📚 Year 8 CIE Statistics: Formula & Theorem Quick Reference Handbook | Year 8 CIE 统计：公式定理速查手册
Welcome to the Year 8 CIE Statistics quick reference handbook. This guide summarises the essential formulas, theorems and graphical techniques you need to master data handling and probability. Keep it handy for revision and homework.
欢迎使用 Year 8 CIE 统计公式定理速查手册。本手册归纳了数据分析和概率部分必须掌握的核心公式、定理和图表技巧，方便你随时复习和完成作业。
Every section below puts the rule first, then explains it in plain language, then shows exactly when to use it in an exam. Read one section at a time, cover the formula, and try to rewrite it from memory before you check.
下面每一节都先给出公式或定理，再用通俗语言解释，最后说明考试中什么时候用它。建议每次只看一节，遮住公式，先自己默写一遍再对照检查。
1. Mean, Median and Mode | 平均数、中位数与众数
The mean is the arithmetic average. To find the mean, add up all the values and divide by the number of values.
平均数即算术平均值。计算方法是将所有数据值相加，再除以数据个数。
Mean = Σx ÷ n
where Σx represents the sum of all data values and n is the total number of values.
其中 Σx 表示所有数据值的和，n 表示数据总个数。
The median is the middle value when the data are arranged in order. If there are two middle values, the median is the mean of those two values.
中位数是将数据按大小顺序排列后位于中间位置的数值。如果有两个中间数，则中位数为这两个数的平均数。
The mode is the value that appears most frequently. A data set may have one mode, more than one mode, or no mode at all when every value occurs the same number of times.
众数是出现次数最多的数值。一组数据可以有一个众数、多个众数，也可能没有众数（当每个数值出现的次数相同时）。
- Always sort the data before finding the median. 求中位数前一定要先排序。
- For an odd number of values, the median is the (n + 1) ÷ 2 th value. 数据个数为奇数时，中位数是第 (n + 1) ÷ 2 个值。
- For an even number of values, the median is halfway between the n ÷ 2 th and the (n ÷ 2 + 1) th values. 数据个数为偶数时，中位数是第 n ÷ 2 个与第 n ÷ 2 + 1 个值的平均数。
- The mode is the only average that works for non-numerical data such as colours or names. 众数是唯一适用于非数值数据（如颜色、姓名）的平均数。
| Average 平均数 | Best used when 适用情形 |
| Mean 平均数 | Data are fairly symmetrical with no extreme values. 数据较对称、无极端值时。 |
| Median 中位数 | Data contain outliers or are skewed. 数据含极端值或明显偏斜时。 |
| Mode 众数 | Data are categorical or you need the most common item. 数据为分类数据，或需要最常见的类别时。 |
2. Range, Quartiles and Interquartile Range | 极差、四分位数与四分位距
The range is the simplest measure of spread. It is the largest value minus the smallest value.
极差是最简单的离散程度度量，等于最大值减去最小值。
Range = largest value − smallest value
Quartiles split an ordered data set into four equal parts. The lower quartile Q1 is the value one quarter of the way through the data, Q2 is the median, and the upper quartile Q3 is the value three quarters of the way through.
四分位数把排好序的数据分成四等份。下四分位数 Q1 位于数据的四分之一处，Q2 就是中位数，上四分位数 Q3 位于数据的四分之三处。
The interquartile range measures the spread of the middle 50% of the data. It is far less affected by extreme values than the range.
四分位距衡量的是中间 50% 数据的离散程度，它受极端值的影响远小于极差。
IQR = Q3 − Q1
The semi-interquartile range is simply half of the interquartile range, and is sometimes used when a smaller number is easier to compare.
半四分位距就是四分位距的一半，有时用它比较数据会更方便。
Semi-interquartile range = (Q3 − Q1) ÷ 2
- Range is quick but sensitive to one extreme value. 极差计算快，但容易被一个极端值影响。
- IQR ignores the smallest 25% and largest 25% of values. 四分位距忽略最小的 25% 和最大的 25% 的数据。
- Always state the units when you give a range or an IQR. 写极差或四分位距时一定要带单位。
3. Frequency Tables and Grouped Data | 频数表与分组数据
When data are given in a frequency table, multiply each value by its frequency, add those products, then divide by the total frequency.
当数据以频数表给出时，先把每个数值乘以它的频数，再把这些乘积相加，最后除以总频数。
Mean from a table = Σfx ÷ Σf
Here x is the data value, f is its frequency, and Σf is the total number of items in the survey.
这里 x 是数据值，f 是它的频数，Σf 是调查数据的总个数。
For grouped data the exact values are hidden, so we use the midpoint of each class interval as x. The answer is therefore an estimate, not an exact mean.
对于分组数据，具体数值已经看不到了，所以我们用每个组区间的中点作为 x。因此得到的是估计值，而不是精确的平均数。
The modal class is the group with the highest frequency, and the median class is the group that contains the middle value. Both are given as intervals, not single numbers.
众数所在组是频数最高的那一组，中位数所在组是包含中间值的那一组。两者都以区间形式给出，而不是单个数字。
| Class 组 | Frequency 频数 | Midpoint x 中点 | fx |
| 0 ≤ t < 10 | 6 | 5 | 30 |
| 10 ≤ t < 20 | 14 | 15 | 210 |
| 20 ≤ t < 30 | 10 | 25 | 250 |
For this table Σf = 30 and Σfx = 490, so the estimated mean is 490 ÷ 30 ≈ 16.3.
对这张表，Σf = 30，Σfx = 490，所以平均数的估计值为 490 ÷ 30 ≈ 16.3。
4. Stem-and-Leaf Diagrams and Dot Plots | 茎叶图与点图
A stem-and-leaf diagram keeps every original value while showing the shape of the distribution. The stem is usually the tens digit and the leaf is the units digit.
茎叶图既保留了每一个原始数据，又能显示数据的分布形状。通常茎是十位数字，叶是个位数字。
A key must always be included, for example 3 | 5 means 35. Without the key the diagram is meaningless.
茎叶图必须写出示例说明，例如 3 | 5 表示 35。没有说明，图就毫无意义。
An ordered stem-and-leaf diagram arranges the leaves in ascending order, which makes the median and quartiles easy to read off.
有序茎叶图把叶按从小到大排列，这样中位数和四分位数就能直接读出来。
A dot plot records each value as a dot above a number line, and is most useful for small data sets where you want to see clusters and gaps quickly.
点图把每个数值用数轴上方的一个点表示，最适合数据量较小、需要快速看出聚集和间隔的情况。
- Count the leaves to check you have not lost data. 数一数叶的个数，确认没有丢数据。
- Back-to-back stem-and-leaf diagrams compare two groups on one picture. 背靠背茎叶图可以在一张图上比较两组数据。
- Both diagrams show individual values, unlike histograms. 这两种图都显示单个数值，而直方图不显示。
5. Bar Charts, Pie Charts and Pictograms | 条形图、饼图与象形图
In a pie chart, the angle of each sector is proportional to the frequency it represents. Multiply the fraction by 360° to get the angle.
在饼图中，每个扇形的圆心角与它代表的频数成正比。把该比例乘以 360° 就得到圆心角。
Sector angle = (frequency ÷ total frequency) × 360°
A bar chart has gaps between the bars and is used for categorical or discrete data. A histogram has no gaps and is used for continuous data.
条形图的柱之间有间隔，用于分类数据或离散数据；直方图的柱之间没有间隔，用于连续数据。
In a pictogram each symbol stands for a fixed number of items, so half a symbol means half that number. Always read the key before counting.
在象形图中，每个符号代表固定数量的项目，所以半个符号就是该数量的一半。数之前一定要先看说明。
- Compound bar charts stack categories within one bar. 复合条形图把不同类别叠在同一个柱里。
- Comparative bar charts place bars side by side for two groups. 对比条形图把两组的柱并排摆放。
- Pie charts compare parts of a whole, not absolute amounts. 饼图比较的是各部分占整体的比例，而不是绝对数量。
6. Histograms and Frequency Density | 直方图与频率密度
When class intervals are not all the same width, you cannot plot frequency on the vertical axis. You must plot frequency density instead.
当各组组距不完全相同时，不能把频数直接画在纵轴上，必须改画频率密度。
Frequency density = frequency ÷ class width
Because of this rule, the area of each bar represents the frequency of that class, not the height.
正因为如此，每个矩形条的面积才代表该组的频数，而不是高度。
Frequency = frequency density × class width
example, the class 10 ≤ t < 20 has width 10, and the class 20 ≤ t < 50 has width 30.
要使用组边界而不是组限。例如 10 ≤ t < 20 这一组组距是 10，而 20 ≤ t < 50 这一组组距是 30。
- Read the question to check whether the vertical axis is frequency or frequency density. 读题时要看清纵轴标的是频数还是频率密度。
- If all class widths are equal, a frequency histogram and a frequency density histogram look the same shape. 如果所有组距都相等，频数直方图与频率密度直方图形状相同。
- Always label the vertical axis with the correct quantity. 纵轴一定要标上正确的量名。
7. Cumulative Frequency Curves and Box Plots | 累积频数曲线与箱线图
A cumulative frequency curve is an S-shaped curve plotted against the upper class boundary of each interval. The points are joined with a smooth curve.
累积频数曲线是一条 S 形曲线，横坐标取每组的组上界，各点用平滑曲线连接。
To read values from the curve, find n ÷ 2 for the median, n ÷ 4 for the lower quartile, and 3n ÷ 4 for the upper quartile, where n is the total frequency.
从曲线上读数时：中位数对应 n ÷ 2，下四分位数对应 n ÷ 4，上四分位数对应 3n ÷ 4，其中 n 是总频数。
A box plot is drawn from the five-number summary: minimum, Q1, median, Q3 and maximum. The box holds the middle 50% of the data and the whiskers stretch to the extremes.
箱线图根据五数概括绘制：最小值、Q1、中位数、Q3、最大值。箱体包含中间 50% 的数据，须线延伸到两端极值。
When comparing two distributions, compare a measure of average (usually the median) and a measure of spread (usually the IQR) and quote the numbers.
比较两组分布时，要比较集中趋势（通常用中位数）和离散程度（通常用四分位距），并引用具体数值。
- A cumulative frequency curve never goes down. 累积频数曲线永远单调不减。
- The curve must start at the lower boundary of the first class with cumulative frequency 0. 曲线应从第一组的下界、累积频数为 0 开始。
- Box plots hide the sample size, so mention n if it matters. 箱线图看不出样本量，若样本量重要就应说明 n。
8. Scatter Diagrams and Correlation | 散点图与相关性
A scatter diagram plots one variable on each axis to show the relationship between them. Each point represents one item.
散点图把两个变量分别放在两个坐标轴上，用来显示它们之间的关系。每个点代表一个个体。
Positive correlation means that as one variable increases the other tends to increase. Negative correlation means that as one increases the other tends to decrease. No correlation means there is no clear pattern.
正相关表示一个变量增大时另一个也趋于增大；负相关表示一个变量增大时另一个趋于减小；不相关则表示没有明显规律。
A line of best fit is drawn through the middle of the points, with roughly equal numbers above and below. It must pass through the mean point.
最佳拟合直线画在点群中间，使上下两侧的点数大致相等，并且应经过平均点。
Correlation does not prove causation. Two variables may both change because of a third, hidden factor.
相关不等于因果。两个变量同时变化，可能是因为第三个隐藏因素在起作用。
- Interpolation, reading inside the data range, is reliable. 内插（在数据范围内读数）比较可靠。
- Extrapolation, reading outside the data range, is risky. 外推（在数据范围外读数）风险较大。
- Describe correlation as strong or weak as well as positive or negative. 描述相关性时，除了正负，还要说明强弱。
9. Time Series and Moving Averages | 时间序列与移动平均
A time series is a set of data recorded at regular intervals, such as every month or every quarter. It usually contains a long-term trend plus seasonal variation.
时间序列是按固定时间间隔记录的数据，例如每月或每季度一次。它通常包含长期趋势和季节性波动。
A moving average smooths out the seasonal ups and downs so that the trend becomes visible.
移动平均可以抹平季节性的上下波动，让趋势变得清晰可见。
For quarterly data, use a four-point moving average. Because four is an even number, you must average two consecutive four-point totals to centre the result.
对于季度数据，使用四点移动平均。因为 4 是偶数，需要再对相邻两个四点总和取平均，才能使结果居中。
Centred 4-point moving average = (first 4 total + next 4 total) ÷ 8
Because each average needs several values, the first and last few data points do not have a moving average. This is expected and is not an error.
由于每个平均值都需要若干数据，序列开头和结尾的几个点没有移动平均值，这是正常现象，不是错误。
- Always check whether the question wants a 3-point, 4-point or 5-point moving average. 一定要看清题目要求的是 3 点、4 点还是 5 点移动平均。
- Subtracting the trend from the original value gives the seasonal variation. 用原值减去趋势值就得到季节变动。
10. Probability Basics: Language, Scale and Sample Space | 概率基础：语言、标尺与样本空间
Probability measures how likely an event is, on a scale from 0 to 1, where 0 is impossible and 1 is certain. It can be written as a fraction, decimal or percentage.
概率衡量事件发生的可能性，取值范围在 0 到 1 之间，0 表示不可能，1 表示必然。可以用分数、小数或百分数表示。
0 ≤ P(A) ≤ 1
If all outcomes are equally likely, the probability of an event is the number of favourable outcomes divided by the total number of possible outcomes.
如果所有结果等可能，事件发生的概率等于有利结果的个数除以所有可能结果的个数。
P(A) = number of favourable outcomes ÷ total number of outcomes
The sample space is the list of all possible outcomes, often shown in a table or a list. Writing it out carefully prevents mistakes in later calculations.
样本空间是所有可能结果的清单，常用表格或列表呈现。把它认真写出来，可以避免后续计算出错。
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