📚 AS Eduqas Psychology: Formula & Theorem Quick Reference Handbook | AS Eduqas 心理学：公式定理速查手册
This quick reference handbook compiles the essential statistical formulas, decision rules, and theorems required for AS Eduqas Psychology. It is designed to help you revise the mathematical procedures used in research methods and inferential testing. Keep it handy for last-minute checks before your exam.
这本速查手册汇总了AS Eduqas心理学考试中必备的统计公式、判定规则和定理，旨在帮助你复习研究方法与推断检验中的数学步骤。考前随手查阅，巩固重点。
1. Measures of Central Tendency | 集中趋势测量
The mean is the arithmetic average of a set of scores. It is calculated by summing all values and dividing by the number of scores. The mean is sensitive to extreme values (outliers).
平均数是所有分数的算术平均值，计算方法是总和除以个数。平均数对极端值（异常值）很敏感。
Mean (x̄) = Σx / n
Where Σx is the sum of all scores, and n is the number of scores.
其中 Σx 为分数总和，n 为分数的个数。
The median is the middle value in an ordered list. If n is odd, it is the central score; if n is even, it is the average of the two middle scores. The mode is the most frequently occurring value.
中位数是排序后位于中间的值。若 n 为奇数，取中间值；若 n 为偶数，取中间两个数的平均值。众数是一组数据中出现次数最多的数值。
2. Range and Variance | 全距与方差
Range provides a simple measure of dispersion: it is the difference between the highest and lowest scores.
全距是最简单的离散度量：即最高分与最低分的差值。
Range = Xmax – Xmin
Variance quantifies how far each score is from the mean. For a sample, the variance (s²) is the average of the squared deviations.
方差表示各数据点与平均数之间偏差的平方的平均值（样本方差）。
s² = Σ(x – x̄)² / (n – 1)
Using (n‑1) corrects the bias in estimating the population variance and is known as Bessel’s correction.
分母使用 n-1 是对总体方差估计的偏误校正，称为贝塞尔校正。
3. Standard Deviation | 标准差
The standard deviation is the square root of the variance and is expressed in the same units as the original data. It indicates the average distance of scores from the mean.
标准差是方差的平方根，单位与原始数据一致，反映了分数与平均数的平均距离。
s = √[ Σ(x – x̄)² / (n – 1) ]
A larger standard deviation means greater variability. In Eduqas exams you may be required to compute s from a small data set and then use it to describe the distribution.
标准差越大，数据变异性越大。Eduqas考试可能要求根据小数据集计算 s 并据此描述分布特征。
4. The Normal Distribution | 正态分布
The normal distribution is a symmetrical, bell‑shaped curve defined by its mean and standard deviation. In a normal distribution, approximately 68% of scores lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
正态分布是一条对称的钟形曲线，由平均数和标准差决定。在正态分布中，约68%的数据落在平均数±1个标准差内，95%落在±2个标准差内，99.7%落在±3个标准差内。
The total area under the curve equals 1, so proportions can be interpreted as probabilities. The theorem that the sampling distribution of the mean approaches normality as sample size increases is the Central Limit Theorem, which underpins many parametric tests.
曲线下总面积为1，因此比例可解读为概率。随着样本量增加，样本平均数的分布趋近正态，这就是中心极限定理，它是许多参数检验的基础。
5. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rho (rₛ) measures the strength and direction of association between two ranked variables. It is suitable for ordinal data and non‑linear monotonic relationships.
斯皮尔曼等级相关系数 (rₛ) 测量两个排序变量之间的关联强度和方向，适用于顺序数据和非线性的单调关系。
rₛ = 1 – (6 Σ d²) / [n(n² – 1)]
Where d is the difference between the ranks of each pair, and n is the number of paired scores. The coefficient ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation).
其中 d 为每一对数据的等级差，n 为配对数据个数。系数范围从 -1（完全负相关）到 +1（完全正相关）。
To test significance, compare the calculated rₛ against the critical value in the Spearman table for the chosen α level and n. If |rₛ| ≥ critical value, reject the null hypothesis.
进行显著性检验时，将计算得到的 rₛ 与斯皮尔曼临界值表中对应显著性水平 α 和 n 的临界值比较。若 |rₛ| ≥ 临界值，则拒绝零假设。
6. Chi-Squared Test (χ² Test) | 卡方检验
The chi-squared test is used with categorical (nominal) data to assess whether there is a significant association between two variables or a significant difference between observed and expected frequencies.
卡方检验用于类别（称名）数据，检验两个变量间是否存在显著关联，或观察频数与期望频数之间是否存在显著差异。
χ² = Σ[ (O – E)² / E ]
O is the observed frequency, E is the expected frequency (calculated from row total × column total / grand total). The degrees of freedom (df) for a contingency table is (number of rows – 1) × (number of columns – 1).
O 为观察频数，E 为期望频数（计算方式：行合计 × 列合计 ÷ 总计）。列联表的自由度 df = （行数 – 1）×（列数 – 1）。
Compare the obtained χ² to the critical value with the appropriate df. If χ² ≥ critical value, the result is statistically significant at the chosen level (usually p < 0.05).
将计算得到的 χ² 值与相应自由度下的临界值比较。若 χ² ≥ 临界值，则在选定的显著水平（通常为 p < 0.05）下结果具有统计学意义。
7. Mann-Whitney U Test | 曼‑惠特尼U检验
The Mann-Whitney U test is a non‑parametric test for independent groups design, used to compare differences between two sets of ordinal (or non‑normal interval) data. It tests
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