📚 Year 9 WJEC Statistics: Formula & Theorem Quick Reference Handbook | 九年级 WJEC 统计：公式定理速查手册
This handbook collates the key formulas, definitions, and theorems required for Year 9 WJEC Statistics. Use it as a rapid reference when solving problems, revising for assessments, or consolidating your understanding of data handling, probability, and statistical measures. Each section pairs concise English explanations with Chinese translations, ensuring clarity for bilingual learners.
本手册汇编了九年级 WJEC 统计所需的关键公式、定义和定理。可在解题、备考复习或巩固数据处理、概率与统计度量理解时用作快速参考。每个部分均将简洁的英文解释与中文翻译配对，确保双语学习者清晰掌握。
1. Measures of Central Tendency | 集中趋势的度量
The three principal measures of central tendency are the mean, median, and mode. They summarise a dataset with a single representative value.
三个主要的集中趋势度量是均值、中位数和众数。它们用一个代表性数值概括数据集。
Mean (x̄) = Σx / n
均值等于所有数据值之和除以数据个数。Σx 表示总和，n 表示频数。
The median is the middle value when data are ordered. For an odd number of observations, it is the centre value; for an even number, it is the average of the two middle values.
中位数是数据排序后的中间值。若观测值个数为奇数，则取正中间的数值；若为偶数，则取中间两个数值的平均数。
The mode is the value that appears most frequently. A dataset may have no mode, one mode, or more than one mode.
众数是出现次数最多的数值。数据集可能没有众数、有一个众数或多个众数。
2. Measures of Spread | 离散程度的度量
Spread tells us how concentrated or dispersed the data are around the centre. Five-number summaries and interquartile range are vital tools.
离散程度表明数据在中心周围的集中或分散程度。五数概括和四分位距是重要工具。
Range = Maximum − Minimum
极差等于最大值减去最小值。
Q₁ position = (n + 1)/4, Q₃ position = 3(n + 1)/4
下四分位数位置 = (n+1)/4，上四分位数位置 = 3(n+1)/4 (其中 n 为数据总数)。若位置不是整数，则通过插值或就近原则确定 Q₁ 和 Q₃。
Interquartile Range (IQR) = Q₃ − Q₁
四分位距等于上四分位数减去下四分位数。IQR 表示中间 50% 数据的跨度。
The five-number summary consists of: Minimum, Q₁, Median, Q₃, Maximum. It is often displayed in box plots.
五数概括包括：最小值、下四分位数、中位数、上四分位数、最大值。通常用箱线图展示。
3. Frequency Distributions | 频数分布
Frequency distributions organise raw data into groups (classes) to reveal patterns. For grouped continuous data, frequency density is used in histograms.
频数分布将原始数据分组以揭示模式。对于分组连续数据，直方图中使用频数密度。
Frequency Density = Frequency / Class Width
频数密度 = 频数 ÷ 组距。在直方图中，条形的面积与频数成正比。
The class midpoint is the average of the upper and lower class boundaries. It is used to estimate the mean from grouped data.
组中点是组上限与组下限的平均值，用于从分组数据中估算均值。
table, the estimated mean is Σ(f × midpoint) / Σf, where f is the frequency of each class.
对于分组频数表，估计均值为 Σ(f × 组中点) ÷ Σf，其中 f 为各组的频数。
4. Cumulative Frequency and Percentiles | 累积频数与百分位数
Cumulative frequency is the running total of frequencies. A cumulative frequency graph (ogive) helps locate medians, quartiles, and percentiles without raw data.
累积频数是频数的累加总和。累积频数图有助于在没有原始数据的情况下确定中位数、四分位数和百分位数。
To estimate the median from a cumulative frequency curve, find the value corresponding to total frequency / 2 on the vertical axis.
从累积频数曲线估计中位数时，在纵轴上找到总频数的一半，读取对应的横坐标值。
Percentile rank: kth percentile position = (k/100) × n
百分位数等级：第 k 百分位数位置 = (k/100) × n。可使用插值法从累积频数图中读取。
The lower quartile (Q₁) is the 25th percentile, the median is the 50th percentile, and the upper quartile (Q₃) is the 75th percentile.
下四分位数是第 25 百分位数，中位数是第 50 百分位数，上四分位数是第 75 百分位数。
5. Probability Rules | 概率法则
Probability quantifies the chance that an event occurs. The basic probability scale ranges from 0 (impossible) to 1 (certain). All outcomes must be equally likely to use classical probability.
概率量化事件发生的可能性。基本概率尺度从 0 (不可能) 到 1 (必然)。使用古典概率时所有结果必须等可能。
P(A) = Number of favourable outcomes / Total number of outcomes
事件 A 的概率等于有利结果数目除以所有可能结果的总数。
Addition Rule: P(A or B) = P(A) + P(B) − P(A and B)
加法法则：P(A 或 B) = P(A) + P(B) − P(A 且 B)。若 A 与 B 互斥，则 P(A and B) = 0，公式简化为 P(A) + P(B)。
Multiplication Rule: P(A and B) = P(A) × P(B|A)
乘法法则：P(A 且 B) = P(A) × P(B|A)，其中 P(B|A) 是给定 A 发生时 B 的条件概率。若事件独立，则 P(B|A) = P(B)，所以 P(A and B) = P(A) × P(B)。
6. Independent and Mutually Exclusive Events | 独立事件与互斥事件
Mutually exclusive events cannot occur at the same time. Independent events have no influence on each other: the occurrence of one does not affect the probability of the other.
互斥事件不能同时发生。独立事件彼此无影响：一个事件的发生不影响另一个事件发生的概率。
For mutually exclusive events A and B, P(A and B) = 0. For independent events, P(A|B) = P(A) and P(B|A) = P(B).
对于互斥事件 A 和 B，P(A 且 B) = 0。对于独立事件，P(A|B) = P(A) 且 P(B|A) = P(B)。
ce, check whether P(A and B) = P(A) × P(B). If equality holds, the events are independent.
检验独立性时，验证 P(A 且 B) 是否等于 P(A) × P(B)。若等式成立，则事件独立。
A common misconception is that mutually exclusive events are also independent. This is false because if one occurs, the probability of the other immediately becomes zero.
常见误解是互斥事件也是独立事件。这是错误的，因为若一个发生，另一个的概率立即变为零。
7. Tree Diagrams | 树状图
Tree diagrams illustrate sequential events and their probabilities. Each branch is labelled with a probability; multiplying along branches gives the joint probability, while adding relevant end probabilities gives the total probability of an outcome.
树状图展示顺序事件及其概率。每条分枝标有概率；沿分枝相乘得到联合概率，相加有关的最终概率则得到某个结果的总概率。
Probabilities on branches stemming from a single node always sum to 1. Replace items with or without replacement accordingly, as this changes the probabilities.
从同一节点伸出的所有分枝的概率之和为 1。需根据是否放回调整条件概率，因为放回与否会改变概率。
P(Two events in sequence) = P(1st) × P(2nd|1st)
两个连续事件的概率 = P(第一个事件) × P(第二个事件|第一个事件发生)。利用树状图可清晰展示所有可能路径。
8. Scatter Diagrams and Correlation | 散点图与相关性
A scatter diagram (scatter plot) displays the relationship between two numerical variables. Correlation describes the direction and strength of the linear association.
散点图展示两个数值变量之间的关系。相关性描述线性关联的方向和强度。
Positive correlation: as one variable increases, the other tends to increase. Negative correlation: as one increases, the other tends to decrease. No correlation shows a random pattern.
正相关：一个变量增大时另一个也趋于增大。负相关：一个增大时另一个趋于减小。无相关则呈现随机模式。
Outliers are points that lie far from the general pattern. They can influence the interpretation of correlation and the line of best fit.
离群值是远离总体模式的点，会影响相关性的解读和最佳拟合线。
Correlation does not imply causation. An observed association may be due to a third factor or coincidence.
相关性并不意味着因果关系。观察到的关联可能因为第三因素或巧合。
9. Equation of the Line of Best Fit | 最佳拟合线方程
the deviations between observed points and the line. Its equation has the form y = mx + c.
最佳拟合线使观测点与直线之间的偏差最小化。其方程形式为 y = mx + c。
Gradient m = (nΣxy − Σx Σy) / (nΣx² − (Σx)²)
斜率 m = (nΣxy − Σx Σy) ÷ (nΣx² − (Σx)²)，其中 Σxy 是每对 x 与 y 乘积的总和，Σx² 是 x 平方的总和。
Intercept c = ȳ − m x̄
截距 c = ȳ − m x̄，其中 ȳ 是 y 的均值，x̄ 是 x 的均值。回归线始终通过点 (x̄, ȳ)。
For a rough hand-drawn line, pass the line through (x̄, ȳ) and follow the trend. The gradient can be estimated as change in y ÷ change in x.
手绘拟合线时，让直线穿经点 (x̄, ȳ) 并贴近整体趋势。斜率可估计为 y 的变化量 ÷ x 的变化量。
The coefficient of determination (r²) indicates how well the line fits; Year 9 students should be aware that values closer to 1 represent a stronger fit.
决定系数 (r²) 显示直线拟合程度；九年级学生须了解数值越接近 1 表示拟合越好。
10. 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 time intervals. Moving averages help smooth out short-term fluctuations and highlight trends.
时间序列是按规律时间间隔记录的数据集。移动平均有助于平滑短期波动并突出趋势。
k-point moving average = (sum of k consecutive values) / k
k 点移动平均值 = (连续的 k 个数值之和) ÷ k。将该平均值置于时间中点有助于描绘趋势线。
When calculating a 4-point moving average, the first average is centred between points 2 and 3. Further centring (e.g. 2-point average of two moving averages) may be needed to align with actual time periods.
计算 4 点移动平均时，第一个平均值位于第 2 与第 3 点之间。可能需要对两个移动平均值再取平均以实现中心化，从而与实际时间段对齐。
Seasonal variation can be identified after removing the trend. Year 9 contexts often involve simple interpretation rather than extensive calculations.
剔除趋势后可识别季节性变动。九年级的内容通常侧重简单解读，而非繁杂计算。
Published by TutorHao | Statistics Revision Series | aleveler.com
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