📚 AS CIE Psychology: Formula & Theorem Quick Reference Handbook | AS CIE 心理学：公式定理速查手册
This quick-reference handbook compiles all the essential statistical formulae and theorems required for the AS CIE Psychology (9990) course. Mastering these is crucial for the Research Methods component, covering descriptive statistics, inferential tests, and probability. Use it for rapid revision before exams and to build confidence in handling quantitative data interpretation.
本速查手册汇集了 AS CIE 心理学（9990）课程所需的所有核心统计公式与定理。掌握这些内容对研究方法部分至关重要，涵盖描述性统计、推断检验及概率理解。可用于考前快速复习，帮助你从容应对量化数据解读。
1. Mean (Arithmetic Average) | 平均数（算术平均值）
The mean is the most commonly used measure of central tendency. It represents the average score in a data set and is sensitive to all values, including outliers. In psychological research, the mean is often reported alongside standard deviation when data are normally distributed.
平均数是最常用的集中趋势量数，代表数据集的平均得分，并对所有值（包括异常值）都敏感。在心理学研究中，当数据呈正态分布时，平均数常与标准差一起报告。
x̄ = Σx / n
Where x̄ = sample mean, Σx = sum of all scores, n = number of scores. Always distinguish sample mean (x̄) from population mean (μ), which is a theoretical value usually unknown.
其中 x̄ 为样本均值，Σx 为所有分数之和，n 为分数的个数。务必区分样本均值 (x̄) 与总体均值 (μ)，后者通常是一个未知的理论值。
2. Variance and Standard Deviation | 方差与标准差
Variance and standard deviation measure the dispersion of scores around the mean. The greater the spread, the larger these values. Standard deviation is the square root of variance and has the same unit as the original data, making it more interpretable in psychological reports.
方差和标准差衡量分数围绕均值的离散程度。离散度越大，这些值越大。标准差是方差的平方根，与原始数据单位相同，因此更容易在心理学报告中进行解释。
Variance: s² = Σ(x − x̄)² / (n − 1)
Standard Deviation: s = √[ Σ(x − x̄)² / (n − 1) ]
Use (n − 1) for sample variance and standard deviation (Bessel’s correction) to provide an unbiased estimate of the population parameter. In AS exams you may be given or asked to compute these values from raw data or summary statistics.
计算样本方差与标准差时使用 (n − 1)（贝塞尔矫正），以得出总体参数的无偏估计值。AS 考试中可能直接给你数据或概要统计量，并要求你计算出这些值。
3. Range and Interquartile Range (IQR) | 全距与四分位距
The range is a simple measure of dispersion, while the interquartile range describes the spread of the middle 50% of data and is resistant to outliers.
全距是一种简单的离散量数，而四分位距描述了中间 50% 数据的散布情况，且不受异常值影响。
Range = Xmax − Xmin
IQR = Q₃ − Q₁
Q₁ is the first quartile (25th percentile), Q₃ is the third quartile (75th percentile). A larger IQR indicates greater variability in the central half of the data. Always consider the range together with other measures, as it can be drastically affected by a single extreme score.
Q₁ 是第一四分位数（第 25 百分位数），Q₃ 是第三四分位数（第 75 百分位数）。IQR 越大，表明数据集中间一半的变异程度越大。永远要将全距和其他量数一起考量，因为它极易受单个极端分数的影响。
4. Pearson’s Product–Moment Correlation (r) | 皮尔逊积差相关 (r)
Pearson’s r measures the strength and direction of a linear relationship between two continuous variables. Values range from −1 to +1, with 0 indicating no linear correlation. It requires interval/ratio data and assumes normal distributions.
皮尔逊 r 衡量两个连续变量之间线性关系的强度和方向。其值介于 −1 到 +1 之间，0 表示不存在线性相关。该系数要求定距/定比数据并假定正态分布。
r = Σ[(x − x̄)(y − ȳ)] / √[ Σ(x − x̄)² × Σ(y − ȳ)² ]
Here x̄ and ȳ are the means of the two variables. The numerator is the covariance, and the denominator scales it. In AS Psychology, you will use critical value tables to determine whether r is significant at a chosen significance level.
这里 x̄ 和 ȳ 是两个变量的均值。分子为协方差，分母对其进行缩放。在 AS 心理学中，你会使用临界值表来判断在所选显著性水平下 r 是否显著。
5. Spearman’s Rank Correlation (rₛ) | 斯皮尔曼等级相关 (rₛ)
Spearman’s rho is a non-parametric alternative to Pearson’s r, used for ordinal data or when assumptions of normality are violated. It assesses monotonic relationships, not necessarily linear. It is frequently used in AS studies with ranked data.
斯皮尔曼等级相关系数是皮尔逊 r 的非参数替代方法，适用于顺序数据或正态性假设不成立的情形。它评估的是单调关系，不一定线性。在 AS 研究中常与等级数据一同使用。
rₛ = 1 − [ 6 Σd² / n(n² − 1) ]
d is the difference between the ranks of each pair of values, n is the number of pairs. When there are tied ranks, a correction is applied, but the AS specification normally expects you to understand the principle and interpret given values.
d 是每对数值的等级差，n 为对子的个数。存在相同等级时需要修正，但 AS 考纲通常要求理解原理并能解读给出的数值。
6. Mann–Whitney U Test | 曼–惠特尼 U 检验
The Mann–Whitney U test is a non-parametric test for independent groups designs. It compares the ranks of two independent samples to decide whether they come from the same distribution. It is appropriate when the dependent variable is ordinal or when interval data violate normality.
曼–惠特尼 U 检验是一种适用于独立组设计的非参数检验。它比较两个独立样本的等级，判断它们是否来自相同的分布。当因变量为顺序数据或定距数据违反正态性时适用。
U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁
U₂ = n₁n₂ + n₂(n₂+1)/2 − R₂
U = min(U₁, U₂)
R₁ and R₂ are the sums of ranks for groups 1 and 2. The smaller U is compared against critical values. If U is equal to or smaller than the critical value, the result is statistically significant at that level.
R₁、R₂ 分别为两组等级和。取较小的 U 值与临界值比较。如果 U 等于或小于临界值，该结果在该水平上具有统计显著性。
7. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验
The Wilcoxon test is used for repeated measures or matched pairs designs. It analyses the signed ranks of difference scores between two conditions. It is the non-parametric counterpart to the paired-samples t-test.
威尔科克森检验用于重复测量或配对设计。它分析两种条件下差值分数的符号等级，是配对样本 t 检验的非参数替代方法。
W = min( ΣR₊ , ΣR₋ )
ΣR₊ is the sum of ranks of positive differences, ΣR₋ is the sum of ranks of negative differences (ignoring zero differences). The smaller of these two sums is the test statistic W. Compare W to the critical Wilcoxon T value: the result is significant if W is less than or equal to the critical value.
ΣR₊ 为正差值的等级和，ΣR₋ 为负差值的等级和（忽略差值为零的对子）。取两者中较小者为检验统计量 W。将 W 与威尔科克森 T 临界值比较：若 W 小于或等于临界值，则结果显著。
8. Chi-Square Test (χ²) | 卡方检验 (χ²)
The chi-square test is used with categorical (nominal) data to examine whether observed frequencies differ significantly from expected frequencies. It can test for independence (association) or goodness of fit. It is one of the inferential tests required in the AS CIE Psychology curriculum.
卡方检验用于类别（名义）数据，以检验观察频数与期望频数之间是否存在显著差异。可检验独立性（关联）或拟合优度。它是 AS CIE 心理学课程要求的推断检验之一。
χ² = Σ [ (O − E)² / E ]
O = observed frequency, E = expected frequency. The sum is taken over all categories. A larger χ² value indicates a greater discrepancy. Degrees of freedom (df) depend on the design: for a 2×2 contingency table, df = 1. Use critical chi-square tables for interpretation.
O 为观察频数，E 为期望频数。对所有类别求和。χ² 值越大表示差异越大。自由度 (df) 取决于设计：对于 2×2 列联表，df = 1。用卡方临界值表进行解读。
9. Effect Size: Cohen’s d | 效应量：Cohen’s d
Effect size quantifies the magnitude of an experimental effect or the difference between two conditions, independent of sample size. Cohen’s d is widely used in AS psychology to assess how meaningful a finding is beyond statistical significance.
效应量量化实验效应或两种条件间差异的大小，独立于样本量。Cohen’s d 在 AS 心理学中被广泛用来评估研究发现在统计显著性之外的实际意义。
d = (M₁ − M₂) / spooled
M₁ and M₂ are the means of the two groups, and spooled is the pooled standard deviation. When group sizes and variances are similar, spooled = √[ (s₁² + s₂²) / 2 ]. A d of 0.2 is considered small, 0.5 medium, and 0.8 large.
M₁ 和 M₂ 为两组均值，spooled 为合并标准差。当样本量和方差相似时，spooled = √[ (s₁² + s₂²) / 2 ]。d = 0.2 为小效应，0.5 为中效应，0.8 为大效应。
10. Normal Distribution: The Empirical Rule | 正态分布：经验法则
Many psychological variables approximate a normal distribution when sampled adequately. The empirical rule (68–95–99.7) describes the percentage of values lying within one, two, and three standard deviations of the mean. This underpins significance testing and confidence intervals.
许多心理变量在充分取样时接近正态分布。经验法则（68–95–99.7）描述了位于均值 1、2、3 个标准差范围内的数据百分比。这为显著性检验和置信区间奠定了基础。
- 68% of data fall within μ ± 1σ | 数据落在 μ ± 1σ 内
- 95% within μ ± 2σ | 在 μ ± 2σ 内
- 99.7% within μ ± 3σ | 在 μ ± 3σ 内
In AS Psychology, if a study’s result falls beyond 1.96 standard deviations from the mean under the null hypothesis, it is usually significant at the 5% level. Remember that sample mean distributions also tend toward normality (central limit theorem).
在 AS 心理学中，若研究结果在原假设下超出均值 1.96 个标准差，通常在 5% 水平上显著。记住样本均值分布也趋于正态（中心极限定理）。
11. Statistical Decision Making: p-values, Type I & Type II Errors | 统计决策：p 值、第一类及第二类错误
The p-value is the probability of obtaining the observed result, or one more extreme, assuming the null hypothesis is true. In AS CIE Psychology, the conventional significance level is p ≤ 0.05. If p is below this threshold, the null hypothesis is rejected.
p 值是在假定原假设为真的情况下，得到当前观察结果或更极端结果的概率。在 AS CIE 心理学中，常规显著性水平为 p ≤ 0.05。若 p 低于该阈值，则拒绝原假设。
| Decision / 决策 | Null Hypothesis True (H₀ true / 原假设为真) | Null Hypothesis False (H₀ false / 原假设为假) |
|---|---|---|
| Reject H₀ / 拒绝原假设 | Type I Error (false positive) / 第一类错误（假阳性） | Correct decision (power) / 正确决策（统计功效） |
| Fail to reject H₀ / 未拒绝原假设 | Correct decision / 正确决策 | Type II Error (false negative) / 第二类错误（假阴性） |
A Type I error occurs when a researcher claims an effect that does not actually exist (probability = α). A Type II error happens when a real effect is missed (probability = β). Understanding these errors helps evaluate the reliability of psychological findings.
第一类错误指研究者宣称并不真实存在的效应（概率 = α）。第二类错误则指漏掉了真实效应（概率 = β）。理解这两种错误有助于评估心理学研究结果的可靠性。
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