Year 13 WJEC Psychology: Formula & Theorem Quick Reference Handbook | Year 13 WJEC 心理学：公式定理速查手册

📚 Year 13 WJEC Psychology: Formula & Theorem Quick Reference Handbook | Year 13 WJEC 心理学：公式定理速查手册

This quick reference handbook brings together every essential formula, statistical test, and conceptual theorem required in Year 13 WJEC Psychology. Whether you are analysing data for Component 2 or revising key inferential tests, having a concise summary of central-tendency measures, dispersion, significance testing, effect size, and error types will save you time and boost your confidence. Master these tools and you will be ready to handle any research-methods question with clarity.

这本速查手册汇集了 Year 13 WJEC 心理学中所有必备的公式、统计检验和概念定理。无论你是在为 Component 2 分析数据，还是复习关键推断性检验，拥有一份关于集中趋势、离散程度、显著性检验、效应量和错误类型的简明总结，都能帮你节省时间、增强信心。掌握这些工具，你就能从容应对任何研究方法类题目。

1. Descriptive Statistics: Central Tendency | 描述统计：集中趋势

The three measures of central tendency summarise a data set with a single representative value. The mean (M) is the arithmetic average, calculated as M = (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; for an even number of scores, take the average of the two middle numbers. The mode is the most frequently occurring value – a dataset can have more than one mode or none at all. The mean is sensitive to outliers, whereas the median is robust against extreme scores, so selecting the right measure depends on the shape of the distribution.

三种集中趋势量数用一个代表值概括数据集。均值（M）是算术平均数，计算公式为 M = (Σx)/n，其中 Σx 为所有分数之和，n 为分数个数。中位数是排序后居中的数值；若个数为偶数则取中间两数的平均值。众数是出现频率最高的值——数据集可以有多个众数，也可以没有众数。均值易受极端值影响，而中位数对异常值具有稳健性，因此选择何种量数取决于分布形态。

Mean formula: M = Σx / n
均值公式：M = Σx / n

2. Descriptive Statistics: Dispersion | 描述统计：离散程度

Dispersion tells us how spread out the scores are. The range is simply the highest score minus the lowest score, but it can be distorted by a single extreme value. Variance measures the average squared deviation from the mean. For a sample, variance s² = Σ(x − M)² / (n − 1). The standard deviation (SD) is the square root of variance: s = √[ Σ(x − M)² / (n − 1) ]. A larger SD indicates greater spread. SD is expressed in the original units, making it more interpretable than variance.

离散程度反映分数的分散情况。全距即最高分减最低分，但易受单一极端值扭曲。方差衡量的是与均值离差平方的平均数。对于样本，方差 s² = Σ(x − M)² / (n − 1)。标准差（SD）是方差的平方根：s = √[ Σ(x − M)² / (n − 1) ]。标准差越大，数据越分散。SD 以原始单位表示，因此比方差更易解释。

Variance (sample): s² = Σ(x − M)² / (n − 1)
方差（样本）：s² = Σ(x − M)² / (n − 1)
Standard deviation: s = √s²
标准差：s = √s²

3. The Normal Distribution & Standard Scores | 正态分布与标准分数

Many psychological variables are normally distributed: the curve is bell-shaped and symmetrical, with most scores clustering around the mean and the frequency dropping off towards the tails. In a normal distribution, approximately 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. A z-score transforms a raw score into the number of standard deviations it lies from the mean: z = (X − μ) / σ (population) or z = (X − M) / s (sample). Z-scores allow comparison across different distributions.

许多心理变量呈正态分布：曲线呈钟形且对称，大多数分数围绕均值集中，尾部频率逐渐降低。在正态分布中，约 68% 的数据落在均值 ±1 SD 内，95% 落在 ±2 SD 内，99.7% 落在 ±3 SD 内。z 分数把原始分数转换为距离均值若干标准差的值：z = (X − μ) / σ（总体）或 z = (X − M) / s（样本）。z 分数可用于不同分布之间的比较。

z = (X − μ) / σ

4. Standard Error & Confidence Intervals | 标准误与置信区间

The standard error of the mean (SEM) estimates how much a sample mean is likely to differ from the population mean. SEM = s / √n, where s is the sample SD and n is the sample size. Larger samples produce smaller standard errors, giving a more precise estimate. A confidence interval (CI) is built around the sample mean: for a 95% CI, the interval is M ± (1.96 × SEM) if the population SD is known and normality can be assumed. A 95% CI means that if we repeated sampling many times, 95% of the intervals would contain the true population mean.

标准误（SEM）估计样本均值与总体均值之间可能存在的差异。SEM = s / √n，其中 s 为样本标准差，n 为样本量。样本越大，标准误越小，估计越精确。置信区间（CI）围绕样本均值构建：当总体标准差已知且可假定正态分布时，95% CI 为 M ± (1.96 × SEM)。95% CI 的含义是：如果重复抽样很多次，95% 的区间会包含真实的总体均值。

SEM = s / √n | 95% CI = M ± 1.96 × SEM

5. Chi-Square Test of Association | 卡方关联性检验

The chi-square test (χ²) determines whether there is a significant association between two categorical variables. It compares observed frequencies (O) in each cell of a contingency table with the frequencies expected (E) if the null hypothesis were true. The test statistic is χ² = Σ [(O − E)² / E]. Degrees of freedom (df) = (number of rows − 1) × (number of columns − 1). The calculated χ² is compared against a critical value at the chosen significance level (usually p < 0.05). If χ² exceeds the critical value, the result is significant, suggesting an association exists.

卡方检验（χ²）检验两个类别变量之间是否存在显著关联。它比较列联表中每个单元格的观察频数（O）与在零假设成立时的期望频数（E）。检验统计量为 χ² = Σ [(O − E)² / E]。自由度 df =（行数 − 1）×（列数 − 1）。将计算得到的 χ² 值与选定显著性水平（通常 p < 0.05）下的临界值比较。若 χ² 大于临界值，则结果显著，提示存在关联。

χ² = Σ [(O − E)² / E]

Example: In a study of 100 participants, 30 of 50 males and 20 of 50 females reported anxiety. Calculate expected frequencies (e.g., male-anxious: 25) and apply the formula.

示例：在一项 100 人的研究中，50 名男性中有 30 人报告焦虑，50 名女性中有 20 人。计算期望频数（如男性焦虑：25）后代入公式即可。

6. Mann-Whitney U Test | 曼-惠特尼U检验

The Mann-Whitney U test is a non-parametric test for an independent-groups design when data are ordinal (or interval but not normally distributed). It ranks all scores together, then sums the ranks for each group (R₁ and R₂). The U statistic for group 1 is U₁ = n₁n₂ + [n₁(n₁+1)]/2 − R₁. U₂ is calculated similarly. The smaller of U₁ and U₂ is compared with a critical U-value (from tables). For significance, the obtained U must be less than or equal to the critical value.

曼-惠特尼U检验是一种用于独立组设计的非参数检验，适用于顺序数据（或等距数据但非正态分布）。它将所有分数统一排序，然后计算每组秩和（R₁ 与 R₂）。第一组的U统计量为 U₁ = n₁n₂ + [n₁(n₁+1)]/2 − R₁。同理可得 U₂。取两者中较小的 U 值与临界 U 值（查表）进行比较。若要达到显著性，所得 U 值必须小于或等于临界值。

U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁

7. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验

For repeated-measures or matched-pairs designs, the Wilcoxon signed-rank test uses ordinal data. Compute the difference score (D) for each pair, ignore zeros, then rank the absolute values of D, assigning the sign (+ or −) back to each rank. Sum the positive ranks (T₊) and negative ranks (T₋). The test statistic T is the smaller of these two sums. Compare T against the critical T-value; the obtained T must be less than or equal to the critical value for significance.

对于重复测量或配对设计，威尔科克森符号秩检验使用顺序数据。为每一对数据计算差值 D，忽略零差值，然后对 |D| 进行排序，并将正负号重新赋给各个秩。分别计算正秩和（T₊）与负秩和（T₋），检验统计量 T 取其中较小者。将 T 与临界 T 值比较；达到显著性要求 T 小于或等于临界值。

T = the smaller of ΣR₊ and ΣR₋

8. Spearman’s Rank-Order Correlation | 斯皮尔曼等级相关

Spearman’s r_s (or rho) measures the strength and direction of a monotonic relationship between two ordinal variables. Rank each variable separately, then for each pair calculate d (difference between ranks). The formula is r_s = 1 − [6 Σ d²] / [n(n² − 1)], where n is the number of pairs. The coefficient ranges from −1 (perfect negative correlation) to +1 (perfect positive correlation). A value close to 0 indicates no monotonic association. Compare r_s against a critical value; if your r_s exceeds the critical value, the correlation is significant.

斯皮尔曼等级相关系数 r_s（或 ρ）衡量两个顺序变量之间单调关系的强度与方向。分别对每个变量排序，然后为每对数据计算秩差 d。公式为 r_s = 1 − [6 Σ d²] / [n(n² − 1)]，其中 n 为数据对数。系数取值范围从 −1（完全负相关）到 +1（完全正相关）。接近 0 表示无单调关系。将 r_s 与临界值比较；若所得 r_s 大于临界值，则相关性显著。

r_s = 1 − 6 Σ d² / [n(n² − 1)]

9. Pearson’s Product-Moment Correlation | 皮尔逊积矩相关

When data are interval/ratio and normally distributed, Pearson’s r quantifies a linear relationship. The calculation uses deviation scores: r = Σ (x − M_x)(y − M_y) / √[ Σ (x − M_x)² Σ (y − M_y)² ]. An equivalent computational formula avoids the separate deviation steps. Like Spearman’s r_s, r ranges from −1 to +1. The coefficient of determination, r², indicates the proportion of shared variance. For significance, compare r with a critical r-value at df = n − 2.

当数据为等距/等比且正态分布时，皮尔逊 r 用于量化线性关系。计算采用离差分数：r = Σ (x − M_x)(y − M_y) / √[ Σ (x − M_x)² Σ (y − M_y)² ]。等效的计算公式可省去逐步求离差的过程。与斯皮尔曼 r_s 一样，r 取值范围为 −1 至 +1。决定系数 r² 表示共享变异的比例。显著性检验时，将 r 与自由度 df = n − 2 下的临界 r 值进行比较。

r = Σ (x − M_x)(y − M_y) / √[ Σ (x − M_x)² Σ (y − M_y)² ]

10. Student’s t-Test (Independent & Paired) | t检验（独立样本与配对样本）

Independent t-test compares the means of two unrelated groups when data are interval and normally distributed. The formula is t = (M₁ − M₂) / SED, where SED (standard error of the difference) uses a pooled variance estimate: SED = √[ s_p² (1/n₁ + 1/n₂) ] and s_p² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁+n₂−2). The paired t-test uses the mean difference (D̄) and standard deviation of differences (s_D): t = D̄ / (s_D/√n). In both cases, compare the obtained t with a critical t-value using the appropriate degrees of freedom.

独立样本 t 检验比较两组无关联的等距数据均值，要求正态分布。公式为 t = (M₁ − M₂) / SED，其中 SED（差异标准误）使用合并方差估计：SED = √[ s_p² (1/n₁ + 1/n₂) ]，而 s_p² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁+n₂−2)。配对 t 检验则使用差值的均值（D̄）和差值标准差（s_D）：t = D̄ / (s_D/√n)。两种情况下均需将所得 t 值与相应自由度下的临界 t 值比较。

Independent: t = (M₁ − M₂) / √[ s_p²(1/n₁ + 1/n₂) ]

Paired: t = D̄ / (s_D/√n)

11. Effect Size & Cohen’s d | 效应量与Cohen’s d

Statistical significance does not tell us about the magnitude of a difference. Cohen’s d is an effect-size measure for independent-group means: d = (M₁ − M₂) / SD_pooled, where SD_pooled = √[ (s₁² + s₂²) / 2 ] for equal sample sizes. Conventionally, d = 0.2 is small, 0.5 medium, and 0.8 large. For correlational studies, effect size is simply r or r_s; 0.1 small, 0.3 medium, 0.5 large. Reporting effect size is essential for evaluating practical importance and is a WJEC requirement.

统计显著性并不能告诉我们差异的大小。Cohen’s d 是独立组均值比较的效应量指标：d = (M₁ − M₂) / SD_pooled，当样本量相等时 SD_pooled = √[ (s₁² + s₂²) / 2 ]。惯例上，d = 0.2 为小效应，0.5 为中等，0.8 为大效应。对于相关研究，效应量即 r 或 r_s 本身；0.1 为小，0.3 为中等，0.5 为大。报告效应量对于评价实际重要性至关重要，也是 WJEC 考试要求。

d = (M₁ − M₂) / SD_pooled

12. Errors & Statistical Power | 错误与统计效力

In null-hypothesis significance testing, two types of errors can occur. A Type I error (false positive) happens when we reject a true null hypothesis; the probability is set by the significance level α (usually 0.05). A Type II error (false negative) occurs when we fail to reject a false null hypothesis; its probability is β. Power (1 − β) is the probability of correctly rejecting a false null. Power increases with larger sample sizes, larger effect sizes, and higher α. In WJEC questions you may be asked to identify error types or to explain how to increase power.

在零假设显著性检验中，可能发生两种错误。第一类错误（假阳性）指拒绝了正确的零假设；其概率由显著性水平 α（通常 0.05）设定。第二类错误（假阴性）指未能拒绝错误的零假设；其概率为 β。统计效力（1 − β）是正确拒绝错误零假设的概率。增大样本量、增大效应量或提高 α 值均能提高效力。WJEC 考题可能要求识别错误类型或解释如何提高效力。

Decision	Null True	Null False
Reject H₀	Type I Error (α)	Correct Decision (Power = 1−β)
Do not reject H₀	Correct Decision	Type II Error (β)

决策	零假设为真	零假设为假
拒绝 H₀	第一类错误（α）	正确决策（效力 = 1−β）
不拒绝 H₀	正确决策	第二类错误（β）

Published by TutorHao | Psychology Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Year 13 WJEC Psychology: Formula & Theorem Quick Reference Handbook | Year 13 WJEC 心理学：公式定理速查手册