KS3 CAIE Psychology: Formula & Theorem Quick Reference | KS3 CAIE 心理学：公式定理速查手册

📚 KS3 CAIE Psychology: Formula & Theorem Quick Reference | KS3 CAIE 心理学：公式定理速查手册

Psychology is often seen as a subject of theories and case studies, but it also relies heavily on quantitative methods. Whether you are designing an experiment, analysing data, or evaluating the significance of results, a solid grasp of the key formulas and statistical theorems is essential. This quick reference handbook brings together all the important mathematical tools you need for KS3 CAIE Psychology, explained simply with bilingual notes and ready-to-use examples.

心理学常被视为一门充满理论和案例研究的学科，但它同样高度依赖定量方法。无论你是在设计实验、分析数据还是评估结果的意义，扎实掌握关键公式和统计定理都是必不可少的。这份速查手册汇集了 KS3 CAIE 心理学所需的所有重要数学工具，通过双语注释和现成的例子，让你轻松理解、快速查阅。

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1. Mean (Average) | 平均数

The mean is the sum of all scores divided by the number of scores. It is the most commonly used measure of central tendency in psychology research.

平均数是所有分数之和除以分数的个数。它是心理学研究中最常用的集中趋势度量。

The formula for a population mean (μ) is:

总体平均数（μ）的公式为：

μ = (x₁ + x₂ + … + xₙ) / N = Σx / N

For a sample mean (x̄), the notation changes to x̄ = Σx / n, where n is the sample size. When reporting descriptive statistics in an experiment, always state the mean to one more decimal place than the original data.

对于样本平均数（x̄），符号变为 x̄ = Σx / n，其中 n 为样本容量。在实验报告中报告描述性统计时，平均数的小数位数应比原始数据多一位。

• Σ (sigma) means ‘sum of’.
• Σ (西格玛) 表示“总和”。
• N is the total number of values in the population; n in the sample.
• N 是总体中的数值总数；n 是样本中的数值总数。
• The mean is sensitive to extreme values (outliers), which may distort the picture of a typical score.
• 平均数对极端值（异常值）敏感，可能扭曲典型分数的情况。

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2. Median | 中位数

The median is the middle value when all scores are arranged in order. Unlike the mean, it is not affected by outliers, making it a better measure of central tendency for skewed distributions.

中位数是将所有分数按顺序排列后位于中间的值。与平均数不同，它不受异常值影响，因此对于偏态分布是更好的集中趋势度量。

For an odd number of scores: median = the (n+1)/2 th value. For an even number, it is the mean of the two middle values.

当分数个数为奇数时：中位数 = 第 (n+1)/2 个值。当个数为偶数时，它是中间两个值的平均数。

Psychologists often report both the median and the mean when the data set contains very high or very low scores, such as reaction times in a cognitive task.

当数据集中包含极高或极低的分数时，例如认知任务中的反应时间，心理学家通常会同时报告中位数和平均数。

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3. Mode | 众数

The mode is the most frequently occurring score in a data set. It is the only measure of central tendency that can be used with nominal (categorical) data, such as favourite colour or type of attachment.

众数是数据集中出现频率最高的分数。它是唯一可用于名义（分类）数据的集中趋势度量，例如最喜欢的颜色或依恋类型。

A data set may have one mode (unimodal), two modes (bimodal), or more. When response categories are labelled, the mode identifies the most common category chosen by participants.

一个数据集可能有一个众数（单峰）、两个众数（双峰）或更多。当反应类别有标签时，众数可以识别参与者选择的最常见类别。

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4. Range | 全距

The range is the simplest measure of dispersion, calculated as the difference between the highest and lowest scores.

全距是最简单的离散度量，计算方法为最高分与最低分之间的差值。

Range = Highest value – Lowest value

Although easy to compute, the range only considers two scores and ignores the spread of the remaining data. It is usually reported alongside the standard deviation to give a fuller picture of variability.

虽然全距易于计算，但它只考虑两个分数，而忽略了其余数据的分布情况。通常与标准差一起报告，以便更全面地反映变异性。

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5. Standard Deviation | 标准差

The standard deviation (SD) quantifies how much scores deviate from the mean. A small SD indicates scores are clustered closely around the mean; a large SD signals wide variability.

标准差（SD）量化了分数偏离平均数的程度。较小的标准差表示分数紧密聚集在平均数周围；较大的标准差则表明变异性很大。

The formula for the standard deviation of a population is:

总体标准差的公式为：

σ = √[ Σ(x – μ)² / N ]

For a sample, psychologists often use the unbiased estimator:

对于样本，心理学家通常使用无偏估计量：

s = √[ Σ(x – x̄)² / (n – 1) ]

In the sample SD, dividing by n–1 (Bessel’s correction) corrects the tendency of a sample to underestimate the population SD. Always accompany the mean with the standard deviation when summarising interval or ratio data in psychology reports.

在样本标准差中，除以 n–1（贝塞尔校正）纠正了样本低估总体标准差的倾向。在心理学报告中总结等距或比率数据时，始终将平均数与标准差一起提供。

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6. Variance | 方差

Variance is simply the square of the standard deviation. It represents the average squared deviation from the mean and is fundamental to many advanced statistical tests, including ANOVA.

方差就是标准差的平方。它表示与平均数之差的平方的平均值，是许多高级统计检验（包括方差分析）的基础。

σ² = Σ(x – μ)² / N (population), s² = Σ(x – x̄)² / (n – 1) (sample)

Because variance is expressed in squared units (e.g., ms² for reaction time), it is less intuitive to interpret than the standard deviation, but it is vital for understanding effect sizes and partitioning variability.

由于方差的单位是平方单位（例如反应时间的 ms²），它的解释不如标准差直观，但对于理解效应大小和分解变异性至关重要。

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7. Percentiles and Interquartile Range | 百分位数与四分位距

Percentiles indicate the relative standing of a score within a distribution. The Pth percentile is the value below which P% of the observations fall. The 50th percentile is the median.

百分位数表示一个分数在分布中的相对位置。第 P 百分位数是指有 P% 的观察值低于该值。第 50 百分位数就是中位数。

The interquartile range (IQR) is the range of the middle 50% of scores:

四分位距（IQR）是中间 50% 分数的范围：

IQR = Q₃ – Q₁ (Q₁ = 25th percentile, Q₃ = 75th percentile)

IQR is resistant to outliers and is often used alongside the median to summarise skewed data, such as time taken to complete a stressful task in health psychology.

IQR 不受异常值影响，通常与中位数一起用于总结偏态数据，例如健康心理学中完成压力任务所需的时间。

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8. Normal Distribution Theorem | 正态分布定理

Many psychological variables (e.g., intelligence test scores) are roughly normally distributed. The normal distribution is symmetrical and bell-shaped, with identical mean, median, and mode.

许多心理学变量（例如智力测验分数）大致呈正态分布。正态分布是对称的钟形曲线，平均数、中位数和众数相同。

The empirical rule states that in a normal distribution:

经验法则指出，在正态分布中：

• About 68% of scores lie within ±1 SD of the mean.
• 大约 68% 的分数在平均数 ±1 个标准差范围内。
• About 95% lie within ±2 SD.
• 大约 95% 在 ±2 个标准差范围内。
• About 99.7% lie within ±3 SD.
• 大约 99.7% 在 ±3 个标准差范围内。

This theorem helps psychologists make inferences about how unusual a particular score is, and it underpins null hypothesis significance testing using z-scores.

这一定理帮助心理学家推断某个特定分数有多不寻常，并且它是使用 z 分数进行零假设显著性检验的基础。

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9. Z-Score (Standard Score) | Z 分数（标准分数）

A z-score transforms a raw score into a standardised value that indicates how many standard deviations it is away from the mean.

Z 分数将原始分数转化为标准化值，表示该分数与平均数相差多少个标准差。

z = (x – μ) / σ (population) or z = (x – x̄) / s (sample)

If a participant’s memory score yields z = +2.0, that score is 2 SDs above the mean – a very high performance. Negative z-scores indicate below-average performance. Z-scores allow comparison across different tests and distributions.

如果一名参与者的记忆得分得出 z = +2.0，则该分数比平均数高出 2 个标准差——表现非常高。负的 z 分数表示低于平均水平的表现。Z 分数可以在不同测验和分布之间进行比较。

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10. Weber’s Law | 韦伯定律

Weber’s Law is a foundational principle in psychophysics. It states that the just noticeable difference (JND) between two stimuli is proportional to the magnitude of the original stimulus.

韦伯定律是心理物理学的基本原理。它指出，两个刺激之间的最小可觉差（JND）与原始刺激的大小成正比。

ΔI / I = k

Symbol	Meaning
ΔI	Increment needed for JND / 达到最小可觉差所需的增量
I	Initial stimulus intensity / 初始刺激强度
k	Weber fraction (constant for a specific sense) / 韦伯分数（对特定感觉为常数）

For example, in judging lifted weights, k ≈ 0.02; a 100 g weight would need a 2 g increase to be detected as heavier. Weber’s Law helps explain sensory thresholds and is tested in many KS3 psychology practicals on perception.

例如，在判断提举的重量时，k ≈ 0.02；要感觉到 100 克的重量变重需要增加 2 克。韦伯定律有助于解释感觉阈值，并在许多 KS3 心理学关于知觉的实践活动中被考查。

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11. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数

When variables are at least ordinal, psychologists often use Spearman’s rho (ρ) to measure the strength and direction of association.

当变量至少为顺序数据时，心理学家常使用斯皮尔曼等级相关系数（ρ）来测量关联的强度和方向。

rₛ = 1 – [ 6Σd² / n(n² – 1) ]

Here, d = difference between the ranks of each pair, and n = number of paired scores. The result ranges from –1 (perfect negative correlation) to +1 (perfect positive correlation). A value near 0 suggests no monotonic relationship.

其中，d = 每对数据的等级差，n = 配对分数的数量。结果范围从 –1（完全负相关）到 +1（完全正相关）。数值接近 0 则表示不存在单调关系。

An example from developmental psychology: correlating the rank order of children’s age and their performance on a conservation task. Spearman’s rho is robust to outliers and does not assume a normal distribution.

发展心理学中的一个例子：将儿童的年龄排序与其在守恒任务中的表现排序进行相关分析。斯皮尔曼等级相关系数对异常值具有稳健性，且不要求正态分布。

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12. Measures of Effect Size: Cohen’s d | 效应量指标：科恩 d 值

Beyond statistical significance, psychologists report effect size to judge the practical importance of a finding. Cohen’s d is commonly used to compare two means.

除了统计显著性，心理学家还报告效应量以判断研究结果的实际重要性。科恩 d 值常用于比较两个平均数。

d = (M₁ – M₂) / SD_pooled

SD_pooled is the weighted average of the two standard deviations. Cohen suggested benchmarks: d = 0.2 (small), 0.5 (medium), 0.8 (large). For instance, if a memory improvement intervention gives d = 0.6, the effect is medium-to-large.

SD_pooled 是两个标准差的加权平均。科恩提出了参考标准：d = 0.2（小）、0.5（中）、0.8（大）。例如，若一项记忆力提升干预的 d = 0.6，则效应量为中到大的水平。

Understanding d helps KS3 students move beyond p-values and critically evaluate whether a statistically significant result is meaningfully large in the real world.

理解 d 值可以帮助 KS3 学生超越 p 值，批判性地评估一个统计上显著的结果在现实世界中是否具有足够大的意义。

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