📚 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 is designed for Year 13 WJEC Psychology students who need a concise, at-a-glance summary of the key statistical formulas and conceptual ‘theorems’ (decision rules) encountered in research methods and inferential testing. From measures of central tendency to choosing the correct non-parametric test, this guide consolidates everything you need for quick revision of the mathematical backbone of Component 2.
本速查手册专为 Year 13 WJEC 心理学学生编写，简洁汇总了研究方法和推断性检验中可能遇到的关键统计公式和概念性“定理”（决策规则）。从集中趋势量数到正确选择非参数检验，本手册整合了 Component 2 数学核心所需的一切，便于快速复习。
1. Descriptive Statistics: Mean, Median, Mode | 描述统计：均值、中位数、众数
The mean is the arithmetic average of all scores and is used with interval/ratio data that are normally distributed. It is calculated by summing all values and dividing by the number of observations.
均值是所有分数的算术平均值，适用于正态分布的等距/比率数据。其计算方法是将所有数值相加再除以观测值的数量。
Mean (x̄) = Σx / n
The median is the middle score when data are arranged in order; it is less affected by outliers and is used with ordinal data or skewed distributions. The mode is the most frequently occurring score, useful for nominal data.
中位数是将数据按顺序排列后位于中间的数值；它受异常值影响较小，适用于顺序数据或偏态分布。众数则是出现频率最高的分数，适用于名义数据。
2. Measures of Dispersion: Variance & Standard Deviation | 离散量数：方差与标准差
Variance describes how much scores differ from the mean. Standard deviation is the square root of variance and indicates the average distance of a score from the mean in the original units. A larger SD indicates greater spread.
方差描述各个分数与均值的差异程度。标准差是方差的平方根，表示分数与均值的平均距离（以原始单位计）。标准差越大，离散程度越高。
Variance (s²) = Σ(x − x̄)² / (n − 1)
Standard Deviation (s) = √[ Σ(x − x̄)² / (n − 1) ]
For the WJEC exam, you may be asked to calculate variance or interpret standard deviation from given raw data, always using the ‘n − 1’ formula when estimating population parameters from a sample.
在 WJEC 考试中，你可能会被要求根据给定原始数据计算方差或解释标准差。当从样本估计总体参数时，务必使用“n − 1”公式。
3. Normal Distribution & Z-Scores | 正态分布与Z分数
A normal distribution is a bell-shaped curve where the mean, median and mode coincide. In psychology, many parametric tests assume the data are normally distributed. The proportion of scores within ±1 SD is about 68%, within ±2 SD about 95%.
正态分布是一种钟形曲线，均值、中位数和众数重合。在心理学中，许多参数检验都假定数据呈正态分布。约68%的分数落在均值±1个标准差内，约95%落在±2个标准差内。
A z-score expresses how many standard deviations a raw score is from the mean. It allows comparison across different distributions.
Z分数表示一个原始分数距离均值有多少个标准差，可用于在不同分布之间进行比较。
z = (x − x̄) / s
In WJEC Component 2, z-scores are linked to the standard normal distribution and probability; for example, an absolute z-score greater than 1.96 corresponds to a two-tailed significance level of p < 0.05.
在 WJEC Component 2 中，Z分数与标准正态分布和概率相关联；例如，绝对值大于1.96的Z分数对应于双尾显著性水平 p < 0.05。
4. Hypothesis Testing & Significance | 假设检验与显著性水平
Hypothesis testing is a decision rule used to determine whether an observed effect is statistically significant. The null hypothesis (H₀) predicts no effect or no difference; the alternative hypothesis (H₁) predicts an effect or a difference.
假设检验是一种决策规则，用于确定观察到的效应是否具有统计显著性。零假设（H₀）预测没有效应或差异；备择假设（H₁）预测有效应或差异。
The significance level (α, alpha) is the probability of rejecting a true null hypothesis. In psychology, α is conventionally set at 0.05 (5%) or 0.01 (1%). If the calculated p-value is less than or equal to α, the result is significant and H₀ is rejected.
显著性水平（α，阿尔法）是拒绝一个真实的零假设的概率。心理学中通常将α设为0.05（5%）或0.01（1%）。如果计算出的p值小于或等于α，结果即显著，拒绝H₀。
In WJEC, you must state whether to use a one-tailed (directional) or two-tailed (non-directional) test, which affects the critical values used for comparison with the calculated test statistic.
在 WJEC 中，你必须说明使用单尾（方向性）还是双尾（非方向性）检验，这将影响用于比较计算出的检验统计量的临界值。
5. Type I and Type II Errors | I型与II型错误
A Type I error occurs when a researcher rejects a true null hypothesis – a ‘false positive’. The probability of making a Type I error is equal to the alpha level (α).
当研究者拒绝真实的零假设时，会发生I型错误——即“假阳性”。犯I型错误的概率等于阿尔法水平 (α)。
A Type II error occurs when a false null hypothesis is not rejected – a ‘false negative’. The probability of a Type II error is symbolised as β (beta). Power is the probability of correctly rejecting a false H₀, calculated as 1 − β.
当错误的零假设未被拒绝时，会发生II型错误——即“假阴性”。II型错误的概率用β（贝塔）表示。统计功效是正确拒绝一个错误H₀的概率，计算公式为 1 − β。
- Increasing sample size reduces the risk of both Type I and Type II errors.
- 增加样本量可以同时降低I型和II型错误的风险。
- A more stringent alpha (e.g. 0.01) reduces Type I errors but increases the chance of Type II errors.
- 更严格的阿尔法水平（如0.01）会减少I型错误，但会增加II型错误的几率。
6. Parametric Tests: Pearson’s r Correlation | 参数检验：皮尔逊积差相关
Pearson’s product-moment correlation coefficient (r) measures the strength and direction of a linear relationship between two interval/ratio variables. Assumptions include normally distributed data, homoscedasticity and a linear relationship.
皮尔逊积差相关系数 (r) 衡量两个等距/比率变量之间线性关系的强度和方向。假设包括数据正态分布、方差齐性和线性关系。
r = Σ[(x − x̄)(y − ȳ)] / √[ Σ(x − x̄)² × Σ(y − ȳ)² ]
The value of r ranges from −1 (perfect negative correlation) to +1 (perfect positive correlation); an r of 0 indicates no linear relationship. You must compare the obtained r against a critical value table with degrees of freedom df = N − 2.
r值的范围从−1（完全负相关）到+1（完全正相关）；r为0表示没有线性关系。你需要将计算出的r值与自由度为 df = N − 2 的临界值表进行对比。
For WJEC, be prepared to interpret correlation coefficients in context and recognise that correlation does not imply causation.
对于WJEC，请准备好结合具体情境解释相关系数，并明确相关关系不代表因果关系。
7. Parametric Tests: Related & Independent t-tests | 参数检验：相关与独立样本t检验
A related t-test (paired samples) compares two means from the same participants, for example before and after an intervention. The formula uses the mean of the difference scores (D̄).
相关样本t检验（配对样本）比较同一批参与者的两个均值，例如干预前后。公式使用差值分数（D̄）的均值。
t = D̄ / (s_D / √n)
where s_D is the standard deviation of the difference scores.
其中 s_D 是差值分数的标准差。
An independent t-test compares the means of two unrelated groups. Assuming equal variances, a simplified formula is calculated and checked against critical values for df = n₁ + n₂ − 2.
独立样本t检验比较两个不相关组的均值。假设方差齐性，使用简化公式计算检验统计量，并与自由度为 df = n₁ + n₂ − 2 的临界值进行比较。
t = (x̄₁ − x̄₂) / √[ (s₁²/n₁) + (s₂²/n₂) ]
In WJEC you are not required to hand-calculate complex t tests from raw data, but you must understand the logic, report the t value and assess significance against the correct critical table.
在WJEC考试中，你不必从原始数据手动计算复杂的t检验，但必须理解其逻辑，报告t值，并根据正确的临界值表评估显著性。
8. Non-Parametric Tests: Spearman’s Rho & Chi-Square | 非参数检验：斯皮尔曼秩相关与卡方检验
Spearman’s rank correlation coefficient (ρ or rₛ) is the non-parametric alternative to Pearson’s r, used with ordinal data or when parametric assumptions are violated. It assesses the monotonic relationship between two ranked variables.
斯皮尔曼秩相关系数（ρ 或 rₛ）是皮尔逊r的非参数替代方法，适用于顺序数据或当参数假设不满足时。它评估两个秩次变量之间的单调关系。
rₛ = 1 − [ 6Σd² / n(n² − 1) ]
where d is the difference between ranks for each participant. Compare obtained rₛ to the Spearman critical value table. df = number of pairs.
其中d是每位参与者的秩次差。将得出的rₛ与斯皮尔曼临界值表进行比较，自由度等于配对数量。
The chi-square test is used for nominal (categorical) data. The test for independence examines whether two categorical variables are associated, while goodness-of-fit tests a single variable against expected frequencies.
卡方检验用于名义（分类）数据。独立性检验考察两个分类变量是否关联，而适合度检验则考察单一变量是否符合预期频数。
χ² = Σ[ (O − E)² / E ]
O = observed frequency, E = expected frequency. Degrees of freedom for independence: (rows − 1) × (columns − 1). Results are compared to the chi-square distribution table.
O = 观察频数，E = 期望频数。独立性检验的自由度为：(行数 − 1) × (列数 − 1)。将结果与卡方分布表进行比较。
9. Non-Parametric Tests: Mann-Whitney U & Wilcoxon | 非参数检验：Mann-Whitney U与Wilcoxon
The Mann-Whitney U test is the non-parametric equivalent of the independent t-test, used when data are at least ordinal and the independent groups design is employed. It compares the ranked data of two separate groups.
Mann-Whitney U检验是独立样本t检验的非参数等价方法，用于至少为顺序数据的独立组设计。它比较两组独立数据的秩次。
U₁ = n₁n₂ + n₁(n₁ + 1)/2 − R₁
U₂ is calculated similarly; the smaller U value is used as the test statistic. Compare with the critical value from the Mann-Whitney table.
U₂的计算方式类似；取较小的U值作为检验统计量，并与Mann-Whitney表的临界值进行比较。
The Wilcoxon signed-rank test is the non-parametric counterpart of the related t-test, used for repeated measures or matched pairs designs. It considers the magnitude as well as the direction of differences.
Wilcoxon符号秩检验是相关样本t检验的非参数对应方法，适用于重复测量或配对组设计。它同时考虑差值的幅度和方向。
T = smaller sum of signed ranks (∑R⁺ or ∑R⁻)
After ranking the absolute differences (ignoring zero-differences), the sum of ranks for positive differences and negative differences are computed. The test statistic T is the smaller sum and is checked against the critical Wilcoxon T table.
在忽略零差值后对绝对差值进行排序，计算正差值的秩和与负差值的秩和。检验统计量T为较小的秩和，并与Wilcoxon临界值T表进行比对。
10. Effect Size: Cohen’s d | 效应量：Cohen’s d
Effect size quantifies the magnitude of a finding, independent of sample size. Cohen’s d is commonly reported for t-tests and represents the standardized difference between two means. It helps assess practical significance beyond mere p-values.
效应量用于量化研究发现的幅度，不受样本量影响。Cohen’s d通常用于t检验，表示两个均值的标准化差异。它有助于评估超越单纯p值的实际显著性。
d = (x̄₁ − x̄₂) / s_pooled
where s_pooled is the pooled standard deviation. In WJEC, knowing the interpretation thresholds is essential: d = 0.2 (small), d = 0.5 (medium), d = 0.8 (large).
其中 s_pooled 是合并标准差。在WJEC中，掌握解释阈值至关重要：d = 0.2（小），d = 0.5（中），d = 0.8（大）。
If the standard deviation of the population is not known, you can also compute a simple effect size from a correlation coefficient: r itself is an effect size, with benchmarks of ±0.1 (small), ±0.3 (medium), ±0.5 (large).
如果总体标准差未知，你也可以根据相关系数计算简单的效应量：r本身就是一种效应量，其基准为 ±0.1（小），±0.3（中），±0.5（大）。
11. Decision Flowchart: Choosing the Right Test | 决策流程图：选择正确的检验
WJEC candidates are expected to justify the choice of an inferential statistical test. Use this rapid decision chain: Identify the level of measurement, the experimental design, and whether parametric assumptions are met.
WJEC考生需要能够论证推断性统计检验的选择理由。使用以下快速决策链：确定测量水平、实验设计类型以及是否满足参数检验假设。
| Data type / Relationship | Design | Parametric (if assumptions met) | Non-Parametric alternative |
|---|---|---|---|
| Difference between two conditions | Repeated measures / Matched pairs | Related t-test | Wilcoxon signed-rank |
| Difference between two groups | Independent groups | Independent t-test | Mann-Whitney U |
| Correlation / Association | Pairs of scores (interval/ratio) | Pearson’s r | Spearman’s rho |
| Nominal data (frequencies) | Independence / Goodness-of-fit | Chi-square (χ²) | |
Always check the normality assumption via inspection of histograms or the Kolmogorov-Smirnov test, though WJEC often supplies summary information rather than requiring full hand calculation.
始终通过直方图或Kolmogorov-Smirnov检验检查正态性假设，不过WJEC通常提供汇总信息，而非要求完整的手动计算。
12. Reporting and Interpreting Results | 结果呈现与解释
Correct APA-style reporting is crucial for Component 2. A standard statement includes: the test statistic, degrees of freedom, sample size, exact p-value (or in relation to α), the direction of effect, and effect size if calculated.
正确的APA风格报告对Component 2至关重要。标准陈述应包括：检验统计量、自由度、样本量、精确p值（或与α的关系）、效应方向，以及计算出的效应量（如有）。
For example: “A related t-test found a significant increase in recall scores from Time 1 (M = 12.3, SD = 2.1) to Time 2 (M = 15.8, SD = 2.4), t(19) = 4.56, p < 0.001, d = 0.92." The WJEC mark scheme rewards explicit connection of the statistical outcome to the psychological context of the study.
例如：“相关样本t检验发现，从时间1（M = 12.3, SD = 2.1）到时间2（M = 15.8, SD = 2.4）的记忆成绩显著提高，t(19) = 4.56, p < 0.001, d = 0.92。”WJEC评分方案鼓励将统计结果与研究背景中的心理学意义明确联系起来。
Remember that a non-significant result (p > 0.05) does not confirm the null hypothesis; it merely suggests insufficient evidence to reject it, which links back to Type II error reasoning.
请记住，不显著的结果 (p > 0.05) 并不能证实零假设成立；它仅仅表示没有足够证据拒绝零假设，这与II型错误的推理息息相关。
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