📚 AS CAIE Psychology: Formula & Theorem Quick Reference Handbook | AS CAIE 心理学：公式定理速查手册
This handbook gathers the most important quantitative formulas, inferential test statistics, and psychological laws you must know for the AS CAIE Psychology exam. Use it for quick revision before tackling data-response questions and research methods assessments.
本手册汇集了 AS CAIE 心理学考试必须掌握的最重要的量化公式、推断检验统计量以及心理学定律。适用于应对数据分析题和研究方法考核前的快速复习。
1. Measures of Central Tendency | 集中趋势的度量
Central tendency summarises a data set with a single representative value. The three main measures are the mean, median, and mode.
集中趋势用单个代表性数值概括数据集。三种主要测量是均值、中位数和众数。
The mean is the arithmetic average: M = ΣX / N, where ΣX is the sum of all scores and N is the number of scores. It uses every data point but is sensitive to outliers.
均值是算术平均数：M = ΣX / N，其中 ΣX 为所有分数之和，N 为分数的个数。它利用了每一个数据点，但对异常值敏感。
The median is the middle value when scores are arranged in order. For an even number of scores, it is the average of the two middle values. The median is robust against extreme scores.
中位数是将分数排序后位于中间的值。对于偶数个分数，它是两个中间值的平均数。中位数不受极端分数的影响。
The mode is the most frequently occurring score. It is the only measure suitable for nominal data, but can be uninformative if all values appear once.
众数是出现次数最多的分数。它是唯一适用于名称量表的度量，但如果所有值都只出现一次，众数便没有意义。
2. Measures of Dispersion: Range & Variance | 离散度量：全距与方差
Dispersion describes how spread out the scores are. A simple index is the range: Range = highest score – lowest score. It is easy to compute but ignores the internal distribution.
离散程度描述分数的分散情况。一个简单的指标是全距：全距 = 最高分 – 最低分。它容易计算，但忽略了内部分布。
Variance indicates the average squared deviation from the mean. For a sample, the unbiased estimator uses N – 1 in the denominator (Bessel’s correction).
方差表明偏离均值的平均平方离差。样本的无偏估计量使用分母N – 1（贝塞尔校正）。
s² = Σ(X – M)² / (N – 1)
A larger variance means the data points are more dispersed; a smaller variance shows they cluster tightly around the mean.
方差越大，数据点越分散；方差越小，数据点越紧密地聚集在均值附近。
3. Standard Deviation & Standard Error | 标准差与标准误
The standard deviation (s) is the positive square root of the variance, expressed in the original measurement units. It is the most widely reported measure of spread.
标准差(s)是方差的正平方根，以原始测量单位表示。它是报告最广泛的离散程度指标。
s = √[ Σ(X – M)² / (N – 1) ]
Approximately 68% of observations in a normal distribution fall within ±1 SD of the mean, and 95% within ±2 SD.
在正态分布中，大约68%的观测值落在均值±1个标准差以内，95%落在±2个标准差以内。
The standard error of the mean (SE) estimates the variability of sample means if you drew many samples. It is used to construct confidence intervals.
均值的标准误(SE)估算如果抽取许多样本，样本均值的变异性。它用于构建置信区间。
SE = s / √n
A smaller SE indicates a more precise estimate of the population mean.
标准误越小，对总体均值的估计越精确。
4. Normal Distribution & Z-Scores | 正态分布与 Z 分数
The normal distribution is a symmetrical, bell-shaped curve defined by the mean and standard deviation. Many psychological variables approximate this distribution.
正态分布是由均值和标准差决定的一条对称钟形曲线。许多心理学变量近似服从该分布。
To compare scores from different distributions, we convert raw scores into z-scores.
为了比较来自不同分布的分数，我们将原始分数转化为z分数。
z = (X – M) / s
A z-score of +1.0 means the score is one standard deviation above the mean; –1.5 means 1.5 SD below the mean. Z-scores allow researchers to calculate percentile ranks using standard normal tables.
z分数为+1.0 表示该分数高于均值一个标准差；–1.5 表示低于均值1.5个标准差。z分数允许研究者通过标准正态表计算百分等级。
5. Spearman’s Rank Correlation |
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