📚 Formula and Theorem Quick Reference Handbook | 公式定理速查手册
Welcome to your Year 8 OCR Statistics quick reference guide. This handbook compiles all the essential formulas, theorems, and rules you need to master data handling, probability, and statistical diagrams. Refer to it whenever you need a quick reminder.
欢迎使用 Year 8 OCR 统计速查手册。本手册汇集了数据处理、概率和统计图表所需的所有基本公式、定理和规则。在需要快速复习时随时查阅。
1. Mean | 平均数
The mean is the average value of a data set. To calculate the mean, add up all the values and then divide by the number of values.
平均数是一组数据的平均值。要计算平均数，将所有数值相加，再除以数值的个数。
Formula in words: Mean = (Sum of all values) ÷ (Number of values)
文字公式：平均数 = (所有数据值之和) ÷ (数据个数)
Mean = ∑x ÷ n
where ∑x represents the total of all data values and n is the count of values.
其中 ∑x 表示所有数据值的总和，n 为数据的个数。
2. Median and Mode | 中位数与众数
To find the median, order the data from smallest to largest. If there is an odd number of values, the median is the middle value.
求中位数时，先将数据从小到大排序。如果数据个数为奇数，中位数就是正中间的那个数。
If there is an even number of values, the median is the mean of the two middle values.
如果数据个数为偶数，中位数是中间两个数的平均数。
The mode is the value that appears most often. A data set can have no mode, one mode (unimodal), or more than one mode (bimodal or multimodal).
众数是出现次数最多的值。一组数据可以没有众数、一个众数（单峰）或多个众数（双峰或多峰）。
3. Range | 极差
The range measures the spread of a data set. It is the difference between the highest value and the lowest value.
极差衡量一组数据的分散程度。它是最大值与最小值之间的差值。
Range = Maximum value − Minimum value
A larger range indicates greater variability in the data.
极差越大，表示数据的变异性越大。
4. Frequency Tables and Estimated Mean | 频率表与估计平均数
When data is grouped into intervals, we estimate the mean using the midpoints of each class interval. Multiply each midpoint by its frequency, sum these products, and divide by the total frequency.
当数据被分成区间时，我们使用每个组区间的组中点来估计平均数。将每个组中点乘以对应频数，求和，再除以总频数。
Estimated Mean = ∑(f × midpoint) ÷ ∑f
Here, f stands for frequency, and midpoint is the value exactly halfway between the class boundaries.
这里 f 代表频数，组中点是正好位于组界中间的值。
Always check that the intervals have equal widths when using this method.
使用此方法时，请始终检查区间是否具有等宽。
5. Basic Probability | 概率基础
Probability is a number that describes how likely an event is to happen. It always lies between 0 (impossible) and 1 (certain), inclusive.
概率是描述事件发生可能性的数值，始终介于 0（不可能）和 1（必然）之间，包括这两个值。
P(event) = Number of favourable outcomes ÷ Total number of possible outcomes
All outcomes must be equally likely for this formula to apply directly.
所有结果必须等可能出现，该公式才可直接应用。
The sum of probabilities of all possible outcomes of an experiment is always 1.
一个实验所有可能结果的概率之和总是 1。
6. Sample Spaces | 样本空间
A sample space is the set of all possible outcomes of a probability experiment. It can be listed, shown in a table, or drawn as a diagram.
样本空间是概率实验中所有可能结果的集合。它可以用列表、表格或图表表示。
For a single coin flip, the sample space is {Heads, Tails}.
抛一枚硬币时，样本空间为 {正面, 反面}。
For rolling a fair six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.
掷一枚公平的六面骰子时，样本空间为 {1, 2, 3, 4, 5, 6}。
When two coins are flipped, the sample space is {HH, HT, TH, TT}, where H stands for Head and T for Tail.
抛两枚硬币时，样本空间为 {HH, HT, TH, TT}，其中 H 代表正面，T 代表反面。
7. Mutually Exclusive Events and the Addition Rule | 互斥事件与加法法则
Two events are mutually exclusive if they cannot happen at the same time. For example, rolling a 2 and rolling a 5 on a single die are mutually exclusive.
如果两个事件不可能同时发生，则它们是互斥事件。例如，掷一粒骰子时，掷出 2 和掷出 5 是互斥的。
For mutually exclusive events A and B: P(A or B) = P(A) + P(B)
This is the addition rule. It works only for mutually exclusive events.
这就是加法法则。它仅适用于互斥事件。
If events are not mutually exclusive, you must subtract the probability of both happening: P(A or B) = P(A) + P(B) − P(A and B).
如果事件不是互斥的，则必须减去两者同时发生的概率：P(A 或 B) = P(A) + P(B) − P(A 且 B)。
8. Tree Diagrams | 树形图
A tree diagram shows all the possible outcomes of two or more events along branches. It is very useful for calculating probabilities of combined independent events.
树形图用分支展示两个或多个事件所有可能的结果。它对于计算独立事件的复合概率非常有用。
Along each branch, write the probability of that outcome. To find the probability of a path, multiply the probabilities along the branches.
在每条分支上，写出该结果的概率。要计算某条路径的概率，将沿分支的概率相乘。
For two independent coin flips, the probability of two heads is P(H and H) = ½ × ½ = ¼.
对于两次独立的抛硬币，得到两次正面的概率为 P(H 且 H) = ½ × ½ = ¼。
If there are several paths leading to the same final outcome, add their probabilities.
如果有多条路径导致相同最终结果，则将它们的概率相加。
9. Scatter Graphs and Correlation | 散点图与相关
A scatter graph displays the relationship between two variables. Each point represents a pair of values (x, y).
散点图显示两个变量之间的关系。每个点代表一对数值 (x, y)。
Positive correlation: as x increases, y also increases (points slope upwards).
正相关：随着 x 增加，y 也增加（点向上倾斜）。
Negative correlation: as x increases, y decreases (points slope downwards).
负相关：随着 x 增加，y 减小（点向下倾斜）。
No correlation: there is no obvious pattern; the points are scattered randomly.
无相关：无明显模式；点随机分布。
The strength of correlation can be described as strong, moderate, or weak. A line of best fit can be drawn through the points to help make predictions.
相关性强度可描述为强、中、弱。可以通过这些点画一条最佳拟合线来帮助做出预测。
10. Data Representation Charts | 数据图表表示
Bar charts use bars of equal width, with heights proportional to frequency. They are used for discrete or categorical data.
条形图使用等宽的条形，高度与频数成正比。它们用于离散或分类数据。
Pie charts show proportions as sectors of a circle. The angle of each sector is calculated using:
饼图用圆的扇形表示比例。每个扇形的角度按下式计算：
Sector angle = (Frequency ÷ Total frequency) × 360°
Always check that your angles sum to 360°.
一定要检查所有角度之和为 360°。
Pictograms use symbols to represent a certain number of data units. A key must explain what each symbol stands for.
象形图使用符号来表示一定数量的数据单元。必须用图例说明每个符号代表的含义。
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