📚 Year 9 OCR Statistics: Formulae & Theorems Quick Reference Handbook | Year 9 OCR 统计：公式定理速查手册
This quick reference handbook brings together all the essential formulae, key theorems and must-know methods for the Year 9 OCR Statistics course. Use it to revise averages, spread, probability and data handling techniques as you prepare for end-of-topic tests or end-of-year examinations. Each section gives you a concise English explanation followed by the Chinese translation, ensuring you build both subject knowledge and bilingual confidence.
这本速查手册汇集了 Year 9 OCR 统计课程中所有重要的公式、核心定理和必会方法。无论是准备单元测验还是学年考试，你都可以用它来复习平均数、离散程度、概率和数据处理技巧。每个小节先用简洁的英文讲解，再配上中文翻译，帮助你在建立学科知识的同时提升双语能力。
1. Mean, Median, Mode & Range | 平均数、中位数、众数和极差
The three averages and the range are the first tools used to summarise a data set.
三个平均数和极差是总结数据集最基础的工具。
Mean: add all the values together and divide by how many values there are.
平均数：将所有数值相加，再除以数值的个数。
Mean = (Sum of values) ÷ (Number of values) or x̄ = ∑x / n
Median: the middle value when the data are arranged in order. If there are two middle numbers, find their mean.
中位数：将数据按大小排序后位于中间的数。如果中间有两个数，则取这两个数的平均数。
Mode: the value that appears most often. A set of data can have one mode, more than one mode, or no mode at all.
众数：出现次数最多的数值。一组数据可能有一个众数、多个众数，也可能没有众数。
Range: the difference between the largest and the smallest value. It is a simple measure of spread.
极差：最大值与最小值之差。它是一个简单的离散程度度量。
Range = Largest value − Smallest value
2. Mean from a Frequency Table | 由频数表求平均数
When data are presented in a frequency table, multiply each distinct value by its frequency, sum these products, then divide by the total frequency.
当数据用频数表（频率表）表示时，将每个不同的数值乘以对应的频数，把这些乘积相加，再除以总频数。
Mean = ∑(f × x) / ∑f
Here f stands for frequency and x stands for the data value. The symbol ∑ means 'sum of'.
其中 f 代表频数，x 代表数据值，符号 ∑ 表示“求和”。
Always check that you have multiplied every row before adding.
一定要确保在相加之前已经将每一行都乘好。
3. Estimating the Mean from Grouped Data | 分组数据估算平均数
For grouped data we do not know the exact values, so we use the midpoint of each class interval as an estimate.
对于分组数据，我们不知道确切数值，因此用每个组距的组中值作为估计值。
Estimated Mean = ∑(f × midpoint) / ∑f
boundaries of the interval and dividing by 2: midpoint = (lower bound + upper bound) ÷ 2.
组中值通过将该组的下限与上限相加后除以2得到：组中值 =（下限 + 上限）÷ 2。
This method gives an approximation; the true mean could be different if the raw data were available.
这种方法只能给出近似值；如果有原始数据，真实平均数可能不同。
4. Median and Mode from a Frequency Table | 频数表求中位数与众数
To find the median from a frequency table, add a cumulative frequency column. Locate the position of the median using (n + 1) ÷ 2, then read off the data value where the cumulative frequency first reaches or exceeds this position.
要从频数表求中位数，需要添加累积频数列。先用 (n + 1) ÷ 2 确定中位数的位置，然后找出累积频数首次达到或超过该位置时所对应的数据值。
For large data sets some courses use n ÷ 2; OCR Year 9 usually introduces the (n + 1) ÷ 2 rule.
对于大数据集，有些教材使用 n ÷ 2；OCR Year 9 通常先介绍 (n + 1) ÷ 2 规则。
The mode from a frequency table is simply the value with the highest frequency.
频数表中的众数就是频数最高的那个数值。
5. Probability Basics | 概率基础
Probability measures how likely an event is to happen. It always lies between 0 (impossible) and 1 (certain).
概率衡量某个事件发生的可能性大小。它总是在 0（不可能）和 1（必然）之间。
P(Event) = Number of favourable outcomes / Total number of equally likely outcomes
Probabilities can be written as fractions, decimals or percentages.
概率可以用分数、小数或百分数来表示。
A fair coin has P(Head) = ½, a fair six-sided die has P(rolling a 3) = ⅙.
一枚均匀硬币 P(正面) = ½，一颗均匀的六面骰子 P(掷出3) = ⅙。
6. Expected Number of Outcomes | 预期结果数
If you repeat an experiment several times, the expected frequency of an event is found by multiplying the probability by the number of trials.
如果多次重复一个试验，某个事件的预期频数等于概率乘以试验次数。
Expected number = P(Event) × Number of trials
For example, if you roll a fair die 60 times, you would expect about 60 × ⅙ = 10 sixes. This does not guarantee exactly ten – it is a long-term average.
例如，将一颗均匀骰子掷 60 次，你预期约有 60 × ⅙ = 10 次出现 6。这并不保证恰好十次——它是一个长期的平均值。
7. Sample Space Diagrams | 样本空间图
A sample space is a list or diagram showing all possible outcomes of an experiment. For two events, a two-way table can be very useful.
样本空间是列出或画出某个试验所有可能结果的清单或图示。对于两个事件，双向表非常有用。
When tossing two coins, the sample space is {HH, HT, TH, TT}, where H = head, T = tail.
抛两枚硬币时，样本空间是 {HH, HT, TH, TT}，其中 H 表示正面，T 表示反面。
all 36 equally likely outcomes, helping you calculate probabilities such as the sum being 7.
掷两颗骰子时，用一个 6×6 的表格可以给出全部 36 种等可能结果，帮助你计算诸如点数和为 7 的概率。
8. Probability Trees | 概率树图
A probability tree diagram shows sequences of events. Each branch is labelled with its probability. The probabilities on branches from the same point always add up to 1.
概率树图显示事件的序列。每条分支上都标有相应的概率。从同一点出发的各分支概率之和总是等于 1。
To find the probability of a combination of outcomes, multiply along the branches. To find the probability of an event that can happen in several ways, add the probabilities of the relevant end branches.
要计算一连串结果的概率，沿着分支相乘。如果一个事件可以通过多种方式发生，就把相关末支的概率相加。
P(A and B) = P(A) × P(B) (for independent events)
At Year 9, trees are mainly used for independent events, such as picking a counter, replacing it, and picking again.
在 Year 9 阶段，树图主要用于独立事件，比如取出一个筹码、放回后再取一次。
9. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. For mutually exclusive events, the probability that 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，则它们是互斥的。对于互斥事件，其中一个或另一个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, getting a '3' and getting an 'even number' are mutually exclusive because you cannot get both on a single roll.
例如，掷一颗骰子时，“掷出 3”和“掷出偶数”是互斥的，因为你不可能在一次投掷中两者同时出现。
10. The Sum of All Probabilities | 概率总和
In any probability experiment, the sum of the probabilities of all possible mutually exclusive outcomes is always 1.
在任何一个概率试验中，所有可能的互斥结果的概率之和总是等于 1。
P(Outcome 1) + P(Outcome 2) + … + P(Outcome n) = 1
This leads to the complement rule: the probability of an event NOT happening is 1 minus the probability that it does happen.
由此引出互补规则：一个事件不发生的概率等于 1 减去它发生的概率。
P(not A) = 1 − P(A)
11. Using Averages to Compare Data Sets | 用平均数比较数据集
Averages on their own can be misleading. When comparing two data sets, always quote an average and a measure of spread, such as the range.
单纯看平均数可能具有误导性。比较两组数据时，一定要同时引用一个平均数和一个离散程度指标，比如极差。
A larger mean suggests higher typical values, while a smaller range indicates more consistency. The median is often preferred when data contain outliers.
平均数较大意味着典型值更高，极差较小则表明数据更稳定。当数据含有异常值时，通常更倾向于使用中位数。
For a symmetric distribution the mean and median are close; for a skewed distribution they will differ.
对于对称分布，平均数和中位数接近；对于偏态分布，两者会有差异。
12. Interquartile Range (IQR) | 四分位距
The range only uses the two extreme values, so it can be affected by outliers. The interquartile range (IQR) measures the spread of the middle 50% of the data and is more robust.
极差只用到了两个极端值，所以容易受异常值影响。四分位距（IQR）衡量中间 50% 数据的分散程度，具有更强的稳健性。
IQR = Q3 − Q1
Q1 (lower quartile) is the median of the lower half of the data; Q3 (upper quartile) is the median of the upper half. To find them by hand, first find the median, then split the data into two halves and find the medians of those halves.
Q1（下四分位数）是数据下半部分的中位数；Q3（上四分位数）是数据上半部分的中位数。手工求取时，先找出中位数，再将数据分成两半，分别找出这两半的中位数即可。
A smaller IQR means the middle half of the data is tightly clustered around the median.
四分位距越小，说明中间一半的数据围绕中位数越紧密。
Published by TutorHao | Statistics Revision Series | aleveler.com
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