📚 AS Edexcel Psychology: Formula & Theorem Quick Reference | AS Edexcel 心理学：公式定理速查手册
This quick reference handbook compiles the essential formulas, statistical test procedures, and decision rules required for the AS Edexcel Psychology specification. It covers descriptive statistics, inferential test calculations, and the logic behind choosing and interpreting tests, ensuring you have a compact, exam-focused revision tool.
本速查手册汇编了 AS Edexcel 心理学课程所要求的基本公式、统计检验步骤与决策规则。手册涵盖描述统计、推断检验计算以及选择和解释检验的逻辑，为你提供一份凝炼且紧扣考点的复习工具。
1. The Mean | 平均数
The mean is the arithmetic average of a set of scores. For raw data, sum all values (Σx) and divide by the number of scores (N). For a frequency distribution, multiply each score by its frequency (f), sum these products, and divide by the total frequency (Σf).
平均数是一组分数的算术平均值。对于原始数据，将所有数值求和（Σx）后除以分数的个数（N）。对于频数分布，将每个分数与其频数（f）相乘，将这些乘积求和，再除以总频数（Σf）。
x̄ = Σx / N or x̄ = Σ(fx) / Σf
2. The Median | 中位数
The median is the middle value in an ordered list of scores. Arrange scores from lowest to highest. If N is odd, the median is the score at position (N+1)/2. If N is even, the median is the midpoint between the two central scores.
中位数是一组有序分数中居于中间位置的数值。将分数从低到高排列，如果 N 为奇数，中位数就是位于第 (N+1)/2 个位置的分数；如果 N 为偶数，中位数则为最中间两个分数的平均值。
3. The Mode | 众数
The mode is the most frequently occurring score in a data set. A distribution can have one mode (unimodal), two modes (bimodal), or more. The mode is the only measure of central tendency suitable for nominal data.
众数是数据集中出现频次最高的分数。一个分布可以有一个众数（单峰）、两个众数（双峰）或更多。众数是唯一适用于称名数据的集中趋势量数。
4. The Range | 全距
The range is a simple measure of dispersion, calculated by subtracting the lowest score from the highest score. It gives a quick sense of spread but is easily influenced by extreme values.
全距是一种简单的离散程度指标，用最高分减去最低分计算得出。它能快速反映数据的散布情况，但容易受到极端值的影响。
Range = Highest Score − Lowest Score
5. Standard Deviation | 标准差
Standard deviation measures how much scores deviate from the mean on average. A small standard deviation indicates that scores cluster closely around the mean; a large one indicates greater spread. For a sample, the formula uses n − 1 as the denominator to provide an unbiased estimate of the population standard deviation.
标准差测量分数平均偏离均值的程度。标准差小说明分数紧密围绕均值分布；标准差大则说明分布更分散。对于样本数据，公式分母使用 n − 1，从而给出总体标准差的无偏估计。
s = √[ Σ (x − x̄)² / (n − 1) ]
Where: x = each individual score, x̄ = sample mean, n = number of scores, Σ = sum of.
其中：x = 每个个体的分数，x̄ = 样本均值，n = 分数个数，Σ = 求和。
6. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rho (ρ) measures the strength and direction of association between two ranked (ordinal) variables. It is appropriate for correlational designs and data that are at least ordinal. The coefficient ranges from −1 (perfect negative correlation) to +1 (perfect positive correlation), with 0 indicating no monotonic relationship.
斯皮尔曼等级相关系数 ρ 测量两个排序（等级）变量之间关系的强度和方向。它适用于相关设计和至少为顺序水平的数据。系数取值范围从 −1（完全负相关）到 +1（完全正相关），0 表示不存在单调关系。
ρ = 1 − [ 6 Σ d² / n (n² − 1) ]
Where d = difference between the ranks of each pair, n = number of paired observations.
其中 d = 每对数据的等级之差，n = 成对观测值的个数。
Step-by-step: (1) Rank each variable separately. (2) Calculate d (rank difference) for each pair. (3) Square each d to obtain d² and sum them. (4) Substitute into the formula. Compare the obtained ρ with the critical value from the Spearman table for the chosen significance level.
计算步骤：（1）分别对每个变量排序赋予等级。（2）计算每对数据的等级差 d。（3）将各 d 平方得到 d² 并求和。（4）代入公式。将求得的 ρ 与所选显著性水平下斯皮尔曼临界值表进行比较。
7. Wilcoxon Signed-Ranks Test | 威尔科克森符号秩检验
The Wilcoxon test is a non-parametric test for repeated measures or matched pairs designs, used with ordinal data. It examines whether the median difference between two conditions is zero. The test statistic is the smaller of the sums of the signed ranks.
威尔科克森检验是一种用于重复测量或配对设计的非参数检验，适用于顺序数据。它检验两个条件下的差值中位数是否为零。检验统计量是符号秩和中较小的那个。
Procedure: (1) Compute the difference between each pair of scores. (2) Ignore any zero differences. (3) Rank the absolute differences from smallest to largest. (4) Attach the original sign to each rank. (5) Sum the ranks with positive signs (T+) and the ranks with negative signs (T−). (6) The observed T is the smaller of T+ and T−. Compare T with the critical value: a significant result occurs when T is less than or equal to the table value (for a directional hypothesis).
步骤：（1）计算每对分数之差。（2）忽略所有零差异。（3）将差值的绝对值由小到大排序赋予秩次。（4）给每个秩次附上原差值的符号。（5）分别求正秩和（T+）与负秩和（T−）。（6）观测值 T 取 T+ 和 T− 中较小的那个。将 T 与临界值比较：当 T 小于或等于查表值（对于方向性假设）时，结果显著。
8. Mann-Whitney U Test | 曼-惠特尼 U 检验
The Mann-Whitney U test is a non-parametric alternative to the independent t-test, suitable for an independent groups design with ordinal data. It tests whether there is a difference between the central tendency of two independent groups.
曼-惠特尼 U 检验是独立样本 t 检验的非参数替代方法，适用于独立组设计和顺序数据。它检验两个独立组的集中趋势是否存在差异。
Procedure: (1) Combine scores from both groups and rank them from 1 for the smallest score to N for the largest. (2) Sum the ranks for group 1 (R₁) and group 2 (R₂). (3) Calculate U for each group using the formulas:
步骤：（1）将两组的分数合并，从最小值到最大值依次赋予秩次 1 至 N。（2）分别求出组 1 的秩和 R₁ 与组 2 的秩和 R₂。（3）用下列公式计算每个组的 U 值：
U₁ = R₁ − [ n₁ (n₁ + 1) / 2 ]
U₂ = R₂ − [ n₂ (n₂ + 1) / 2 ]
(4) The observed U is the smaller of U₁ and U₂. Compare this U value with the critical Mann-Whitney U table. The result is significant if U is less than or equal to the critical value (directional test).
（4）观测 U 值取 U₁ 和 U₂ 中较小的那个。将求得的 U 值与曼-惠特尼临界值表比较。若 U 小于或等于临界值（方向性检验），则结果显著。
9. Chi-Squared Test | 卡方检验
The chi-squared (χ²) test is used with nominal (categorical) data. In AS Psychology, it appears in two forms: the chi-squared test for independence (association) and the chi-squared goodness-of-fit test. Both compare observed frequencies (O) with expected frequencies (E).
卡方检验适用于称名（类别）数据。在 AS 心理学中，它有两种形式：卡方独立性（关联性）检验和卡方拟合优度检验。两者都通过比较观察频数（O）与期望频数（E）来进行判断。
χ² = Σ [ (O − E)² / E ]
For the test of independence: degrees of freedom df = (number of rows − 1) × (number of columns − 1). For a goodness-of-fit test: df = number of categories − 1. Compare the calculated χ² with the critical value from the chi-squared table; a significant result occurs when χ² is greater than or equal to the critical value.
独立性检验：自由度 df =（行数 − 1）×（列数 − 1）。拟合优度检验：df = 类别数 − 1。将计算得到的 χ² 值与卡方分布表中的临界值比较；当 χ² 大于或等于临界值时，结果显著。
10. Choosing a Statistical Test | 统计检验的选择
Selecting the correct test depends on three factors: the level of measurement (nominal, ordinal, interval/ratio), the research design (independent groups, repeated measures/matched pairs, or correlation), and whether the investigation tests a difference or an association.
选择正确的检验取决于三个因素：测量水平（称名、顺序、等距/等比）、研究设计（独立组、重复测量/配对，或相关设计）以及研究是在检验差异还是关联。
| Design | Nominal data | Ordinal data | Interval data |
|---|---|---|---|
| Independent groups | Chi-squared | Mann-Whitney U | Unrelated t‑test (mentioned conceptually) |
| Repeated measures / Matched pairs | Sign test (not always required in AS) | Wilcoxon | Related t‑test (conceptual) |
| Correlation | — | Spearman’s rank | Pearson’s r (conceptual) |
In AS Edexcel, the main focus is on non-parametric tests for ordinal data (Mann-Whitney, Wilcoxon, Spearman) and the chi‑squared test for nominal data. Parametric tests are typically covered only in the context of their assumptions.
在 AS Edexcel 课程中，重点在于适用于顺序数据的非参数检验（曼-惠特尼、威尔科克森、斯皮尔曼）以及适用于称名数据的卡方检验。参数检验通常仅涉及其前提假设。
11. Interpreting p-Values and Significance | p 值与显著性的解读
In psychology, the conventional significance level is p ≤ 0.05. If the observed test statistic is more extreme than the critical value, the null hypothesis can be rejected and the result is deemed statistically significant. A significant result does not prove the experimental hypothesis; it merely suggests that the observed pattern is unlikely to be due to chance alone.
心理学中，惯用的显著性水平为 p ≤ 0.05。如果观测统计量比临界值更极端，便可拒绝零假设，结果被认为具有统计显著性。显著结果并不证明实验假设，它仅说明观测到的模式不太可能仅由偶然因素造成。
A Type I error occurs when a researcher falsely rejects a true null hypothesis (a false positive). A Type II error occurs when a researcher fails to reject a false null hypothesis (a false negative).
Ⅰ 类错误指错误地拒绝了真实的零假设（假阳性）；Ⅱ 类错误指未能拒绝一个错误的零假设（假阴性）。
12. Key Notation Summary | 核心符号速览
Familiarity with statistical notation speeds up calculation and interpretation. Below are the most frequently used symbols in AS Edexcel Psychology research methods.
熟悉统计符号能加快计算和解读。以下为 AS Edexcel 心理学研究方法中最常用的符号。
- n — number of scores in a sample
- n — 样本中分数的个数
- x̄ — sample mean
- x̄ — 样本均值
- Σ — sum of
- Σ — 求和
- s — sample standard deviation
- s — 样本标准差
- d — difference between ranks or paired scores
- d — 秩次或配对分数之差
- ρ — Spearman’s rank correlation coefficient
- ρ — 斯皮尔曼等级相关系数
- T — Wilcoxon test statistic
- T — 威尔科克森检验统计量
- U — Mann-Whitney test statistic
- U — 曼-惠特尼检验统计量
- χ² — chi‑squared value
- χ² — 卡方值
- O — observed frequency
- O — 观察频数
- E — expected frequency
- E — 期望频数
- df — degrees of freedom
- df — 自由度
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