📚 SQA Year 10 Psychology: Quick Reference to Key Formulas & Theorems | SQA 十年级心理学：公式定理速查手册
This quick reference handbook brings together the essential formulas, statistical equations and foundational laws that you will encounter throughout the SQA Year 10 Psychology course. Whether you are analysing data for a research investigation or recalling key psychophysical principles, having these tools at your fingertips will strengthen your understanding and exam performance.
本速查手册汇集了 SQA 十年级心理学课程中将遇到的基本公式、统计方程和基础定律。无论你是在为研究调查分析数据，还是在回忆核心的心理物理学原理，掌握这些工具都能帮助你加深理解并提升考试成绩。
1. Mean (Arithmetic Average) | 算术平均值
The mean is the most widely used measure of central tendency. It is calculated by adding up all the individual scores in a data set and then dividing by the total number of scores. This single value represents the mathematical ‘centre’ of the data, though it can be distorted by extreme outliers.
平均值是最常用的集中趋势度量。它通过将数据集中所有单个分数相加，再除以分数总数求得。这一数值代表了数据的数学“中心”，但容易受极端异常值的影响。
x̄ = Σx / n
符号说明： x̄ 代表样本均值，Σx 表示所有观测值 x 的总和，n 为样本量（即观测值的个数）。
2. Median (Midpoint) | 中位数
The median is the middle score when all values are arranged in ascending order. If the number of scores is odd, the median is simply the centre value. If the number of scores is even, you must take the two central numbers, add them together and divide by two. The median is unaffected by outliers and is often used alongside the mean to give a fuller picture of the data’s central tendency.
中位数是将所有数值按升序排列后位于中间的分数。如果数据个数为奇数，中位数就是正中央的那个值。如果数据个数为偶数，则需要取中间两个数相加后除以二。中位数不受异常值影响，常与均值一起使用，以便更全面地反映数据的集中趋势。
3. Mode (Most Frequent Score) | 众数
The mode is the score that appears most frequently in a data set. A set may have one mode (unimodal), two modes (bimodal) or more. The mode is particularly useful for categorical data but can sometimes lack stability in small samples. It is the only measure of central tendency that can be used with nominal data.
众数是数据集中出现频率最高的分数。一组数据可能有一个众数（单峰），两个众数（双峰）或更多。众数对分类数据特别有用，但在小样本中有时不够稳定。它是唯一可用于名义数据的集中趋势度量。
4. Range (Spread of Scores) | 全距
The range is the simplest measure of dispersion. It is found by subtracting the smallest value from the largest value in the data set. Although quick to calculate, the range is heavily influenced by a single extreme score and does not reflect how the data are distributed around the centre.
全距是最简单的离散度指标。它由数据集中最大值减去最小值求得。虽然计算迅速，但全距极易受到单个极端分数的影响，且无法反映数据围绕中心的分布情况。
Range = Maximum value − Minimum value
符号说明： 最大值减去最小值。例如，分数为 3, 7, 8, 12, 15，全距为 15 − 3 = 12。
5. Standard Deviation (Variation Around the Mean) | 标准差
Standard deviation is a more robust measure of dispersion that tells you how much, on average, each score deviates from the mean. A small standard deviation indicates that data points cluster closely around the mean, whereas a large standard deviation indicates they are spread out. In SQA Year 10 Psychology you will often work with the sample standard deviation (using n − 1) because research data are drawn from samples.
标准差是一种更稳健的离散度指标，它告诉你每个分数平均偏离均值的程度。标准差小表示数据点紧密聚集在均值周围，标准差大则表示数据分散。在 SQA 十年级心理学中，你常会用到样本标准差（除以 n − 1），因为研究数据来自样本。
s = √[ Σ(x − x̄)² / (n − 1) ]
符号说明： s 为样本标准差，x 为每个原始分数，x̄ 为样本均值，n 为样本量。计算步骤：先求均值，再求每一个分数与均值的离差，平方后求和，除以 n−1，最后开平方根。
6. Percentage Change | 百分比变化
Percentage change is widely used in psychology to express differences between conditions or over time. It shows the increase or decrease as a proportion of the original value. This formula is especially handy when comparing pre-test and post-test scores or changes in behaviour intensity.
百分比变化在心理学中广泛用于表示不同条件之间或随时间的变化。它以原始值为参照，显示出增长或下降的比例。当你比较前测与后测分数或行为强度的变化时，这个公式特别方便。
% Change = (New value − Original value) / Original value × 100
示例： 焦虑分数从 40 降至 32，百分比变化 = (32 − 40)/40 × 100 = −20%，表明下降了 20%。
7. Z-Score (Standard Score) | Z 分数
A z-score tells you how many standard deviations a raw score lies above or below the mean. It allows you to compare scores from different distributions or tests on a common scale. In SQA Psychology, z-scores are useful for understanding where an individual’s performance stands relative to the group.
Z 分数表示一个原始分数处于均值之上或之下多少个标准差。它使你能够在共同的尺度上比较来自不同分布或测验的分数。在 SQA 心理学中，z 分数有助于理解个体的表现相对于群体所处的位置。
z = (x − μ) / σ or for samples: z = (x − x̄) / s
符号说明： x 为个体分数，μ 为总体均值，σ 为总体标准差；用样本时以 x̄ 替代 μ，以 s 替代 σ。z 为正值表示高于均值，负值表示低于均值。
8. Pearson’s r (Product-Moment Correlation) | 皮尔逊积差相关系数
Pearson’s r measures the strength and direction of a linear relationship between two continuous variables. The value of r always lies between −1 and +1. An r of +1 indicates a perfect positive correlation, −1 a perfect negative correlation, and 0 no linear relationship. You may not have to compute it by hand in Year 10, but you should recognise the formula and understand what it tells you about covariation.
皮尔逊 r 衡量两个连续变量之间线性关系的强度和方向。r 的值始终介于 −1 和 +1 之间。r 为 +1 表示完全正相关，−1 表示完全负相关，0 表示没有线性关系。在十年级你可能不需要手动计算，但应认识该公式并理解它对于共变关系的意义。
r = Σ(x − x̄)(y − ȳ) / √[ Σ(x − x̄)² Σ(y − ȳ)² ]
符号说明： x 和 y 为两个变量的成对分数，x̄ 和 ȳ 分别为其均值。分子为协方差，分母为标准差的乘积，确保了标准化的度量。
9. Spearman’s Rank Correlation | 斯皮尔曼等级相关系数
Spearman’s rho (ρ) is a non-parametric alternative to Pearson’s r. It is used when data are ordinal or when the relationship is monotonic but not necessarily linear. You first rank the scores for each variable, find the difference between the ranks for each participant, square those differences and insert them into the formula. Spearman’s ρ is particularly common in SQA research investigations where participants are ranked on questionnaires or observations.
斯皮尔曼等级相关系数 (ρ) 是皮尔逊 r 的非参数替代方法。当数据是顺序变量，或变量间呈单调关系但不一定是线性时使用。你首先对每个变量的分数排序，求出每个参与者等级之差，将差值平方后代入公式。在 SQA 的研究调查中，当受试者在问卷或观察中被排序时，斯皮尔曼 ρ 尤其常用。
ρ = 1 − ( 6 Σd² ) / [ n( n² − 1) ]
符号说明： ρ 为斯皮尔曼等级相关系数，d 表示每对等级之差，n 为配对数量。校正因子 6 使系数范围控制在 −1 到 +1 之间。
10. Weber’s Law (Psychophysics) | 韦伯定律
Weber’s Law describes the relationship between the magnitude of a physical stimulus and the amount of change needed for a person to notice a difference (the just noticeable difference, JND). It states that the JND is a constant proportion of the original stimulus intensity. This law holds well for mid-range intensities across many sensory modalities.
韦伯定律描述了物理刺激的强度与个体刚刚能察觉到变化所需的刺激增量（最小可觉差）之间的关系。定律表明，最小可觉差是原始刺激强度的一个恒定比例。该定律对中等强度的多种感觉通道均适用。
ΔI / I = k
符号说明： ΔI 为最小可觉差（JND），I 为原始刺激的物理强度，k 为韦伯常数（不同感觉通道的 k 值不同，例如亮度辨别约为 0.02，重量辨别约为 0.05）。
11. Fechner’s Law (Psychophysics) | 费希纳定律
Gustav Fechner extended Weber’s findings and proposed a logarithmic relationship between physical stimulus intensity and perceived sensation magnitude. According to Fechner’s Law, as the physical intensity of a stimulus increases geometrically, the perceived sensation increases only arithmetically. This explains why doubling the brightness of a light does not make it appear twice as bright.
古斯塔夫·费希纳扩展了韦伯的发现，提出物理刺激强度与感知感觉量之间存在对数关系。根据费希纳定律，当刺激的物理强度以几何级数增加时，心理感觉量仅以算术级数增加。这解释了为何将灯光亮度加倍并不能让亮度感觉加倍。
S = k log I
符号说明： S 为心理感觉量，I 为物理刺激强度，k 为常数，取决于感觉通道和测量单位。费希纳定律强调了感知相对于物理世界的压缩性质。
12. Yerkes-Dodson Law (Arousal & Performance) | 耶克斯-多德森定律
The Yerkes-Dodson Law describes the relationship between arousal and performance, taking the form of an inverted U-shaped curve. Performance improves with increasing arousal up to an optimal point, after which further arousal leads to deterioration. Additionally, the optimal level of arousal depends on task complexity: simple or well-learned tasks benefit from higher arousal, while complex or novel tasks require lower arousal for peak performance.
耶克斯-多德森定律描述了唤醒与绩效之间的关系，呈倒U形曲线。随着唤醒水平上升，绩效会逐步提高直至一个最优点，此后唤醒进一步升高则导致绩效下降。此外，最佳唤醒水平取决于任务复杂度：简单或熟练的任务在较高唤醒下表现更好，而复杂或新颖任务需要较低唤醒才能达到最佳绩效。
Although this law is not expressed as a single mathematical equation, its principle is fundamental to understanding stress, anxiety and performance in areas such as exam preparation and sports psychology. Remember the inverted-U: moderate arousal yields the best results.
尽管该定律不以单一数学公式表示，但其原理对于理解考试准备和运动心理学等领域的压力、焦虑与绩效至关重要。请记住倒U曲线：中等唤醒带来最佳表现。
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