📚 Year 10 Eduqas Statistics: Formula & Theorem Quick Reference | Year 10 Eduqas 统计：公式定理速查手册
This quick reference handbook brings together all the essential formulas and theorems you need for Year 10 Eduqas Statistics. Use it alongside your class notes to reinforce key concepts, from calculating averages to interpreting correlation and time series.
这份速查手册汇总了 Year 10 Eduqas 统计学所需的所有核心公式与定理。配合课堂笔记使用，能够帮助你巩固从计算均值到解读相关性与时间序列等各个关键知识点。
1. Mean, Median, Mode | 均值、中位数、众数
The mean for raw data is found by summing all values and dividing by the number of data items: Mean = Σx / n.
原始数据的均值是将所有数值相加再除以数据个数：均值 = Σx / n。
For a frequency table, use midpoints x and frequencies f: Mean = Σf x / Σf.
对于频数表，使用组中值 x 与频数 f：均值 = Σf x / Σf。
The median is the middle value once data are sorted. Its position is at (n + 1) / 2. For grouped data, read the median from a cumulative frequency graph.
中位数是排序后数据的中间值，位置在 (n + 1) / 2 处。对分组数据，从累积频数图中读取中位数。
The mode is the value with the highest frequency. A data set can be bimodal or have no mode.
众数是出现次数最多的数值；一组数据可能出现双众数或无众数。
2. Measures of Dispersion | 离散程度的度量
The range is the difference between the largest and smallest values: Range = Max − Min.
极差是最⼤值与最小值之差：极差 = 最大值 − 最小值。
The interquartile range (IQR) measures the spread of the middle 50%: IQR = Q₃ − Q₁, where Q₁ and Q₃ are the lower and upper quartiles.
四分位距 (IQR) 反映中间 50% 数据的分散程度：IQR = Q₃ − Q₁，Q₁ 和 Q₃ 分别为下、上四分位数。
For sample data, the variance is given by s² = Σ(x − x̄)² / (n − 1). The standard deviation is its square root: s = √[ Σ(x − x̄)² / (n − 1) ].
样本方差公式为 s² = Σ(x − x̄)² / (n − 1)，标准差是其平方根：s = √[ Σ(x − x̄)² / (n − 1) ]。
A shortcut for computation uses Sxx = Σx² − (Σx)²/n, then s = √(Sxx / (n − 1)).
一个简便的计算公式为 Sxx = Σx² − (Σx)²/n，进而 s = √(Sxx / (n − 1))。
3. Frequency Distributions and Histograms | 频数分布与直方图
For histograms, the vertical axis shows frequency density, calculated as Frequency Density = Frequency ÷ Class Width.
直方图的纵轴表示频率密度：频率密度 = 频数 ÷ 组距。
Cumulative frequency graphs plot running totals against the upper class boundary. They allow you to estimate the median, quartiles and percentiles.
累积频数图是将累积频数对组上限描点，可用于估计中位数、四分位数和百分位数。
The area of each bar in a histogram is proportional to the frequency. Bars are drawn with no gaps when the data are continuous.
直方图中每个直条的面积与频数成比例。连续数据绘制直方图时，条与条之间不留空隙。
4. Basic Probability | 基本概率
For an event A in a finite sample space, P(A) = n(A) / n(S), where n(S) is the total number of equally likely outcomes.
若事件 A 来自有限样本空间，P(A) = n(A) / n(S)，n(S) 为等可能结果总数。
Mutually exclusive events cannot happen together: P(A or B) = P(A) + P(B).
互斥事件不能同时发生：P(A 或 B) = P(A) + P(B)。
For independent events, P(A and B) = P(A) × P(B). Conditional probability is given by P(A|B) = P(A ∩ B) / P(B).
对于独立事件，P(A 且 B) = P(A) × P(B)。条件概率公式为 P(A|B) = P(A ∩ B) / P(B)。
5. Binomial Distribution | 二项分布
If X follows the binomial distribution B(n, p), the probability of exactly r successes is P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ.
若 X 服从二项分布 B(n, p)，恰有 r 次成功的概率为 P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ。
Here ⁿCᵣ is the binomial coefficient, also written as C(n, r) or nCr. The conditions for a binomial model are: fixed number of trials, two outcomes per trial, constant probability p, and independence.
这里 ⁿCᵣ 为二项式系数，也可记作 C(n, r) 或 nCr。二项分布的条件为：试验次数固定、每次只有两种结果、概率 p 恒定、各次试验独立。
6. The Normal Distribution | 正态分布
The normal curve is bell‑shaped, symmetric about the mean μ. The standard deviation σ determines the spread. The empirical rule states: about 68% of data lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
正态曲线呈钟形，关于均值 μ 对称。标准差 σ 决定分散程度。经验法则：约 68% 的数据落在 μ ± σ 内，95% 在 μ ± 2σ 内，99.7% 在 μ ± 3σ 内。
Standardising converts any normal observation to a z‑score: z = (x − μ) / σ. The z‑score tells how many standard deviations x is above or below the mean.
标准化转换将任意正态观测值变为 z 分数：z = (x − μ) / σ，z 分数表示 x 比均值高出或低几个标准差。
7. Scatter Graphs and Correlation | 散点图与相关
A scatter graph shows the relationship between two variables. Correlation can be positive, negative, or non‑existent. The strength is described by how closely the points follow a straight line.
散点图显示两个变量之间的关系。相关可以为正、负或无相关；强度取决于点接近直线的程度。
A line of best fit drawn by eye should pass through the mean point (x̄, ȳ) and reflect the overall trend.
目测绘制的最佳拟合线应穿过均值点 (x̄, ȳ) 并反映整体趋势。
8. Spearman’s Rank Correlation | 斯皮尔曼秩相关
Spearman’s rank coefficient measures the strength and direction of monotonic association: rₛ = 1 − (6 Σd²) / (n(n² − 1)), where d is the difference between ranks of each pair.
斯皮尔曼秩相关系数衡量单调关联的强度与方向：rₛ = 1 − (6 Σd²) / (n(n² − 1))，其中 d 是每对数据的秩次差。
The value rₛ ranges from −1 to 1. A value near +1 indicates strong positive rank correlation; near −1 indicates strong negative rank correlation.
rₛ 的取值范围为 −1 到 1。接近 +1 表示强正秩相关；接近 −1 表示强负秩相关。
9. Least Squares Regression Line | 最小二乘回归线
The regression line of y on x has equation y = a + b x. The gradient b is computed from summary statistics: b = Sxy / Sxx, and the intercept a = ȳ − b x̄.
y 对 x 的回归线方程为 y = a + b x。斜率 b 由以下统计量计算：b = Sxy / Sxx，截距 a = ȳ − b x̄。
Here Sxy = Σxy − (Σx Σy) / n and Sxx = Σx² − (Σx)² / n. The line always passes through the mean point (x̄, ȳ).
其中 Sxy = Σxy − (Σx Σy) / n，Sxx = Σx² − (Σx)² / n。该直线必过均值点 (x̄, ȳ)。
Use the regression line only within the range of observed data to estimate values; extrapolation beyond the data may be unreliable.
回归线只应在观测数据范围内用于估计；外推可能导致不可靠的结果。
10. Index Numbers | 指数
A simple price index measures the price change of a single item relative to a base period: Index = (P₁ / P₀) × 100.
简单价格指数衡量单个项目相对于基期的价格变化：指数 = (P₁ / P₀) × 100。
When several items are involved, a weighted index combines price relatives using importance weights. The weighted aggregate price index formula is Index = Σ(P₁ × q₀) / Σ(P₀ × q₀) × 100 (Laspeyres) or similar.
涉及多个项目时，加权指数利用权重将价格比综合起来。加权综合价格指数公式可用拉氏指数：指数 = Σ(P₁ × q₀) / Σ(P₀ × q₀
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