Year 11 CAIE Psychology: Formula & Theorem Rapid Revision Guide | CAIE Year 11 心理学：公式定理速查手册

📚 Year 11 CAIE Psychology: Formula & Theorem Rapid Revision Guide | CAIE Year 11 心理学：公式定理速查手册

This rapid revision guide covers all essential statistical formulas, research methods theorems, and key psychological laws required for the CAIE IGCSE Psychology (0457) syllabus. It is designed to help you check definitions, apply calculations, and interpret findings accurately in the exam. Use it alongside past papers for maximum benefit.

这本速查手册覆盖了 CAIE IGCSE 心理学（0457）大纲要求的所有核心统计公式、研究方法定理和关键心理学定律。它能帮助你在考试中准确核对定义、运用计算并解释研究结果。建议结合历年真题使用，效果更佳。

---

1. Measures of Central Tendency | 集中趋势测量

Central tendency refers to the typical score around which data cluster. The three main measures are mean, median, and mode.

集中趋势是指数据聚集的中心值。三个主要测量指标是平均数、中位数和众数。

Mean: The arithmetic average, sensitive to extreme scores. Used with interval or ratio data.

平均数：算术平均值，易受极端值影响。适用于间隔或比率数据。

Mean (x̄) = Σx / n

where Σx is the sum of all scores and n is the number of scores.

其中 Σx 为所有分数的总和，n 为分数的个数。

Median: The middle value when scores are ordered from smallest to largest. If n is even, it is the average of the two middle scores. Not affected by outliers.

中位数：将分数从小到大排列后的中间值。如果 n 为偶数，则取中间两个值的平均数。不受极端值影响。

Mode: The most frequently occurring score. A data set may be unimodal, bimodal, or multimodal. Used with nominal data.

众数：出现频率最高的分数。数据集可以是单峰、双峰或多峰的。适用于称名数据。

---

2. Measures of Dispersion | 离散程度测量

Dispersion shows how spread out the scores are. The main measures are range and standard deviation.

离散程度反映数据的分散情况。主要测量指标包括全距和标准差。

Range: The difference between the highest and lowest scores. Quick to calculate but highly influenced by outliers.

全距：最高分与最低分之间的差值。计算简便，但极易受极端值影响。

Range = Highest score − Lowest score

Standard Deviation (s): A more precise measure of dispersion that shows the average distance of each score from the mean. A small s means scores are tightly clustered around the mean.

标准差（s）：一种更精确的离散量数，表示各分数与平均数的平均距离。标准差小意味着分数紧密聚集在平均数周围。

s = √[ Σ(x − x̄)² / (n − 1) ]

This is the formula for the sample standard deviation, where x̄ is the mean and n is the number of scores.

这是样本标准差的计算公式，x̄ 为平均数，n 为分数的个数。

---

3. 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 the relationship between two sets of ordinal data or ranked interval data. It is a non-parametric test.

斯皮尔曼等级相关系数（ρ）用于衡量两组顺序数据或排序后的间隔数据之间的关联强度与方向。它是一种非参数检验。

ρ = 1 − (6 ΣD²) / [n (n² − 1)]

where D is the difference between the ranks of each pair, and n is the number of pairs.

其中 D 是每对数据的秩次差，n 是数据对的数量。

Steps: 1. Rank the scores for each variable separately. 2. Calculate D (difference between the two ranks for each participant). 3. Square D and sum to obtain ΣD². 4. Substitute into the formula. 5. Compare the calculated ρ to the critical value from the Spearman table (for a given n and significance level, typically p≤0.05). If ρ is equal to or greater than the critical value, the correlation is significant.

步骤：1. 分别对每个变量进行排序。2. 计算每名参与者两个秩次的差值 D。3. 求 D 的平方，并计算总和 ΣD²。4. 代入公式。5. 将计算得到的 ρ 值与斯皮尔曼临界值表（给定 n 和显著性水平，通常 p≤0.05）进行比较。若 ρ 大于或等于临界值，则相关显著。

n	Critical value (two-tailed, p≤0.05)
5	1.000
6	0.886
7	0.786
8	0.738
9	0.700
10	0.648

The null hypothesis states there is no correlation in the population. A significant ρ allows you to reject the null hypothesis.

零假设认为总体中不存在相关。若 ρ 显著，则可拒绝零假设。

---

4. Wilcoxon Signed-Ranks Test | 威尔科克森符号秩检验

The Wilcoxon test is a non-parametric test for a repeated measures or matched pairs design. It uses ordinal data to decide if there is a significant difference between two conditions.

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。它利用顺序数据判断两种条件之间是否存在显著差异。

Steps: 1. Calculate the difference between each pair of scores. 2. Ignoring the sign, rank the absolute differences (smallest difference = rank 1). 3. Assign the sign (+ or −) of the difference to each rank. 4. Sum the positive ranks (T+) and the negative ranks (T−). 5. The test statistic T is the smaller of T+ and T−. 6. Compare T to the critical value from the Wilcoxon table. If T is less than or equal to the critical value, the difference is significant (reject the null hypothesis).

步骤：1. 计算每对分数的差值。2. 忽略符号，将差值的绝对值排序（最小差值秩次为1）。3. 将差值的正负号分配给相应的秩次。4. 分别计算正秩次总和（T+）与负秩次总和（T−）。5. 检验统计量 T 取 T+ 与 T− 中较小的一个。6. 将 T 与威尔科克森临界值表进行比较。若 T 小于或等于临界值，则差异显著（拒绝零假设）。

T = smaller of T⁺ and T⁻

Any pairs with a difference of zero are excluded from the analysis and the sample size n is reduced accordingly.

差值为零的数据对会被剔除，样本量 n 相应减少。

---

5. Mann-Whitney U Test | 曼-惠特尼 U 检验

The Mann-Whitney U test is the non-parametric alternative to the independent t-test. It is used with an independent groups design and ordinal data to test for a significant difference between two groups.

曼-惠特尼 U 检验是独立样本 t 检验的非参数替代。适用于独立组设计和顺序数据，检验两个组别之间是否存在显著差异。

Steps: 1. Combine all scores from both groups and rank them from 1 (lowest) to N (highest). 2. Sum the ranks for group 1 (R₁) and group 2 (R₂). 3. Calculate U₁ and U₂ using:

步骤：1. 将两组的分数合并，从1（最低）到 N（最高）排序。2. 分别计算组1的秩和 R₁ 与组2的秩和 R₂。3. 按以下公式计算 U₁ 和 U₂：

U₁ = n₁n₂ + [n₁(n₁+1)/2] − R₁

U₂ = n₁n₂ − U₁

where n₁ and n₂ are the sample sizes of the two groups.

其中 n₁ 和 n₂ 分别为两组的样本量。

4. The test statistic U is the smaller of U₁ and U₂. 5. Compare the smaller U with the critical value from the Mann-Whitney table. If U is less than or equal to the critical value, reject the null hypothesis; the difference is significant.

4. 检验统计量 U 取 U₁ 与 U₂ 中较小的值。5. 将较小的 U 值与曼-惠特尼临界值表进行比较。若 U 小于或等于临界值，则拒绝零假设，差异显著。

---

6. Chi-Squared Test for Independence | 卡方独立性检验

The chi-squared (χ²) test is used with nominal (categorical) data in an independent groups design to see if there is a significant association between two variables.

卡方（χ²）检验用于独立组设计中的称名（类别）数据，以检验两个变量之间是否存在显著关联。

χ² = Σ [ (O − E)² / E ]

where O = observed frequency, E = expected frequency. Expected frequency for each cell = (row total × column total) / grand total.

其中 O 为观察频数，E 为期望频数。每个单元格的期望频数 =（行合计 × 列合计）/ 总计。

Steps: 1. Construct a contingency table and fill in observed frequencies. 2. Calculate expected frequencies for each cell. 3. Apply the formula to obtain χ². 4. Determine degrees of freedom: df = (number of rows − 1) × (number of columns − 1). 5. Compare calculated χ² with the critical value from the chi-squared table. If χ² is greater than or equal to the critical value, the result is significant (p≤0.05) and the variables are associated.

步骤：1. 构建列联表并填入观察频数。2. 计算每个单元格的期望频数。3. 代入公式计算 χ²。4. 确定自由度：df =（行数 − 1）×（列数 − 1）。5. 将计算出的 χ² 值与卡方临界值表比较。若 χ² 大于或等于临界值，则结果显著（p≤0.05），变量之间存在关联。

At IGCSE level, a 2×2 table is frequently used, so df = 1. The null hypothesis states there is no association between the variables.

IGCSE 阶段常用 2×2 表格，此时自由度为 1。零假设认为变量之间没有关联。

---

7. Sign Test | 符号检验

The sign test is the simplest non-parametric test for a repeated measures design. It uses nominal data (the direction of change: + or −) to see if there is a significant difference between two conditions.

符号检验是最简单的非参数检验，用于重复测量设计。它使用称名数据（变化的方向：+ 或 −）来检验两种条件之间是否存在显著差异。

Steps: 1. For each participant, record whether condition B shows an increase (+) or decrease (−) compared to condition A. Ignore ties. 2. Count the number of pluses and minuses. 3. The test statistic S is the frequency of the less frequent sign. 4. Compare S with the critical value from the sign test table. If S is less than or equal to the critical value, the result is significant. The null hypothesis (no difference) is rejected.

步骤：1. 对每位参与者，记录条件 B 相对于条件 A 是增加（+）还是减少（−）。平局不纳入计算。2. 分别统计正号和负号的个数。3. 检验统计量 S 取出现次数较少的那个符号的频数。4. 将 S 与符号检验临界值表比较。若 S 小于或等于临界值，则结果显著，拒绝无差异的零假设。

This test is very quick but loses a lot of information because it only uses the direction of difference, not its size.

此检验非常快捷，但由于只利用了差异的方向而忽略了差异的大小，会丢失大量信息。

---

8. Levels of Measurement and Data Types | 测量水平与数据类型

Understanding levels of measurement helps you choose the correct descriptive statistics and inferential test. The four levels are:

理解测量水平有助于选择合适的描述统计量和推断检验。共有四个水平：

Nominal: data are categories with no order (e.g., eye colour, yes/no). Mode is the only appropriate measure of central tendency. Chi-squared and sign test are used.

称名数据：无顺序的类别数据（例如眼睛颜色、是/否）。众数是唯一合适的集中趋势量数。适用于卡方检验和符号检验。

Ordinal: data are ordered but intervals between ranks are not equal (e.g., rating scales, class positions). Use median and mode. Spearman, Wilcoxon, Mann-Whitney are appropriate.

顺序数据：有顺序但秩次之间的间隔不相等（例如评分量表、班级名次）。使用中位数和众数。适用于斯皮尔曼、威尔科克森和曼-惠特尼检验。

Interval: equal intervals but no true zero (e.g., temperature in Celsius, IQ). Mean, median, mode can be used. Parametric tests (t-test, Pearson) may be used if data are normally distributed.

间隔数据：有相等间隔但无绝对零点（例如摄氏温度、智商）。可使用平均数、中位数和众数。若数据正态分布，可用参数检验（t 检验、皮尔逊相关）。

Ratio: equal intervals and a true zero (e.g., reaction time, height, weight). All measures available, and parametric tests can be applied if assumptions are met.

比率数据：有相等间隔且有绝对零点（例如反应

Published by TutorHao | Year 11 Psychology Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）