📚 Pre-U CCEA Psychology: Formula & Theorem Quick Reference Handbook | Pre-U CCEA 心理学：公式定理速查手册
This concise handbook brings together the essential formulae, laws and theorems you need for CCEA Pre-U Psychology. Whether you are computing statistical tests, interpreting psychophysical scales or recalling classic cognitive limits, these equations and principles provide a quick reference for both revision and examination practice. Each entry is explained in simple terms, with the symbol definitions you need to apply them correctly.
这本速查手册汇集了 CCEA Pre-U 心理学所需的核心公式、定律和定理。无论你是在计算统计检验、解释心理物理量表，还是回顾经典的认知极限，这些方程和原理都能为复习和考试提供快速参考。每个条目都用简明语言说明，并给出了正确应用所需的符号定义。
1. Weber’s Law | 韦伯定律
Weber’s Law describes the just noticeable difference (JND) between two stimuli, stating that the ratio of the JND to the original stimulus intensity is constant for a given sensory modality.
韦伯定律描述了两个刺激之间的最小可觉差（JND），指出对于特定感觉通道，JND 与原始刺激强度的比值是一个常数。
ΔI / I = k
where ΔI is the increment required for a JND, I is the initial stimulus intensity, and k is the Weber fraction (constant).
其中 ΔI 是产生最小可觉差所需的增量，I 是初始刺激强度，k 是韦伯分数（常数）。
The smaller the Weber fraction, the more sensitive the sensory system is to changes in that dimension. For example, for lifted weights k ≈ 0.02, meaning a 2 % increase is just noticeable.
韦伯分数越小，感觉系统对该维度变化越敏感。例如，对提举重物 k ≈ 0.02，即增加 2 % 才刚好被觉察。
2. Fechner’s Law | 费希纳定律
Fechner’s Law extends Weber’s work by proposing a logarithmic relationship between physical stimulus intensity and perceived sensation magnitude.
费希纳定律在韦伯工作的基础上提出物理刺激强度与感知觉大小之间存在对数关系。
S = k log I
where S is the perceived sensation magnitude, I is the physical stimulus intensity, and k is a scaling constant that depends on the Weber fraction.
其中 S 是感知觉大小，I 是物理刺激强度，k 是一个取决于韦伯分数的标度常数。
This law implies that equal ratios of stimulus intensity produce equal increments in sensation. Sensation grows more slowly as intensity increases. It holds reasonably well for brightness and loudness at moderate intensities.
该定律意味着相等的刺激强度比率产生相等的感受增量。感觉随强度增加而增长得越来越慢。在中等强度下，它对亮度和响度的预测效果较好。
3. Stevens’ Power Law | 史蒂文斯幂定律
Stevens challenged Fechner’s logarithmic law by proposing that sensation magnitude is a power function of stimulus intensity, with an exponent that differs by sensory dimension.
史蒂文斯挑战费希纳的对数定律，提出感觉大小是刺激强度的幂函数，其指数因感觉维度而异。
S = k Iⁿ
where S is perceived magnitude, I is physical intensity, k is a scaling constant, and n is the exponent characteristic of the sensory modality (e.g. n ≈ 0.33 for brightness, n ≈ 3.5 for electric shock).
其中 S 是感知大小，I 是物理强度，k 是标度常数，n 是该感觉模态的特征指数（如亮度 n ≈ 0.33，电击 n ≈ 3.5）。
When n < 1, sensation grows slower than intensity (compressive); when n > 1, it grows faster (expansive). This formula is used in magnitude estimation tasks.
当 n < 1 时，感觉比强度增长慢（压缩）；当 n > 1 时，感觉增长更快（扩张）。该公式用于数量估计任务。
4. Signal Detection Theory: d’ and Criterion | 信号检测论：d’ 和判断标准
Signal detection theory separates sensitivity from response bias. The sensitivity index d’ (d-prime) measures how distinguishable a signal is from noise.
信号检测论将感受性与反应偏向分开。感受性指标 d’（d 撇）衡量信号与噪声的可区分程度。
d’ = z(Hit) − z(False Alarm)
where z(Hit) is the z-score of the hit rate and z(False Alarm) is the z-score of the false alarm rate. Larger d’ values indicate greater sensitivity.
其中 z(Hit) 是击中率的 z 分数，z(False Alarm) 是虚报率的 z 分数。d’ 值越大，感受性越高。
The criterion (c) measures response bias:
判断标准 (c) 测量反应偏向：
c = −0.5 [z(Hit) + z(False Alarm)]
Positive c indicates a conservative bias (saying ‘no’ more often); negative c indicates a liberal bias (saying ‘yes’ more often).
c 为正表示保守偏向（更常说“无信号”）；c 为负表示自由偏向（更常说“有信号”）。
5. Standard Scores (z-scores) | 标准分数 (z 分数)
A z-score expresses a raw score’s distance from the mean in units of standard deviation. It is essential for comparing values across different distributions and for many hypothesis tests.
z 分数以标准差为单位表示原始分数与平均值的距离。它对跨分布比较数值以及许多假设检验至关重要。
z = (X − μ) / σ
for population data, or
对于总体数据，或
z = (X − M) / s
for sample data. A z-score of 1.5 means the score is 1.5 standard deviations above the mean. In a normal distribution, about 68 % of scores lie within z = ±1, and about 95 % within ±1.96.
用于样本数据。z 分数为 1.5 意味着该分数比平均值高 1.5 个标准差。在正态分布中，约 68 % 的分数落在 z = ±1 之间，约 95 % 落在 ±1.96 之间。
6. Intelligence Quotient (IQ) Formula | 智商 (IQ) 公式
The classical ratio IQ formula, used in early intelligence tests, compares mental age to chronological age.
传统的比率智商公式用于早期智力测验，将心理年龄与实际年龄进行比较。
IQ = (MA / CA) × 100
where MA is mental age and CA is chronological age. A child with an MA of 10 years and a CA of 8 years would have an IQ of (10/8) × 100 = 125.
其中 MA 是心理年龄，CA 是实际年龄。一个心理年龄 10 岁、实际年龄 8 岁的孩子 IQ 为 (10/8) × 100 = 125。
Modern IQ tests use deviation IQs based on a normal distribution with a mean of 100 and a standard deviation of 15: IQ = 15z + 100, where z is the individual’s z-score.
现代 IQ 测验使用离差智商，基于均值为 100、标准差为 15 的正态分布：IQ = 15z + 100，其中 z 是个体的 z 分数。
7. t-Test Formula (Independent Samples) | t 检验公式（独立样本）
The independent-samples t-test compares the means of two unrelated groups. Its formula, assuming equal variances, is:
独立样本 t 检验比较两个无关组的均值。在假设方差齐性的条件下，其公式为：
t = (M₁ − M₂) / √[ (sₚ² / n₁) + (sₚ² / n₂) ]
where M₁ and M₂ are the group means, sₚ² is the pooled variance, and n₁ and n₂ are the sample sizes. The pooled variance combines the variances of both groups.
其中 M₁ 和 M₂ 是两组均值，sₚ² 是合并方差，n₁ 和 n₂ 是样本量。合并方差综合两组的方差。
The degrees of freedom (df) = n₁ + n₂ − 2. If the calculated t exceeds the critical value at a chosen alpha level (e.g. 0.05), the null hypothesis is rejected.
自由度 (df) = n₁ + n₂ − 2。若计算所得的 t 值超过所选 α 水平（如 0.05）的临界值，则拒绝零假设。
8. Chi-Square (χ²) Formula | 卡方 (χ²) 公式
The chi-square test for independence evaluates whether two categorical variables are associated. The test statistic is:
卡方独立性检验评估两个类别变量是否有关联。检验统计量为：
χ² = Σ (O − E)² / E
where O is the observed frequency in each cell and E is the expected frequency under the null hypothesis. Expected frequencies are calculated as (row total × column total) / grand total.
其中 O 是每个单元格中的观察频数，E 是在零假设下的期望频数。期望频数计算为（行合计 × 列合计）/ 总合计。
Degrees of freedom = (number of rows − 1) × (number of columns − 1). If χ² is larger than the critical value, the variables are significantly associated.
自由度 =（行数 − 1）×（列数 − 1）。若 χ² 大于临界值，则变量间存在显著关联。
A special case is the goodness-of-fit χ², which compares observed frequencies to a theoretical distribution using the same formula.
一个特例是拟合优度 χ² 检验，用相同公式比较观察频数与理论分布的期望频数。
9. Pearson’s r Correlation Coefficient | 皮尔逊相关系数 r
Pearson’s r measures the strength and direction of a linear relationship between two continuous variables. The computational formula is:
皮尔逊 r 衡量两个连续变量之间线性关系的强度和方向。计算公式为：
r = [ Σ (X − Mₓ)(Y − Mᵧ) ] / [ √Σ (X − Mₓ)² · √Σ (Y − Mᵧ)² ]
where X and Y are the raw scores and Mₓ, Mᵧ are their means. The numerator is the covariance, and the denominator standardises the measure so that r always lies between −1 and +1.
其中 X 和 Y 是原始分数，Mₓ、Mᵧ 是它们的均值。分子为协方差，分母将度量标准化，使得 r 始终落在 −1 到 +1 之间。
A positive r indicates that as one variable increases, the other tends to increase; a negative r indicates an inverse relationship. The coefficient of determination, r², tells us the proportion of variance shared by the two variables.
正 r 表示一个变量增大时另一个也趋向增大；负 r 表示反向关系。决定系数 r² 表示两个变量共享的方差比例。
10. Miller’s Law (7 ± 2) | 米勒定律 (7 ± 2)
George Miller (1956) proposed that the capacity of short-term memory is about seven items, plus or minus two. This ‘magical number’ applies to the number of chunks of meaningful information we can hold at once.
乔治·米勒 (1956) 提出短时记忆的容量约为 7 ± 2 个项目。这个“神奇的数字”适用于我们一次能保持的有意义信息组块的数量。
C = 7 ± 2 chunks
Chunking allows us to recode information into larger meaningful units, effectively expanding the amount of information we can remember. For instance, the letter sequence ‘C I A F B I U S A’ can be chunked into three familiar acronyms: CIA, FBI, USA.
组块化将信息重新编码为更大的有意义的单元，有效扩展了可记住的信息量。例如，字母序列 C I A F B I U S A 可被组块为三个熟悉的缩写：CIA、FBI、USA。
While more recent research suggests a smaller true capacity of about 3–4 chunks, Miller’s law remains a cornerstone concept in cognitive psychology for understanding working memory limitations.
尽管较新的研究提示真正的容量可能更小（约 3–4 个组块），米勒定律仍是认知心理学中理解工作记忆限制的核心概念。
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