数据个数为 5，因此平均数为 35 ÷ 5 = 7。
2. The Median | 中位数
The median is the middle value when all data are arranged in order. If there are two middle numbers, take their mean. The median splits the data into two equal halves.
中位数是将所有数据按大小顺序排列后位于中间的值。如果有两个中间数，则取它们的平均数。中位数将数据分成数量相等的两半。
For an odd number of values, median position = (n + 1) ÷ 2. For an even number, it is the average of the n/2 th and (n/2 + 1)th values.
对于奇数个数据，中位数的位置 = (n + 1) ÷ 2。对于偶数个数据，则是第 n/2 个和第 (n/2 + 1) 个值的平均数。
3. The Mode | 众数
The mode is the value that appears most frequently in a data set. A set may have one mode, more than one mode (bimodal or multimodal), or no mode at all if all values are equally frequent.
众数是数据集中出现频率最高的值。一个数据集可以有一个众数、多个众数（双众数或多众数），如果所有值出现次数相等则没有众数。
The mode is particularly useful for categorical data, such as the most common eye colour in a survey.
众数对于分类数据特别有用，例如调查中最常见的眼睛颜色。
4. The Range (Measure of Spread) | 极差（离散度的度量）
The range is the difference between the largest and smallest values. It tells you how spread out the data are in the simplest possible way.
极差是最大值和最小值之间的差值。它用最简单的方式告诉你数据的分散程度。
Range = Largest value – Smallest value
A small range means the data are clustered closely together; a large range shows greater variability.
极差小意味着数据紧密地聚集在一起；极差大则表明变异性较大。
5. Mean from a Frequency Table | 从频率表求平均数
When data are presented in a frequency table, each value is multiplied by its frequency before summing. This avoids adding the same number many times individually.
当数据以频率表的形式呈现时，先将每个值乘以其频率，然后再求和。这样可以避免多次单独累加同一个数。
Estimated Mean = Σ(f × x) ÷ Σf
Here f is the frequency and x is the data value. Total the ‘f × x’ column and divide by the total frequency.
其中 f 为频率，x 为数据值。计算‘f × x’这一列的总和，再除以总频率。
6. Estimated Mean for Grouped Data | 分组数据的估算平均数
With grouped data, you do not know the exact values, so you use the midpoint of each class interval as an estimate for x.
对于分组数据，你不知道确切的值，因此使用每个组区间的中点作为 x 的估计值。
Midpoint = (Lower bound + Upper bound) ÷ 2
Then apply the same formula: Estimated mean = Σ(f × midpoint) ÷ Σf. This gives a reasonable approximation of the true mean.
然后应用相同公式：估算平均数 = Σ(f × 中点) ÷ Σf。这样能合理地近似真实平均数。
7. Probability Scale and Basics | 概率标度与基础知识
Probability measures how likely an event is to happen. It always lies between 0 (impossible) and 1 (certain). It can be written as a fraction, decimal, or percentage.
概率衡量一个事件发生的可能性。它总是在 0（不可能）和 1（必然）之间。可以用分数、小数或百分数来表示。
P(Event) = Number of favourable outcomes ÷ Total number of equally likely outcomes
If you toss a fair coin, P(Head) = 1/2 = 0.5 = 50%.
如果抛一枚公平硬币，P(正面) = 1/2 = 0.5 = 50%。
8. Sample Space Diagrams | 样本空间图
A sample space is the set of all possible outcomes of an experiment. Listing outcomes in a table or a two-way grid helps you count them accurately and calculate probabilities.
样本空间是一个实验所有可能结果的集合。用表格或双向网格列出结果有助于准确计数并计算概率。
For example, when rolling a fair six‑sided die and tossing a coin, there are 12 equally likely outcomes, shown perfectly in a 6 × 2 sample space table.
例如，当抛一枚硬币并掷一个公平六面骰子时，有 12 种等可能结果，这在一个 6 × 2 的样本空间表中可以完美展现。
9. Mutually Exclusive Events (Addition Rule) | 互斥事件（加法法则）
Two events are mutually exclusive if they cannot happen at the same time. The probability that either one or the other occurs is the sum of their individual probabilities.
如果两个事件不可能同时发生，那么它们就是互斥的。其中任一事件发生的概率等于它们各自概率之和。
P(A or B) = P(A) + P(B) (if A and B are mutually exclusive)
For example, when rolling a die, the probability of getting a 1 or a 2 is 1/6 + 1/6 = 1/3.
例如，掷一个骰子时，得到 1 或 2 的概率为 1/6 + 1/6 = 1/3。
10. Independent Events (Multiplication Rule) | 独立事件（乘法法则）
Two events are independent if the outcome of one does not affect the outcome of the other. The probability that both occur is the product of their individual probabilities.
如果两个事件中一个事件的结果不影响另一个事件的结果，那么这两个事件是独立的。两者同时发生的概率是它们各自概率的乘积。
P(A and B) = P(A) × P(B) (if A and B are independent)
Flipping a fair coin twice: P(Two heads) = 1/2 × 1/2 = 1/4.
抛一枚公平硬币两次：P(两次正面) = 1/2 × 1/2 = 1/4。
11. Calculating Angles for Pie Charts | 计算饼图的角度
A pie chart displays proportions as sectors of a circle. The whole circle is 360°, so each category’s angle is determined by its frequency relative to the total.
饼图用圆的各个扇形来表示比例。整个圆是 360°，因此每个类别的角度由其频率相对于总数的比例决定。
Angle = (Frequency ÷ Total frequency) × 360°
If 30 out of 120 people prefer apples, the angle for apples is (30/120) × 360° = 90°, a quarter of the pie.
如果 120 人中有 30 人喜欢苹果，那么苹果对应的角度为 (30/120) × 360° = 90°，即饼图的四分之一。
12. Interquartile Range (IQR) | 四分位距
The interquartile range is a measure of spread that ignores extreme values. It is the difference between the upper quartile (Q₃) and the lower quartile (Q₁), covering the middle 50% of the data.
四分位距是一种忽略极端值的离散度量。它是上四分位数（Q₃）与下四分位数（Q₁）的差值，涵盖了中间 50% 的数据。
IQR = Q₃ – Q₁
The lower quartile is the median of the lower half of the data; the upper quartile is the median of the upper half. The IQR tells you how spread out the central portion is.
下四分位数是数据下半部分的中位数；上四分位数是数据上半部分的中位数。IQR 告诉你核心部分的分散程度。
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