📚 Year 10 CCEA Statistics: Formula & Theorem Quick Reference Handbook | Year 10 CCEA 统计：公式定理速查手册
This quick reference handbook covers the essential formulas and theorems for Year 10 CCEA Statistics. It includes descriptive statistics, probability, data representation, index numbers, time series, and correlation. Every rule is explained with its formula and interpretation to support revision and problem-solving.
本速查手册涵盖 Year 10 CCEA 统计的核心公式与定理，包括描述统计、概率、数据呈现、指数、时间序列与相关性。每条规则均配有公式与说明，助力复习与解题。
1. Measures of Central Tendency | 集中趋势度量
For a set of n observations x₁, x₂, …, xₙ, the sample mean is the sum of all the values divided by the number of values. It is the balance point of the data and is the measure most affected by extreme values.
对于 n 个观测值 x₁, x₂, …, xₙ，样本均值为所有数值之和除以数值个数。它是数据的平衡点，也是受极端值影响最大的度量。
x̄ = (∑x) / n = (x₁ + x₂ + … + xₙ) / n
For grouped data, the estimated mean is x̄ m is the class midpoint. The midpoint is the average of the lower and upper class boundaries, so the mean calculated from grouped data is an estimate, not an exact value.
对于分组数据，平均值估计为 x̄ = ∑(f × m) / ∑f，其中 f 为频数，m 为组中值。组中值为组下限与组上限的平均数，因此由分组数据算出的均值是估计值，并非精确值。
x̄ = ∑(f × m) / ∑f
When two data sets are combined, the overall mean is a weighted mean of the two separate means. Each mean is weighted by the size of its own group.
当两组数据合并时，总体均值是两个各自均值的加权平均，权重为各自样本的容量。
x̄combined = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)
The median is the middle value when data are ordered. Its position is (n + 1)/2. If n is odd the median is an actual data value; if n is even it is the mean of the two middle values.
中位数是排序后位于中间的值，位置为 (n + 1)/2。若 n 为奇数，中位数是实际数据值；若 n 为偶数，则取中间两个值的平均。
Position of median = (n + 1) / 2
For grouped data, interpolation using the cumulative frequency graph gives the median. The same result can be obtained from the linear interpolation formula below, where L is the lower class boundary of the median class, F is the cumulative frequency before that class, f is the frequency of that class and w is the class width.
分组数据可通过累积频率曲线插值得到中位数。也可用下面的线性插值公式，其中 L 为中位数组的下组界，F 为该组之前的累积频数，f 为该组频数，w 为组距。
Median = L + [ (n/2 − F) / f ] × w
The mode is the most frequently occurring value. In grouped data, the modal class is the interval with the highest frequency density, since class widths may differ. The modal value inside that class can be estimated with the formula below, where d₁ is the gap to the previous class frequency and d₂ is the gap to the next class frequency.
众数是出现次数最多的值。在分组数据中，众数类是频率密度最高的区间，因为各组组距可能不同。该区间内部的众数可用下式估计，其中 d₁ 为与前一组频数之差，d₂ 为与后一组频数之差。
Mode = L + [ d₁ / (d₁ + d₂) ] × w
The mean uses every value, the median resists outliers and the mode identifies the most common category. In an exam, always state which average you are using and justify the choice in the context of the question.
均值利用所有数值，中位数抗极端值影响，众数则指出最常见的类别。考试中务必说明你使用哪种平均数，并结合题目情境说明选择理由。
2. Measures of Dispersion | 离散程度度量
Q₁. The range is quick but sensitive to outliers, while the IQR describes the spread of the middle 50 percent of the data.
极差 = 最大值 − 最小值。四分位距 IQR = Q₃ − Q₁。极差计算快捷但易受极端值影响，而四分位距描述中间 50% 数据的离散程度。
Range = max − min IQR = Q₃ − Q₁
The sample variance measures the average squared deviation from the mean, using n − 1 in the denominator to correct for the bias of a sample estimate. The sample standard deviation is its square root and is expressed in the same units as the original data.
样本方差衡量各数值与均值偏差平方的平均值，分母用 n − 1 以修正样本估计的偏差。样本标准差是其平方根，单位与原始数据相同。
s² = ∑(x − x̄)² / (n − 1) s = √[ ∑(x − x̄)² / (n − 1) ]
and is usually faster in timed conditions, provided you keep careful track of ∑x and ∑x².
另一种计算式为 s = √[(∑x² − (∑x)²/n) / (n − 1)]。该形式无需逐个计算偏差，在限时作答中通常更快，但要小心记录 ∑x 与 ∑x²。
s = √[ (∑x² − (∑x)²/n) / (n − 1) ]
For grouped data, standard deviation can be approximated using s = √[∑f(m − x̄ grouped mean, this is an estimate because individual values inside each class are unknown.
对于分组数据，标准差近似为 s = √[∑f(m − x̄)² / (∑f − 1)]，其中 m 为组中值，∑f 为总频数。与分组均值一样，由于组内具体数值未知，这只是一个估计值。
s = √[ ∑f(m − x̄)² / (∑f − 1) ]
The coefficient of variation expresses the standard deviation as a percentage of the mean. It allows you to compare variability between data sets measured in different units or with very different means.
变异系数把标准差表示为均值的百分比，可用于比较单位不同或均值差异很大的数据集之间的离散程度。
CV = (s / x̄) × 100%
ue is coded by y = (x − a)/b, the mean and standard deviation change in a predictable way: ȳ = (x̄ − a)/b and sy = sx / |b|. Coding never changes the shape of the distribution or the value of the correlation coefficient.
当所有数值按 y = (x − a)/b 进行编码时，均值与标准差的变化可预测：ȳ = (x̄ − a)/b，sy = sx / |b|。编码不改变分布形状，也不改变相关系数的数值。
3. Quartiles and Percentiles | 四分位数与百分位数
The lower quartile Q₁ is at position (n + 1)/4, the median Q₂ at (n + 1)/2, and the upper quartile Q₃ at 3(n + 1)/4. For grouped data use the cumulative frequency curve or linear interpolation.
下四分位数 Q₁ 的位置是 (n + 1)/4，中位数 Q₂ 在 (n + 1)/2，上四分位数 Q₃ 在 3(n + 1)/4。分组数据用累积频率曲线或线性插值确定。
Q₁: (n + 1)/4 Q₂: (n + 1)/2 Q₃: 3(n + 1)/4
The kth percentile is the value below which k percent of the ordered data lie. Percentiles generalise quartiles: Q₁ is the 25th percentile, the median is the 50th percentile and Q₃ is the 75th percentile.
第 k 百分位数是指有 k% 的有序数据小于或等于该值。百分位数是四分位数的推广：Q₁ 是第 25 百分位数，中位数是第 50 百分位数，Q₃ 是第 75 百分位数。
Position of Pₖ ≈ k(n + 1) / 100
For grouped data, the kth percentile is estimated by interpolation in the same way as the median, replacing n/2 with kn/100 in the numerator.
对于分组数据，第 k 百分位数用与中位数相同的插值方法估计，只需把分子中的 n/2 换成 kn/100。
Pₖ = L + [ (kn/100 − F) / f ] × w
The semi-interquartile range is half the IQR and gives a simple measure of spread that ignores the most extreme quarter of values at each end.
半四分位距是四分位距的一半，是一个忽略两端极端四分之一数值的简易离散度量。
Semi-IQR = (Q₃ − Q₁) / 2
1.5 × IQR. Values outside this fence should be checked rather than automatically deleted.
异常值通常定义为小于 Q₁ − 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的数值。超出这一界限的数值应加以核查，而不是自动删除。
Outlier: x < Q₁ − 1.5 × IQR or x > Q₃ + 1.5 × IQR
Percentiles appear in real contexts such as growth charts, standardised test scores and waiting-time data. When interpreting a percentile, always write a sentence in the context of the problem, for example: "about 90 percent of students scored below this mark".
百分位数常见于生长曲线、标准化考试成绩与等候时间数据等真实情境。解释百分位数时，务必结合题目情境写一句完整的话，例如："大约 90% 的学生得分低于该分数"。
4. Data Representation | 数据呈现
A frequency table organises raw data into classes. The class width is the difference between consecutive lower class boundaries, and the frequency density is frequency divided by class width.
频数表把原始数据整理为组。组距为相邻组下限之差，频率密度等于频数除以组距。
Class width = upper boundary − lower boundary Frequency density = f / w
In a histogram the horizontal axis is continuous and the vertical axis is frequency density, so the area of each bar is proportional to the frequency. This is essential when class widths are unequal.
直方图的横轴是连续变量，纵轴是频率密度，因此每个矩形的面积与频数成正比。当组距不相等时，这一点尤为关键。
Area of bar = frequency density × class width = frequency
A cumulative frequency curve plots cumulative frequency against upper class boundary. Read the median at n/2, the lower quartile at n/4 and the upper quartile at 3n/4 up the vertical axis, then across to the curve and down to the horizontal axis.
累积频率曲线以组上限为横坐标、累积频数为纵坐标。在纵轴上取 n/2 读中位数、取 n/4 读下四分位数、取 3n/4 读上四分位数，横向对应到曲线，再向下读出横轴数值。
A box plot displays the minimum, Q₁, median, Q₃ and maximum on a single scale. Comparing two box plots reveals differences in both average and spread, and the position of the median within the box indicates skewness.
箱线图在同一数轴上显示最小值、Q₁、中位数、Q₃ 与最大值。比较两个箱线图可以看出平均水平与离散程度的差异，中位数在箱内的位置则反映偏态。
A stem-and-leaf diagram retains the original data values while sorting them, which is why the original values can still be recovered. Back-to-back stem-and-leaf diagrams compare two groups using a shared stem.
茎叶图在排序的同时保留原始数据值，因此仍可还原出原始数据。背靠背茎叶图以共用的茎比较两组数据。
tegorical data, and the scatter diagram for paired numerical data. Choosing the correct display is itself an assessed skill.
其他标准图表包括：饼图，其中每个扇形角度等于 (f/n) × 360°；条形图用于分类数据；散点图用于成对数值数据。选择正确的图表本身就是一项考查技能。
| Display | Best used for | 中文说明 |
| Histogram | Continuous grouped data | 连续型分组数据 |
| Cumulative frequency curve | Finding median and quartiles | 求中位数与四分位数 |
| Box plot | Comparing two distributions | 比较两个分布 |
| Stem-and-leaf | Small data sets, keeping values | 小样本且需保留原值 |
| Scatter diagram | Relationship between two variables | 两变量间的关系 |
5. Probability Fundamentals | 概率基础
For equally likely outcomes, the probability of event A is the number of favourable outcomes divided by the total number of outcomes. Every probability lies between 0 and 1 inclusive.
对于等可能结果，事件 A 的概率等于有利结果数除以总结果数。任何概率都介于 0 与 1 之间（含端点）。
P(A) = number of favourable outcomes / total number of outcomes 0 ≤ P(A) ≤ 1
The complement rule states that the probability of an event not happening is one minus the probability that it does happen. This is often the fastest route to an answer when "at least one" appears in the question.
对立事件法则指出，事件不发生的概率等于 1 减去它发生的概率。当题目出现"至少一个"时，这往往是最快的解题路径。
P(A′) = 1 − P(A)
The addition rule for two events accounts for the overlap, which would otherwise be counted twice. When events are mutually exclusive the overlap is empty and the rule simplifies.
两事件的加法法则要扣除被重复计算的重叠部分。当两事件互斥时，重叠为空集，法则随之简化。
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Mutually exclusive: P(A ∩ B) = 0 so P(A ∪ B) = P(A) + P(B)
Two events are independent when the outcome of one does not affect the probability of the other. Only then may the probabilities be multiplied directly.
当一件事的结果不影响另一件事的概率时，两事件相互独立。只有在独立的前提下，才能直接相乘求交事件概率。
Independent events: P(A ∩ B) = P(A) × P(B)
A set of events is exhaustive if their probabilities sum to 1, and expected frequency is found by multiplying a probability by the number of trials.
若一组事件的概率之和为 1，则称为完备事件组；期望频数等于概率乘以试验次数。
Expected frequency = P(A) × number of trials
A common exam error is multiplying probabilities for events that are not independent, for example when items are taken without replacement. Always ask whether the first outcome changes the second.
常见失分点是对并不独立的事件直接相乘，例如无放回抽取。始终先问自己：第一个结果是否改变了第二个结果的概率。
6. Conditional Probability and Tree Diagrams | 条件概率与树形图
The conditional probability P(A|B) is the probability of A given that B has already occurred. It restricts the sample space to the outcomes in B.
条件概率 P(A|B) 表示在 B 已发生的条件下 A 发生的概率，它把样本空间限制在 B 所包含的结果之内。
P(A|B) = P(A ∩ B) / P(B) (P(B) > 0)
Rearranging gives
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