📚 Year 10 WJEC Psychology: Quick Reference Formula & Theorem Handbook | WJEC 心理学公式定理速查手册
This handbook summarises every formula, equation and statistical theorem you are expected to know for Year 10 WJEC Psychology. It covers descriptive statistics, inferential tests and the key decision rules that turn raw data into meaningful conclusions. Keep this guide handy for revision, homework and exam practice.
本手册总结了 Year 10 WJEC 心理学要求掌握的所有公式、方程和统计定理，涵盖描述统计、推断检验以及将原始数据转化为有意义结论的关键决策规则。复习、做作业和备考时随时查阅。
1. Measures of Central Tendency: Mean, Median, Mode | 集中趋势的度量：平均数、中位数、众数
The mean is the arithmetic average and is used with interval/ratio data that is normally distributed. It is the most sensitive measure because it takes every score into account.
平均数是算术平均值，适用于正态分布的等距/等比数据。它是最灵敏的度量，因为它考虑了每一个分数。
Mean (x̄) = Σx ÷ n
Σ (Sigma) means 'sum of', x represents each individual score, and n is the total number of scores.
Σ（西格玛）表示“求和”，x 代表每一个单独的分数，n 是分数的总个数。
The median is the middle value when scores are arranged in order. Use it when there are extreme outliers or with ordinal data.
中位数是将分数按顺序排列后位于中间的值。当存在极端异常值或处理顺序数据时使用中位数。
The mode is the most frequently occurring score. It is the only measure suitable for nominal data (categories).
众数是出现频率最高的分数。它是唯一适用于称名数据（类别）的度量。
2. Measures of Dispersion: Range | 离差的度量：全距
Range tells you how spread out the data are. It is quick to calculate but easily affected by a single extreme score.
全距告诉你数据的分散程度。计算快捷，但容易受单个极端值影响。
Range = Highest score – Lowest score
Always give the range as a single number, not as 'from X to Y'. For example, if the highest score is 18 and the lowest is 3, the range is 15.
全距始终以单个数字给出，而不是写成“从 X 到 Y”。例如，最高分是 18，最低分是 3，全距为 15。
3. Standard Deviation (SD) | 标准差
Standard deviation is a more precise measure of dispersion than the range. It shows the average distance of each score from the mean. The larger the SD, the greater the spread.
标准差是比全距更精确的离差度量。它显示每个分数与平均数的平均距离。标准差越大，数据越分散。
Sample SD (s) = √[ Σ(x – x̄)² ÷ (n – 1) ]
Steps: (1) Calculate the mean (x̄). (2) Subtract the mean from each score to get deviation (x – x̄). (3) Square each deviation. (4) Sum all squared deviations. (5) Divide by (n – 1). (6) Take the square root.
步骤：(1) 计算平均数 (x̄)。(2) 每个分数减去平均数得到离差 (x – x̄)。(3) 将每个离差平方。(4) 将所有平方离差求和。(5) 除以 (n – 1)。(6) 取平方根。
4. Normal Distribution and the Empirical Rule | 正态分布与经验法则
Many psychological variables are normally distributed: the data form a symmetrical, bell‑shaped curve where the mean, median and mode coincide at the centre.
许多心理学变量呈正态分布：数据形成对称的钟形曲线，平均数、中位数和众数在中心重合。
Empirical Rule: 68% – 95% – 99.7%
About 68% of scores fall within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. This allows you to estimate the rarity of a particular score.
大约 68% 的分数落在平均数 ±1 个标准差之内，95% 落在 ±2 个标准差之内，99.7% 落在 ±3 个标准差之内。这可以让你估计某个分数的罕见程度。
5. Standard Error of the Mean | 平均数的标准误
The standard error (SE) estimates how much the sample mean is likely to differ from the true population mean. A smaller SE indicates a more precise estimate.
标准误 (SE) 估计样本平均数与真实总体平均数可能相差多少。标准误越小，估计越精确。
SE = s ÷ √n
s is the sample standard deviation and n is the sample size. As sample size increases, SE decreases.
s 是样本标准差，n 是样本容量。样本容量越大，标准误越小。
6. Probability and Significance Levels (α) | 概率与显著性水平
In Psychology, we test hypotheses by calculating the probability (p) that the observed results occurred by chance. The significance level (α) is the cut‑off, usually set at 0.05 (5%).
在心理学中，我们通过计算观察到结果是由偶然因素造成的概率 (p) 来检验假设。显著性水平 (α) 是临界值，通常设定为 0.05 (5%)。
If p ≤ 0.05, result is significant → reject null hypothesis.
e I error = falsely rejecting a true null; Type II error = failing to reject a false null.
如果 p ≤ 0.05，结果显著 → 拒绝零假设。如果 p > 0.05，结果不显著 → 接受零假设。I 类错误 = 错误地拒绝了正确的零假设；II 类错误 = 未能拒绝错误的零假设。
7. Chi-Square Test (χ²) for Independence | 卡方独立性检验
Chi-square is a non‑parametric test used with nominal data in independent groups designs. It compares observed frequencies with expected frequencies.
卡方是一种非参数检验，用于独立组设计中的称名数据。它比较观察频数与期望频数。
χ² = Σ [ (O – E)² ÷ E ]
frequency, E = expected frequency = (row total × column total) ÷ grand total. Calculate (O – E for every cell and sum them to obtain χ². The degrees of freedom (df) = (number of rows – 1) × (number of columns – 1). Compare your χ² to the critical value table.
对于列联表中的每一个单元格：O = 观察频数，E = 期望频数 = （行合计 × 列合计）÷ 总计。对每个单元格计算 (O – E)² / E 并求和得到 χ²。自由度 (df) = （行数 – 1）×（列数 – 1）。将你的 χ² 值与临界值表进行对比。
8. Spearman’s Rank Correlation Coefficient (ρ) | 斯皮尔曼等级相关系数
Spearman’s rho measures the strength and direction of a relationship between two ordinal (ranked) variables. It is used with a correlational design.
斯皮尔曼等级相关系数衡量两个顺序（等级）变量之间关系的强度和方向。用于相关设计。
ρ = 1 – [ 6 Σd² ÷ (n(n² – 1)) ]
nsistent). (2) Find the difference (d) between ranks for each pair. (3) Square each d. (4) Sum d². (5) Apply formula. ρ ranges from – tive), with 0 meaning no correlation. Compare ρ with a critical values table for Spearman.
步骤：(1) 分别对两组分数进行排序（1 = 最高或最低，保持一致）。(2) 计算每对等级的差值 (d)。(3) 求每个 d 的平方。(4) 求和 Σd²。(5) 代入公式。ρ 取值范围为 –1（完全负相关）到 +1（完全正相关），0 表示无相关。将 ρ 值与斯皮尔曼临界值表对比。
9. 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. It tests for a difference between two independent groups when data are at least ordinal.
曼‑惠特尼 U 检验是独立样本 t 检验的非参数替代。当数据至少为顺序水平时，用于检验两个独立组之间的差异。
U₁ = n₁n₂ + n₁(n₁ + 1) ÷ 2 – R₁
U₂ = n₁n₂ + n₂(n₂ + 1) ÷ 2 – R₂
n₁ and n₂ are the sample sizes; R₁ and R₂ are the sum of ranks for each group. The smaller of U₁ and U₂ is the test statistic U. Compare U with the critical value table; a significant result means there is a difference between the groups.
n₁ 和 n₂ 是样本容量；R₁ 和 R₂ 是每组的秩和。取 U₁ 和 U₂ 中较小的一个作为检验统计量 U。将 U 值与临界值表对比；显著结果意味着两组之间存在差异。
10. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验
This test is used for a repeated measures or matched pairs design when data are at least ordinal. It compares two sets of scores from the same participants to see if there is a significant difference.
该检验用于重复测量或配对组设计，数据至少为顺序水平。它比较同一参与者的两组分数，判断是否存在显著差异。
T = Sum of ranks of the less frequent sign
allest). (4) Sum the ranks with a positive sign (+), and separately sum those with a negative sign (– ritical value table; T must be equal to or less than the critical value to be significant.
步骤：(1) 计算每对数据的差值。 (2) 忽略差值为零的对。 (3) 对差值的绝对值排秩（1 = 最小）。 (4) 分别对正号秩和负号秩求和。检验统计量 T 是较小的秩和。将 T 与临界值表对比；T 必须小于或等于临界值才显著。
11. Effect Size: Cohen’s d | 效应量：科恩的 d
Statistical significance tells you a result is unlikely to be due to chance, but it does not tell you how large the effect is. Cohen’s d gives a standardised measure of effect size.
统计显著性告诉你结果不太可能由偶发因素造成，但并未告知效应有多大。科恩的 d 提供了一个标准化的效应量度量。
d = (x̄₁ – x̄₂) ÷ spooled
x̄₁ and x̄₂ are the two group means. spooled ou may simply be asked to recognise that small d ≈ 0.2, medium d ≈ 0.5, and large d ≈ 0.8.
x̄₁ 和 x̄₂ 是两组平均数。spooled 是合并标准差（通常是两组标准差的平均值）。对于 WJEC 考试，你可能只需要知道 d ≈ 0.2 为小效应，d ≈ 0.5 为中等效应，d ≈ 0.8 为大效应。
12. Quick Decision Guide for Choosing a Statistical Test | 选择统计检验的速判指南
Always ask three questions: (1) What is the level of measurement? (2) What is the research design? (3) Is the data normally distributed? This leads you to the correct test.
始终问三个问题：(1) 测量水平是什么？(2) 研究设计是什么？(3) 数据是否正态分布？这能引导你选出正确的检验方法。
| Data type / Design | Test |
| Nominal, independent groups | Chi-square (χ²) |
| Ordinal, independent groups | Mann-Whitney U |
| Ordinal, repeated measures / matched pairs | Wilcoxon Signed-Rank |
| Ordinal or interval, correlation | Spearman’s Rank |
For interval/ratio data that meets parametric assumptions (normality, homogeneity of variance), parametric tests are preferred, but at Year 10 level the focus is on non‑parametric alternatives.
对于满足参数假设（正态性、方差齐性）的等距/等比数据，参数检验更优，但在 Year 10 阶段重点是非参数替代检验。
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