📚 Year 9 OCR Statistics: Formula & Theorem Quick Reference Handbook | Year 9 OCR 统计：公式定理速查手册
This handbook brings together all the essential formulas, rules and theorems you need to master for the Year 9 OCR Statistics course. Keep it close for quick revision, homework support and exam preparation. Each section is presented as a clear English explanation followed by its Chinese counterpart, so you can build both subject knowledge and academic language skills at the same time.
本手册汇集了 Year 9 OCR 统计课程中必须掌握的所有核心公式、规则与定理。你可以用它进行快速复习、辅助作业和备考。每个部分都以清晰的英文解释搭配中文对应内容，帮助你在巩固学科知识的同时提升学术语言能力。
1. Measures of Central Tendency | 集中趋势度量
The three main measures of central tendency are mean, median and mode. They each give a 'typical' value for a data set, but are calculated differently and are appropriate under different circumstances. The mean is the arithmetic average, useful for data without extreme outliers. The median is the middle value when data are ordered, and it is not affected by outliers. The mode is the most frequent value, especially useful for categorical data.
三种主要的集中趋势度量是平均数、中位数和众数。它们各自给出数据集的“典型”值，但计算方法不同，适用情形也不同。平均数是算术平均值，适用于没有极端异常值的数据。中位数是排序后处于中间位置的值，不受异常值影响。众数是出现频率最高的值，尤其适用于分类数据。
Mean = (Σx)/n
To find the median, arrange data in order. If n is odd, the median is the ((n+1)/2)th value. If n is even, it is the mean of the (n/2)th and (n/2 + 1)th values.
求中位数时，先将数据排序。若 n 为奇数，中位数是第 ((n+1)/2) 个值；若 n 为偶数，则是第 (n/2) 和第 (n/2+1) 个值的平均数。
Mode = value with highest frequency
A data set can have one mode (unimodal), two modes (bimodal) or more. When every value occurs only once, there is no mode.
一组数据可以有一个众数（单峰）、两个众数（双峰）或更多。如果每个值都只出现一次，则没有众数。
2. Measures of Spread | 离散程度度量
Spread tells us how scattered or concentrated the data are. The simplest measure is the range. A larger range means greater variability. Range is quick to compute but sensitive to outliers.
离散程度告诉我们数据是分散还是集中。最简单的度量是极差。极差越大，变异性越大。极差计算迅速，但对异常值敏感。
Range = Maximum value - Minimum value
For a more robust picture, we use the interquartile range (IQR), which we will cover with quartiles. The range alone is often reported alongside the median to summarise a data set.
为了获得更稳健的描述，我们使用四分位距（IQR），这将在四分位数部分介绍。通常极差与中位数一起报告，以概括数据集。
3. Quartiles & Interquartile Range | 四分位数与四分位距
rdered data set into four equal parts. The lower quartile (Q1) is the value below which 25% of the data lie. The upper quartile (Q3) is the value below which 75% quartile (Q2). The interquartile range IQR = Q3 - Q1 measures the spread of the middle 50% of the data.
四分位数将有序数据集分为四个相等的部分。下四分位数 (Q1) 是有 25% 的数据小于等于它的值；上四分位数 (Q3) 是有 75% 的数据小于等于它的值。中位数即第二四分位数 (Q2)。四分位距 IQR = Q3 - Q1，度量的是中间 50% 数据的分散程度。
To find Q1 and Q3 by hand, use the position formulas: Q1 is at position (n+1)/4; Q3 is at position 3(n+1)/4. If the position is not an integer, interpolate between the neighbouring ordered values.
手工计算时，使用位置公式：Q1 的位置为 (n+1)/4；Q3 的位置为 3(n+1)/4。若位置不是整数，则在相邻的两个有序值之间进行插值。
IQR = Q3 - Q1
- 1.5×IQR or greater than Q3 + 1.5×IQR. Box plots make these boundaries easy to visualise.
IQR 用于识别异常值。常见的规则是：任何小于 Q1 - 1.5×IQR 或大于 Q3 + 1.5×IQR 的值被视为异常值。箱线图可以直观地展示这些边界。
4. Frequency Tables & Estimated Mean | 频数表与估算平均数
When data are grouped, we lose the exact values and can only estimate the mean. First, find the midpoint of each class interval. Multiply each midpoint by its frequency, sum these products and divide by the total frequency.
当数据被分组后，我们失去了确切值，只能估算平均数。首先求出每个组区间的中点值，然后将每个中点值与对应的频数相乘，把这些乘积求和，再除以总频数。
Estimated mean = Σ(midpoint × frequency) / Σfrequency
he last column divided by sum of f gives the estimate. Remember: this is not the exact mean because we assume all data in a class are at the midpoint.
你可以用表格来整理：列包括组区间、频数 (f)、中点 (x) 以及 f×x。最后一列的总和除以 f 的总和就得到估算平均数。请记住：这并非精确平均数，因为我们假设同一组内所有数据都落在中点上。
The modal class is the class interval with the highest frequency. The median class can be found by locating the position n/2 in a cumulative frequency column.
众数组是频数最高的组区间。中位数组可以通过在累积频率列中找到位置 n/2 来确定。
5. Histograms & Frequency Density | 直方图与频率密度
A histogram uses the area of bars to represent frequency. When class widths are unequal, we must calculate frequency density so that area is proportional to frequency. The vertical axis shows frequency density, not frequency.
直方图用条形的面积来表示频率。当组距不相等时，必须计算频率密度，使得面积与频率成正比。纵轴表示频率密度，而非频率本身。
Frequency density = Frequency / Class width
er intervals do not misleadingly look more frequent just because they span a larger range.
对于给定的条形，其高度 = 频率密度，宽度 = 组距，因此面积 = 频率密度 × 组距 = 频率。这确保了较宽的区间不会仅仅因为跨度大而看起来频率更高。
- To draw a histogram: calculate frequency density for each class, label axes clearly, draw bars that touch (no gaps, since data are continuous).
- 绘制直方图时：计算每个组的频率密度，清晰标注坐标轴，绘制相邻的条形（因为数据是连续的，无间隔）。
6. Cumulative Frequency & Box Plots | 累积频率与箱线图
A cumulative frequency table adds a running total of frequencies. Plotting the upper class boundary against cumulative frequency gives a smooth curve (or polygon). From this curve, you can read off quartiles and the median without raw data.
累积频率表逐组累加频率。将组上限值对应累积频率描点，可绘制出光滑的曲线（或多边形）。据此曲线，即使没有原始数据，也能读出四分位数和中位数。
A box plot (box-and-whisker diagram) displays the five-number summary: minimum, lower quartile, median, upper quartile and maximum. It clearly shows the centre, spread and any potential outliers.
箱线图（箱形图）展示了五数概括法：最小值、下四分位数、中位数、上四分位数和最大值。它能清晰地显示数据中心、离散程度以及潜在的异常值。
When constructing a box plot, first find the five numbers. Draw a box from Q1 to Q3 with a line at the median. Extend whiskers to the minimum and maximum within the non-outlier range. Plot any outliers as individual points beyond the whiskers.
构建箱线图时，先求出这五个数值。从 Q1 到 Q3 画一个矩形，在中间画一条中位数线。从矩形两侧延伸须线至非异常值范围内的最小值和最大值。将任何异常值作为单独的点绘制在须线之外。
7. Basic Probability Rules | 基本概率规则
Probability measures how likely an event is to occur, on a scale from 0 (impossible) to 1 (certain). It can be expressed as a fraction, decimal or percentage. The sample space is the set of all possible outcomes.
概率度量事件发生的可能性，范围从 0（不可能）到 1（必然）。它可以用分数、小数或百分数表示。样本空间是所有可能结果的集合。
P(A) = Number of favourable outcomes / Total number of outcomes
pace is 1. For any event A, P(not A) = 1 - P(A). This is often called the complement rule.
样本空间中所有互斥结果的概率之和为 1。对于任何事件 A，P(非 A) = 1 - P(A)。这常被称为互补规则。
an occur together, we use the general addition rule: P(A or B) = P(A) + P(B) - P(A and B).
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 A 和 B，P(A 或 B) = P(A) + P(B)。如果它们可能同时发生，则需使用一般加法规则：P(A 或 B) = P(A) + P(B) - P(A 且 B)。
8. Tree Diagrams for Combined Events | 复合事件的树状图
Tree diagrams help visualise sequences of events and calculate probabilities for combined outcomes. Each branch is labelled with a probability, and the probabilities on branches from the same point sum to 1.
树状图有助于可视化一系列事件并计算复合结果的概率。每条分支都标有概率，从同一点出发的分支概率之和为 1。
To find the probability of a specific path (e.g., A and then B), multiply the probabilities along the branches. When you need the probability of an event that can occur in several ways, add the probabilities of all relevant paths.
求某条特定路径（例如先 A 后 B）的概率时，将沿途分支的概率相乘。当需要求出通过多种方式发生的事件的概率时，将所有相关路径的概率相加。
P(A and B) = P(A) × P(B given A)
For independent events, P(B given A) is simply P(B). This multiplication rule is fundamental for constructing tree diagrams and solving more complex probability problems at GCSE level.
对于独立事件，P(B given A) 就是 P(B)。这条乘法规则是构建树状图和解决更复杂的 GCSE 概率问题的基础。
Always check that the final set of outcomes (the ends of branches) all together sum to 1. This is a useful check for accuracy.
务必检查最终的一系列结果（分支末端）的概率总和是否为 1。这是验证准确性的好方法。
9. Scatter Graphs & Correlation | 散点图与相关性
A scatter graph displays the relationship between two quantitative variables. Each point represents a paired observation (x, y). We look for correlation: positive (as x increases, y tends to increase), negative (as x increases, y tends to decrease) or no correlation.
散点图展示两个定量变量之间的关系。每个点代表一对观测值 (x, y)。我们观察相关性：正相关（x 增加时 y 呈增加趋势）、负相关（x 增加时 y 呈减少趋势）或无相关。
We describe correlation by its type, strength and whether it is linear. Correlation does not imply causation; a relationship may be due to a third factor or coincidence. Always interpret in context.
我们通过类型、强度和是否呈线性来描述相关性。相关性并不表示因果关系；一种关系可能由第三个因素或巧合引起。始终在具体情境下解读。
10. Line of Best Fit (Linear Regression) | 最佳拟合线（线性回归）
When a scatter graph shows linear correlation, we can model the relationship with a straight line to make predictions. The line of best fit is drawn so that the points are as close as possible to the line, with roughly equal numbers above and below.
当散点图呈现线性相关时，我们可以用一条直线对关系进行建模并做出预测。最佳拟合线的绘制要使各点尽可能靠近直线，且直线上下点数大致相等。
d, the least squares regression line of y on x is used. The equation is y = a + bx, where:
更精确的方法是使用 y 对 x 的最小二乘回归线。其方程为 y = a + bx，其中：
b = (nΣxy - Σx Σy) / (nΣx² - (Σx)²)
a = mean of y - b × mean of x
e the gradient formula: m = (y₂ - y₁)/(x₂ - x₁). Then read the y-intercept (c) to write y = mx + c.
在 Year 9，你通常需要描点、目测画线，并通过在线上的两个点（不必是原始数据点）求出方程。使用斜率公式：m = (y₂ - y₁)/(x₂ - x₁)。然后读出 y 截距 (c)，写出 y = mx + c。
Use the line for interpolation (predicting within the data range) but be cautious with extrapolation (predicting outside the range), as the relationship may no longer hold.
使用该直线进行内插（在数据范围内预测）时要谨慎，而外推（预测范围之外）时更要小心，因为这种关系可能不再成立。
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