📚 GCSE Eduqas Statistics: Formula & Theorem Quick Reference Handbook | GCSE Eduqas 统计：公式定理速查手册
This handbook provides a concise reference for all essential formulas and theorems covered in the GCSE Eduqas Statistics specification. Each section presents key equations with clear notation, allowing you to quickly review and apply them during revision and exams.
本手册为 GCSE Eduqas 统计课程中所有重要公式和定理提供简明参考。每个部分都列出了关键方程和清晰符号，便于您在复习和考试中快速查阅和应用。
1. Measures of Central Tendency | 集中趋势测量
The arithmetic mean for raw data is the sum of all values divided by the number of values. For grouped data, multiply each class midpoint (m) by its frequency (f) and divide by the total frequency.
对于原始数据，算术平均值是所有值的总和除以值的个数。对于分组数据，将每个组的中点值 (m) 乘以其频数 (f)，然后除以总频数。
x̄ = (∑x)/n or x̄ = (∑fx)/∑f
The median is the middle value when data are ordered. For n observations, the median position is (n+1)/2. In grouped data, use interpolation: Median = L + ((n/2 – F) / fm) × c, where L is the lower class boundary, F is the cumulative frequency before the median class, fm is the frequency of that class, and c is the class width.
中位数是数据排序后中间的值。对于 n 个观测值，中位数位置为 (n+1)/2。在分组数据中，使用插值公式：中位数 = L + ((n/2 – F) / fm) × c，其中 L 为所在组下限，F 为到上一组为止的累积频数，fm 为该组频数，c 为组距。
The mode is the most frequently occurring value. For grouped data, identify the modal class (highest frequency density) or estimate the mode using the formula: Mode ≈ L + ( (f₁ – f₀) / (2f₁ – f₀ – f₂) ) × c, where f₁ is the frequency of the modal class, f₀ the frequency before and f₂ after.
众数是最常出现的值。对于分组数据，找出众数所在组（最高频率密度）或用公式估算：众数 ≈ L + ( (f₁ – f₀) / (2f₁ – f₀ – f₂) ) × c，其中 f₁ 为众数组频数，f₀ 为前一组频数，f₂ 为后一组频数。
2. Measures of Dispersion | 离散程度测量
The range is the difference between the largest and smallest values: Range = Xmax – Xmin. The interquartile range (IQR) = Q₃ – Q₁, where Q₁ is the lower quartile and Q₃ is the upper quartile.
极差是最大值与最小值的差：极差 = Xmax – Xmin。四分位距 (IQR) = Q₃ – Q₁，其中 Q₁ 为下四分位数，Q₃ 为上四分位数。
Variance measures average squared deviation from the mean. The population variance (σ²) and sample variance (s²) are calculated as follows. For grouped data, use midpoints and frequencies.
方差衡量数据与均值偏差的平方平均值。总体方差 (σ²) 和样本方差 (s²) 计算公式如下。对于分组数据，使用中点值和频数。
σ² = (∑(x – μ)²) / N or s² = (∑(x – x̄)²) / (n – 1)
Standard deviation is the square root of variance: σ = √σ². A simpler computational formula is often used: s = √( (∑x² – (∑x)²/n) / (n-1) ). For grouped data, replace x by class midpoint m and multiply by f.
标准差是方差的平方根：σ = √σ²。通常使用简便计算式：s = √( (∑x² – (∑x)²/n) / (n-1) )。对于分组数据，用组中点 m 代替 x，并乘上频数 f。
3. Frequency Distributions and Histograms | 频率分布与直方图
In histograms, the area of each bar is proportional to the frequency. Frequency density is used when class widths are unequal.
在直方图中，每个条形的面积与频数成正比。当组距不相等时，需要使用频率密度。
Frequency density = Frequency / Class width
频率密度 = 频数 / 组距
When constructing a histogram, plot class boundaries on the horizontal axis and frequency density on the vertical axis. The total area under the histogram equals the total frequency.
绘制直方图时，横轴标出组边界，纵轴标出频率密度。直方图下的总面积等于总频数。
For a frequency polygon, join the midpoints of the tops of the histogram bars with straight lines. For cumulative frequency curves, plot cumulative frequency against upper class boundaries.
对于频率多边形，用直线连接直方图条形顶部的中点。对于累积频率曲线，将累积频数对于组上限值绘制点图。
4. Cumulative Frequency, Quartiles and Box Plots | 累积频率、四分位数与箱线图
Quartiles divide ordered data into four equal parts. Q₁ is the value at position (n+1)/4, Q₂ at (n+1)/2, Q₃ at 3(n+1)/4. From a cumulative frequency graph, quartiles are read off at the corresponding cumulative frequencies.
四分位数将有序数据分为四等份。Q₁ 位于第 (n+1)/4 位，Q₂ 位于第 (n+1)/2 位，Q₃ 位于第 3(n+1)/4 位。从累积频率图中，在相应的累积频数处读取四分位数。
IQR = Q₃ – Q₁ & Semi-interquartile range = IQR / 2
A box plot displays the five-number summary: minimum, Q₁, median, Q₃, maximum. Outliers may be defined as values below Q₁ – 1.5 × IQR or above Q₃ + 1.5 × IQR.
箱线图展示五数总结：最小值、Q₁、中位数、Q₃、最大值。异常值可定义为低于 Q₁ – 1.5×IQR 或高于 Q₃ + 1.5×IQR 的值。
5. Basic Probability Rules | 基础概率规则
Probability of an event A is P(A) = number of favourable outcomes / total number of outcomes, with 0 ≤ P(A) ≤ 1.
事件 A 的概率 P(A) = 有利结果的数量 / 所有可能结果的总数，且 0 ≤ P(A) ≤ 1。
The addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
加法法则：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。对于互斥事件，P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。
The complement rule: P(A’) = 1 – P(A). For independent events, the multiplication rule is P(A ∩ B) = P(A) × P(B).
补集法则：P(A’) = 1 – P(A)。对于独立事件，乘法法则为 P(A ∩ B) = P(A) × P(B)。
6. Conditional Probability and Tree Diagrams | 条件概率与树状图
Conditional probability is the probability of event A given event B has occurred: P(A|B) = P(A ∩ B) / P(B).
条件概率是指在事件 B 已发生的条件下事件 A 发生的概率：P(A|B) = P(A ∩ B) / P(B)。
Tree diagrams help visualise sequential events. Multiply along branches for combined probabilities, and sum probabilities of relevant end points.
树状图有助于可视化连续事件。沿着分支相乘得到联合概率，并将相关终点的概率相加。
P(A ∩ B) = P(A) × P(B|A)
Two events are independent if P(A|B) = P(A) or equivalently P(A ∩ B) = P(A) × P(B).
若 P(A|B) = P(A) 或等价地 P(A ∩ B) = P(A) × P(B)，则两事件独立。
7. Binomial Distribution | 二项分布
If there are n independent trials, each with success probability p and failure probability q = 1 – p, then the number of successes X follows a binomial distribution: X ~ B(n, p).
若有 n 次独立试验，每次成功概率为 p，失败概率为 q = 1 – p，则成功次数 X 服从二项分布：X ~ B(n, p)。
P(X = r) = nCr pr qn-r
The binomial coefficient nCr = n! / (r!(n-r)!). Mean (expected value) = np, variance = npq.
二项系数 nCr = n! / (r!(n-r)!)。均值（期望值）= np，方差 = npq。
For calculations, you can use the formula, a calculator’s binomial PD function, or statistical tables if provided.
计算时可使用公式、计算器的二项概率分布功能，或若提供的统计表。
8. Scatter Graphs and Correlation | 散点图与相关性
Scatter diagrams show the relationship between two variables. Correlation describes the strength and direction of a linear relationship.
散点图展示两个变量之间的关系。相关性描述线性关系的强度和方向。
Pearson’s product-moment correlation coefficient r measures linear correlation:
皮尔逊积矩相关系数 r 衡量线性相关：
r = Sxy / √(Sxx Syy)
where Sxy = ∑xy – (∑x∑y)/n, Sxx = ∑x² – (∑x)²/n, Syy = ∑y² – (∑y)²/n. r ranges from -1 to +1.
其中 Sxy = ∑xy – (∑x∑y)/n，Sxx = ∑x² – (∑x)²/n，Syy = ∑y² – (∑y)²/n。r 的取值范围为 -1 到 +1。
Spearman’s rank correlation coefficient rs is used when data are ranked:
当数据为等级数据时，使用斯皮尔曼等级相关系数 rs：
rs = 1 – (6∑d²) / (n(n² – 1))
where d is the difference between each pair of ranks. It also ranges from -1 to +1, with +1 indicating perfect agreement.
其中 d 为每对等级之差。其取值范围也是 -1 到 +1，+1 表示完全正等级相关。
9. Regression Lines | 回归线
The equation of the least squares regression line of y on x is y = a + bx, where b = Sxy / Sxx and a = ȳ – b x̄. This line minimises the sum of squared vertical deviations.
y 对 x 的最小二乘回归线方程为 y = a + bx，其中 b = Sxy / Sxx，a = ȳ – b x̄。该直线使竖直偏差平方和最小。
b = (n∑xy – ∑x∑y) / (n∑x² – (∑x)²)
The regression line can be used to estimate y for a given x (interpolation within the data range). Beware of extrapolation beyond the data range, which may be unreliable.
回归线可用于对给定 x 值估计 y（数据范围内的插值）。当心外推超出数据范围，可能不可靠。
If x on y regression is needed, the line is x = a’ + b’y with b’ = Sxy / Syy and a’ = x̄ – b’ ȳ.
若需要对 y 求 x 的回归线，方程为 x = a’ + b’y，其中 b’ = Sxy / Syy，a’ = x̄ – b’ ȳ。
10. Index Numbers | 指数
An index number compares a value over time relative to a base period. Simple price index = (Pt / P0) × 100, where Pt is the current price and P0 is the base period price.
指数比较随时间变化的值相对于基期的变化。简单价格指数 = (Pt / P0) × 100，其中 Pt 为当期价格，P0 为基期价格。
Weighted aggregate indexes reflect quantities. The Laspeyres index uses base period quantities (q0):
加权综合指数反映数量。拉氏指数使用基期数量 (q0)：
Laspeyres Price Index = (∑ptq0 / ∑p0q0) × 100
The Paasche index uses current period quantities (qt):
帕氏指数使用当期数量 (qt)：
Paasche Price Index = (∑ptqt / ∑p0qt) × 100
Index numbers can be used to calculate percentage change and to deflate monetary figures to show real change.
指数可用于计算百分比变化，并可对货币数值进行平减以显示实际变化。
11. Time Series and Moving Averages | 时间序列与移动平均
A time series consists of data recorded at successive time intervals. Its components are trend, seasonal variation, cyclical fluctuations, and random irregular movement.
时间序列由连续时间区间记录的数据组成。其构成部分包括趋势、季节变动、循环波动和随机不规则变动。
Moving averages smooth out short-term fluctuations to reveal the trend. For quarterly data, a four-point moving average is often used. The moving average is placed at the centre of the time points.
移动平均能平滑短期波动以揭示趋势。对于季度数据，常使用四项移动平均。移动平均放置于时间点的中心位置。
Four-point moving average = (y₁ + y₂ + y₃ + y₄) / 4
To centre the moving average when an even number of points is used, a further two-point moving average of the moving averages is taken.
若使用偶数个点的移动平均，为了居中，需对移动平均值再进行一次两项移动平均。
Seasonal variation can then be estimated by subtracting the trend from the original data (additive model) or dividing by the trend (multiplicative model).
之后可通过从原始数据中减去趋势（加法模型）或除以趋势（乘法模型）来估算季节变动。
12. Standardised Scores (z-scores) | 标准化得分（z分数）
A standardised score (z-score) indicates how many standard deviations a raw value lies above or below the mean. It is used to compare values from different distributions.
标准化得分（z 分数）表明原始值与均值相差多少个标准差。用于比较不同分布中的值。
z = (x – μ) / σ or z = (x – x̄) / s
If z is positive, the value is above the mean; if negative, below. A z-score of 0 means the value equals the mean.
若 z 为正，表明该值高于均值；若为负，则低于均值。z 分数为 0 表示该值等于均值。
When comparing test results, a higher z-score indicates a relatively better performance regardless of the original scales.
在比较考试成绩时，较高的 z 分数表示相对更好的表现，无论原始标尺如何。
Published by TutorHao | Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导