📚 Year 12 Edexcel Psychology: Quick Reference Handbook of Formulas & Theorems | Edexcel Year 12 心理学公式定理速查手册
Welcome to your one‑stop quick reference for the essential statistical formulas and decision rules required in the Year 12 Edexcel Psychology specification. This handbook covers every calculation you may be asked to perform in AS research methods, along with the inferential principles that underpin hypothesis testing. Use it alongside past papers and your class notes to build confidence with numerical data.
欢迎使用这本针对 Year 12 Edexcel 心理学课程打造的公式定理速查手册。手册涵盖了 AS 研究方法部分所有你可能需要亲自计算的公式，以及假设检验背后的推断原理。请结合历年真题和课堂笔记一起使用，帮助你从容应对数据分析类考题。
1. Measures of Central Tendency | 集中趋势量数
Three values summarise the centre of a data set. The mean (x̄) is the arithmetic average: x̄ = Σx / n, where Σx is the sum of all scores and n is the number of scores. The median is the middle value when data are ordered. The mode is the most frequently occurring score. In Edexcel AS, you must be able to select the most appropriate measure based on the level of measurement and the presence of outliers.
三个值能概括数据集的中心。均值（x̄）是算术平均数：x̄ = Σx / n，其中 Σx 表示所有数值之和，n 表示数据个数。中位数是将数据排序后位于中间位置的数值。众数是出现次数最多的数值。在 Edexcel AS 考试中，你需要根据测量层次和是否存在异常值，选择最合适的集中趋势指标。
2. Measures of Dispersion: Range and Interquartile Range | 离散量数：全距与四分位距
Range captures the spread from the smallest to the largest value: Range = maximum – minimum. For ordinal data or when outliers are present, the interquartile range (IQR) is preferred: IQR = Q3 – Q1, where Q1 is the 25th percentile and Q3 the 75th percentile. Both are simple to calculate and are often the first step in describing variability.
全距反映的是数据从最小值到最大值的跨度：全距 = 最大值 – 最小值。当处理顺序数据或存在异常值时，应优先使用四分位距（IQR）：IQR = Q3 – Q1，其中 Q1 为第 25 百分位数，Q3 为第 75 百分位数。这两种量数计算简便，通常是描述变异性的第一步。
3. Standard Deviation Formula | 标准差公式
Standard deviation tells you how much, on average, each score deviates from the mean. The sample standard deviation formula used in Edexcel AS is:
s = √[ Σ(x – x̄)² / (n – 1) ]
Here, x is each data point, x̄ the mean, and n the number of scores. Always work through the squared deviations, sum them, divide by n – 1, and take the square root. A larger s indicates greater spread around the mean.
标准差反映每个数据点与均值之间的平均偏差。Edexcel AS 阶段使用的样本标准差公式如下：
s = √[ Σ(x – x̄)² / (n – 1) ]
公式中，x 为各数据点，x̄ 为均值，n 为数据个数。计算步骤为：先求出每个数据与均值之差的平方，求和后除以 n – 1，最后取平方根。s 值越大，代表数据围绕均值的离散程度越高。
4. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Used to test for a relationship between two sets of ordinal data, or when the data are not normally distributed. The Spearman’s rho formula is:
rₛ = 1 – (6 ΣD²) / (n(n² – 1))
D is the difference between each pair of ranks, and n is the number of pairs. The result ranges from –1 (perfect negative correlation) to +1 (perfect positive correlation). Compare your observed rₛ with the critical value from the Spearman table at the desired significance level.
该检验用于考察两组顺序数据之间的相关性，或当数据不满足正态分布时使用。斯皮尔曼等级相关系数公式为：
rₛ = 1 – (6 ΣD²) / (n(n² – 1))
D 是每对等级之差，n 是配对数量。结果的范围从 –1（完全负相关）到 +1（完全正相关）。将计算得到的 rₛ 与斯皮尔曼检验表中指定显著性水平下的临界值进行比较。
5. Chi‑Squared Test of Association | 卡方独立性检验
Chi‑squared (χ²) determines whether two categorical variables are associated or independent. The formula is:
χ² = Σ (O – E)² / E
rand total. Sum the contributions from all cells. The degrees of freedom (df) = (number of rows – 1) × (number of columns – 1).
卡方（χ²）用于判断两个类别变量之间是否存在关联还是彼此独立。计算公式为：
χ² = Σ (O – E)² / E
O 代表列联表每个单元格中的观测频数，E 为期望频数，由（行合计 × 列合计）/ 总计得到。将所有单元格的贡献值相加。自由度 df = （行数 – 1）×（列数 – 1）。
6. Mann–Whitney U Test | 曼‑惠特尼 U 检验
This non‑parametric test compares two independent groups on an ordinal dependent variable. Compute the two U values:
U₁ = N₁N₂ + [N₁(N₁ + 1)]/2 – R₁
U₂ = N₁N₂ + [N₂(N₂ + 1)]/2 – R₂
N₁ and N₂ are the group sizes, and R₁ and R₂ are the sum of ranks for each group (rank all scores together, then sum ranks within each group). The observed U is the smaller of U₁ and U₂. Compare it against the Mann–Whitney critical value table.
这种非参数检验用于比较两个独立组在顺序因变量上的差异。计算两个 U 值：
U₁ = N₁N₂ + [N₁(N₁ + 1)]/2 – R₁
U₂ = N₁N₂ + [N₂(N₂ + 1)]/2 – R₂
N₁ 和 N₂ 为各组样本量，R₁ 和 R₂ 为各组秩和（将全部数据统一排序后，分别计算每组的秩总和）。取 U₁ 和 U₂ 中的较小值作为实测 U，再与曼‑惠特尼临界值表对比。
7. Wilcoxon Signed Ranks Test | 威尔科克森符号秩检验
Used for related‑measures designs (repeated measures or matched pairs) with ordinal data. Steps:
- Calculate the difference score for each pair.
- Rank the absolute differences (ignore zeros).
- Sum the ranks for positive differences (T+) and for negative differences (T–).
The test statistic W is the smaller of T+ and T–. Compare W to the critical value for the given number of non‑zero differences (N). H₀ is rejected if W ≤ critical value (for a two‑tailed test with α = 0.05, etc.).
该检验用于相关测量设计（重复测量或配对设计）中的顺序数据。步骤如下：
- 计算每对数据的差值。
- 对差值的绝对值进行排序（忽略差值为零的情况）。
- 分别计算正差值的秩和（T+）与负差值的秩和（T–）。
检验统计量 W 取 T+ 和 T– 中的较小值。将 W 与给定非零差值对数（N）的临界值比较。若 W ≤ 临界值（例如双尾 α = 0.05），则拒绝 H₀。
8. The Sign Test | 符号检验
The sign test is the simplest non‑parametric test for a difference in a related‑measures design with nominal or ordinal data. Count the number of participants showing a positive change and a negative change. The observed value S he critical value from the binomial distribution table for N (total number of pairs showing a difference) and specified significance level.
符号检验是相关测量设计中最简单的非参数检验，适用于称名或顺序数据。统计正/负变化方向的人数。实测值 S 为出现次数较少的符号的频数。例如，若 9 名被试表现改善，2 名变差，则 S = 2。将 S 与基于出现差异的总对数 N 和指定显著性水平的二项分布临界值表进行比较。
9. Probability, Significance Levels and p‑values | 概率、显著性水平与 p 值
In inferential testing, a p‑value is the probability of obtaining the observed result (or one more extreme) if the null hypothesis is true. The significance level (α) is the threshold at which we reject H₀ – usually 0.05 (5%). If your observed test statistic is equal to or more extreme than the critical value, the result is statistically significant. For Spearman and chi‑squared, the observed value must be greater than or equal to the critical value; for Mann‑Whitney, Wilcoxon and Sign tests, the observed value must be less than or equal to the critical value.
在推断性检验中，p 值是指在虚无假设为真的条件下，获得当前结果（或更极端结果）的概率。显著性水平（α）是我们拒绝 H₀ 的阈值，通常取 0.05（5%）。若实测统计量等于或超过临界值，则结果具有统计显著性。注意，对于斯皮尔曼相关与卡方检验，实测值必须大于或等于临界值；而对于曼‑惠特尼 U、威尔科克森和符号检验，实测值必须小于或等于临界值。
10. Decision Rules and Errors in Hypothesis Testing | 假设检验的决策规则与错误
Two types of error can occur. A Type I error is rejecting a true null hypothesis (a false positive); its probability equals α. A Type II error is failing to reject a false null hypothesis (a false negative). Psychologists typically use the 5% level to balance these risks. Remember the rule: if the calculated statistic exceeds the critical value (for tests where larger values are significant) or falls below the critical value (for tests where smaller values are significant), you can reject H₀ and accept the experimental hypothesis.
可能会出现两类错误。I 类错误是指错误地拒绝了真实的虚无假设（假阳性），其概率等于 α。II 类错误是指未能拒绝错误的虚无假设（假阴性）。心理学家通常采用 5% 的水平来平衡这两类风险。牢记判定规则：当统计量大于临界值（适用于较大值显著的检验）或小于临界值（适用于较小值显著的检验）时，就可以拒绝 H₀，接受实验假设。
Published by TutorHao | Psychology Revision Series | aleveler.com
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