📚 Year 11 CCEA Psychology: Formula & Theorem Quick Reference Handbook | Year 11 CCEA 心理学：公式定理速查手册
This handbook brings together the essential formulas, statistical tools, and classical theorems you will meet in the Year 11 CCEA Psychology course. Whether you are preparing to analyse data from a memory experiment, interpret correlation in a class survey, or grasp foundational laws of perception, these quick-reference pages provide clear definitions, worked examples, and the mathematical expressions behind the concepts. Keep it close when completing research methods tasks or revising for end‑of‑unit tests.
本手册汇总了 Year 11 CCEA 心理学课程中必须掌握的核心公式、统计工具与经典定理。无论你准备分析记忆实验的数据、解读课堂调查中的相关系数，还是理解感知觉的基本定律，这份速查手册都给出了清晰的定义、实例演示和概念背后的数学表达。在做研究技能作业或准备单元测试时，将它放在手边随时翻阅。
1. Calculating Means, Medians, and Modes | 计算平均数、中位数与众数
Any data summary in psychology begins with a measure of central tendency. The three most common measures are the mean, the median, and the mode. Each captures a different aspect of a typical score, and choosing the right one depends on the shape of your data and the presence of extreme values.
心理学中的任何数据总结都从集中趋势量数开始。最常见的三种量数是平均数、中位数和众数。它们各自反映了典型分数的不同侧面，选择使用哪一个取决于数据分布形状以及是否存在极端值。
The arithmetic mean (x̄ for a sample) is the sum of all individual scores divided by the number of scores. It is the most widely used measure and is sensitive to every value in the set.
算术平均数（样本记作 x̄）等于所有单个分数之和除以分数个数。它是使用最广的量数，对数据集中的每一个值都敏感。
x̄ = Σxᵢ ÷ n
where xᵢ represents each score and n is the sample size. For a grouped frequency table the same principle applies using class midpoints.
其中 xᵢ 代表每一个分数，n 是样本容量。对于分组频数表，使用组中值后计算原理相同。
The median is the middle score when all values are arranged in order. If n is odd, it is the single central value; if n is even, take the mean of the two central scores. The median is unaffected by outliers, making it the preferred measure for skewed distributions such as reaction times where a few unusually long latencies can distort the mean.
中位数是将所有数值按顺序排列后位于正中的分数。若 n 为奇数，取正中一个值；若 n 为偶数，取正中两个值的平均数。中位数不受异常值影响，因此在偏态分布（例如反应时数据，少数极长的潜伏期就会拉高平均数）中被优先使用。
The mode is simply the most frequently occurring score in a set. A distribution may have one mode (unimodal), two (bimodal), or more. In questionnaire research the mode is often reported for nominal categories, such as the most common attachment style in a class sample.
众数是数据集中出现频率最高的分数。分布可能是单峰的、双峰的或多峰的。在问卷研究中，处理称名类别时常报告众数，例如一个班级样本中最常见的依恋类型。
2. Range and Standard Deviation | 极差与标准差
Measures of dispersion tell us how spread out the scores are. The simplest is the range, which is quick to calculate but strongly affected by extreme values. The standard deviation provides a much more reliable picture of variability because it takes every score into account.
离散量数反映分数的分散程度。最简单的量数是极差，虽然计算快捷，但极易受极端值影响。标准差由于考虑了每一个分数，能够给出更加可靠的变异信息。
The range is the difference between the highest score and the lowest score.
极差等于最高分与最低分之间的差值。
Range = Xₘₐₓ − Xₘᵢₙ
Psychologists often report the range alongside the median to show the spread while acknowledging possible outliers.
心理学家在展示分布时，常常同时报告极差和中位数，以便在描述离散程度时提示可能的异常值。
The standard deviation is the square root of the average squared deviation from the mean. When dealing with population data, use μ and divide by N. When using a sample to estimate the population, the degrees of freedom adjustment (n − 1) gives s, the sample standard deviation.
标准差是各个分数与平均数离差平方的平均数的平方根。处理总体数据时使用 μ 并除以 N；用样本来估计总体时，采用自由度调整（n − 1）可得到样本标准差 s。
σ = √[ Σ(xᵢ − μ)² ÷ N ] s = √[ Σ(xᵢ − x̄)² ÷ (n − 1) ]
A larger standard deviation indicates that individual scores are widely scattered around the mean; a smaller one means they cluster tightly. In cognitive experiments, a consistently small standard deviation across conditions suggests stable performance.
标准差越大，表明个体分数在平均数周围散布得越远；越小则表明分数聚集得越紧密。在认知实验中，不同条件下标准差始终较小说明成绩较为稳定。
root yields s ≈ 1.14. This tells you that most scores fall within roughly one point of the mean.
例如，某班记忆测验分数为 7、8、9、9、10，平均数为 8.6。离差平方和为 5.2，除以 4 得 1.3，开平方得到 s ≈ 1.14。这表明大多数分数大致落在平均数附近 1 分左右的范围内。
3. The Normal Distribution and the 68–95–99.7 Rule | 正态分布与 68–95–99.7 法则
Many psychological variables – height, IQ scores, extraversion ratings – follow a bell‑shaped normal distribution. The precise properties of this curve let researchers predict what proportion of scores lies within a certain distance of the mean.
许多心理变量——身高、智商分数、外向性评分——都服从钟形的正态分布。这条曲线的精确特性能够让研究者预测在距离均值一定范围内分数的比例。
In a perfectly normal distribution the mean, median, and mode coincide at the peak. The curve is symmetrical about the mean, and tails extend to infinity without ever touching the horizontal axis.
在完美的正态分布中，平均数、中位数和众数均在顶峰处重合。曲线关于平均数左右对称，尾部无限延伸却永远不会碰触横轴。
The 68–95–99.7 rule (also called the empirical rule) states:
68–95–99.7 法则（也称经验法则）指出：
Approximately 68% of all scores lie within ±1 SD of the mean.
大约 68% 的分数落在平均数 ±1 个标准差范围内。
Approximately 95% lie within ±2 SD.
大约 95% 落在 ±2 个标准差范围内。
Approximately 99.7% lie within ±3 SD.
大约 99.7% 落在 ±3 个标准差范围内。
These percentages are crucial when setting up hypotheses about how unusual a particular score is. A child obtaining an IQ of 130, where the population mean is 100 and SD is 15, falls exactly 2 SD above the mean – a score achieved by only about 2.5% of the population in the upper tail.
这些百分比在判断某个分数有多罕见时十分关键。假设总体智商平均数为 100、标准差为 15，一名儿童测得智商 130，恰好落在平均数以上 2 个标准差的位置——在右侧尾部仅有约 2.5% 的人达到该分数。
4. z‑Scores and Percentile Ranks | z 分数与百分位等级
A z‑score expresses how far and in which direction a raw score deviates from the mean, measured in units of standard deviation. It is the foundation for comparing performances across different tests and for finding a score’s percentile rank.
z 分数用标准差为单位，表示一个原始分数偏离平均数多远以及偏在哪一方向。它是在不同测验间比较表现并求出百分位等级的基础。
z = (X − μ) ÷ σ
Where X is the raw score, μ the population mean, and σ the population standard deviation. For a sample, replace μ with x̄ and σ with s.
其中 X 是原始分数，μ 是总体平均数，σ 为总体标准差。对于样本，用 x̄ 代 μ、s 代 σ。
A positive z‑score indicates the score is above the mean; a negative z‑score indicates it is below. A z‑score of 0 means the score equals the mean. Using z‑tables (or knowledge of the normal curve), you can convert a z‑score into a percentile rank – the percentage of scores in the distribution that fall below that value.
正的 z 分数表示分数高于平均数；负值表示低于平均数。z = 0 表示分数正好等于平均数。借助 z 表（或正态曲线的性质），你可以将 z 分数转换为百分位等级——即分布中低于该数值的分数所占的百分比。
For grouped frequency data, an approximate percentile rank can be calculated directly:
PR = [(B + 0.5E) ÷ N] × 100
Here B is the count of scores below the score of interest, E is the number of scores exactly equal to it, and N is the total number of scores.
式中 B 是低于目标分数的人数，E 是等于该分数的人数，N 是总人数。
a class of 35, her percentile rank is [(28 + 0.5×2) ÷ 35] × 100 ≈ 82.9, meaning roughly 83% of the class is shorter.
如果一名学生在 35 人的班级中身高高于 28 名同学，并与另外 2 人同高，她的百分位等级为 [(28 + 0.5×2) ÷ 35] × 100 ≈ 82.9，表示大约 83% 的同学比她矮。
5. Skewness and the Pearson Coefficient | 偏度与皮尔逊偏度系数
Not all psychological data fall symmetrically. Reaction times are often positively skewed with a long right tail, while easy tests can produce a negative skew. Skewness describes the direction and degree of this asymmetry, and a simple formula helps quantify it.
并非所有心理数据都是对称的。反应时常呈右偏（正偏态），尾部向右拖长；过于简单的测验则会带来负偏态。偏度描述了这种不对称的方向和程度，一条简易公式助你量化它。
Distributions where the mean is pulled above the median are positively skewed; those where the mean is pulled below the median are negatively skewed.
若平均数被拉到中位数之上，分布是正偏态的；若平均数被拉至中位数之下，则为负偏态。
Pearson's first coefficient of skewness uses the mean, mode, and standard deviation:
Sk = (mean − mode) ÷ s
When the mode is ambiguous, a more reliable version replaces it with the median:
Sk = 3(mean − median) ÷ s
A positive Sk value confirms a long right tail; a negative value confirms a long left tail. Values close to zero suggest a nearly symmetric shape. Recognising skew is vital: for highly skewed data the median is the appropriate central tendency, and non‑parametric statistical tests are often required.
正 Sk 值确认右侧长尾；负值确认左侧长尾。接近零的数值提示接近对称。识别偏度至关重要：对于高度偏态的数据，中位数才是恰当的集中趋势量数，而且常常需要使用非参数统计检验。
6. Spearman's Rank Correlation | 斯皮尔曼等级相关
When two sets of ordinal data are collected from the same participants, Spearman's rank correlation coefficient (rₛ) measures the strength and direction of the association. It is frequently used in CCEA coursework where you rank participants on two variables, such as extraversion rating and number of selfies posted per week.
当从同一批参与者收集到两组顺序数据时，斯皮尔曼等级相关系数（rₛ）可衡量关系的强度与方向。在 CCEA 课程作业中，它常被用于参与者双变量排序的情境，比如外向性评分和每周自拍发布数。
The formula relies on the differences between the paired ranks:
rₛ = 1 − [ 6 Σ D² ÷ n(n² − 1) ]
Here D is the difference between the rank positions of each participant on the two variables, and n is the number of pairs.
式中 D 为每位参与者在两个变量上排名的差值，n 为数据对数。
Steps to calculate rₛ:
Rank the participants on variable A and variable B, giving 1 to the highest (or lowest) consistently.
分别按变量 A 和变量 B 对参与者进行排名，最高或最低赋 1 并保持一致。
Find the difference D for each pair and square it.
计算每对差 D 并平方。
Sum all D² values and substitute into the formula.
将所有 D² 值求和后代入公式。
The value of rₛ always lies between −1 and +1. A positive coefficient means high ranks on one variable tend to go with high ranks on the other; a negative value indicates an inverse relationship. An rₛ close to zero suggests no monotonic link. Always remember that correlation does not imply causation.
rₛ 的值总是在 −1 到 +1 之间。正系数意味着一个变量上的高排名倾向于伴随另一个变量的高排名；负值表示反向关系。rₛ 接近零则暗示不存在单调关系。务必牢记：相关不等于因果。
7. Weber's Law and Just Noticeable Difference | 韦伯定律与最小可觉差
Weber's Law is one of the oldest quantitative laws in psychology, emerging from psychophysics. It states that the size of a just noticeable difference (JND) is a constant proportion of the original stimulus intensity.
韦伯定律是心理学领域最古老的定量定律之一，起源于心理物理学。它指出：最小可觉差（JND）的大小与原刺激强度成恒定的比例关系。
ΔI ÷ I = k
Here I is the physical intensity of a standard stimulus, ΔI is the smallest increase in intensity required for a person to detect a change on 50% of trials, and k is the Weber fraction, which is constant for a particular sensory dimension.
式中 I 为标准刺激的物理强度，ΔI 是无凭意识地觉察到变化（50% 的正确率）所需的最小增量，k 为韦伯分数——对某一特定感觉维度，它保持不变。
For lifted weights the Weber fraction is approximately 0.02, so a 100‑gram standard weight would need about 2 extra grams before a difference is noticed. For sound intensity, k is closer to 0.1. The law holds well for moderate stimulus intensities but breaks down at extremely low or high levels. Understanding Weber's Law helps psychologists design experiments on perception and make sense of everyday phenomena, such as why it is easy to notice a few extra coins in an empty hand but far harder to notice them in a heavy shopping bag.
提起重物的韦伯分数约为 0.02，因此 100 克的标准砝码大约需要增加 2 克才能被察觉差异。对于声音强度，k 接近 0.1。该定律在中等刺激强度范围内成立，但在极低或极高强度下失效。理解韦伯定律有助于心理学家设计感知实验并解释日常现象，比如为何空手里多出几枚硬币很容易察觉，而沉重的购物袋里增加几枚却很难发现。
8. Ebbinghaus Forgetting Curve and Savings | 艾宾浩斯遗忘曲线与节省量
Hermann Ebbinghaus pioneered the scientific study of memory by testing himself with nonsense syllables. His work produced two lasting tools: the forgetting curve that describes the rapid initial decline of memory followed by a levelling‑off, and the savings method for measuring how much information is truly retained.
赫尔曼·艾宾浩斯
Published by TutorHao | Year 11 Psychology Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图