📚 How the Citation per Faculty Ratio Is Calculated: Why Citations Are Divided by Faculty Numbers | 师均引用率的计算方式：为什么引用数要除以教员人数
Whenever you read a university ranking table, a research audit or a school prospectus, you meet the same strange-looking number: citations per faculty. It is a single figure, usually somewhere between 10 and 200, and it is produced by dividing one large number by another. This article explains exactly how that division is set up, what the numerator and the denominator each contain, and why dividing by the number of faculty members is not an arbitrary choice but a deliberate statistical correction. The reasoning is pure ratio mathematics of the kind you meet in A-Level, but its consequences are serious enough to shape international university rankings.
无论你是在看大学排名表、科研审计报告还是学校宣传册，都会遇到同一个看起来有点奇怪的数字：师均引用率。它通常是一个介于 10 到 200 之间的数值，由一个大数除以另一个大数得到。本文将逐步说明这个除法究竟是如何建立的、分子与分母各包含什么，以及为什么”除以教员人数”并非随意选择，而是一种刻意的统计修正。其背后的数学原理正是你在 A-Level 中会学到的比值运算，但它带来的后果足以影响国际大学排名。
1. What “Citations per Faculty” Actually Measures | 师均引用率到底衡量什么
Citations per faculty is a standardised indicator of research intensity. A citation is a reference in one published paper to another published paper; it signals that the cited work was useful to later researchers. Counting all citations received by an institution gives a measure of total research influence, but that measure grows simply because an institution is large. Dividing by faculty turns an absolute quantity into a rate: influence produced per academic staff member.
师均引用率是衡量科研强度的标准化指标。引用（citation）是指一篇已发表论文对另一篇已发表论文的参考文献，它表明被引用的成果对后来的研究者有用。把一所机构获得的所有引用加总，可以得到”科研总影响力”的度量，但这个数字仅仅因为机构规模大就会变大。除以教员人数，就把绝对量转化成了比率：每位学术人员所产生的影响力。
In physics terms, total citations behave like an extensive property (mass, volume), while citations per faculty behaves like an intensive property (density, concentration). Intensive measures can be compared fairly between objects of different size; extensive ones cannot.
用物理学的语言说，引用总数像是一种”广延量”（质量、体积），而师均引用率像是一种”强度量”（密度、浓度）。强度量可以在大小不同的对象之间公平比较，广延量则不能。
2. The Core Formula | 核心公式
At its simplest the indicator is one quotient. Write the number of citations received in a fixed time window as C, and the number of full-time equivalent academic staff as F. Then the ratio is defined as follows.
最简单的形式下，这个指标就是一个商。把在固定时间窗口内获得的引用数记作 C，把全职当量学术人员数记作 F，则比值定义如下。
Citations per Faculty = C ÷ F = C / F
The unit is “citations per faculty member”. Because C is a count with no unit and F is a count of people, the unit of the quotient is effectively citations · person⁻¹. If C = 96,000 and F = 1,600, then the ratio is 96,000 ÷ 1,600 = 60.0 citations per faculty member.
单位是”每条教员对应的引用数”。由于 C 是无单位的计数，F 是人数，所以商的单位实质上是 citations · person⁻¹。若 C = 96,000，F = 1,600，则比值为 96,000 ÷ 1,600 = 60.0 次引用／人。
A useful rearrangement for estimation questions is C = F × ratio. This lets you work out the citation total an institution must have generated if you already know its staff size and its published ratio.
用于估算题的一个有用变形是 C = F × 比值。若已知某校的教员规模与公布的比值，就能反推它必须产生的引用总量。
3. Why Not Just Count Total Citations? | 为什么不能只看引用总数
Total citations are dominated by size. A university with 6,000 research-active staff will almost always out-cite a specialist institute with 300 staff, even if every individual at the institute is more heavily cited. Reporting raw totals would therefore rank wealth and headcount rather than research quality.
引用总数会被规模主导。一所有 6,000 名科研活跃教员的大学，几乎总会超过一所有 300 名教员的小型专业机构，即使后者每一位个体的被引次数都更高。因此，直接公布原始总数，排名排的其实是”体量和人数”，而不是科研质量。
Dividing by F acts as a scaling correction. It answers the question: if all institutions had exactly the same number of academic staff, how would their citation totals compare? This is why the ratio is called a normalised indicator. Normalisation is the same idea as scaling a recipe down to one person, or expressing fuel use as litres per 100 km rather than total litres consumed.
除以 F 起到”尺度修正”的作用。它回答的问题是：假如所有机构都拥有完全相同的教员人数，它们的引用总量会如何比较？这就是该比值被称为”归一化指标”的原因。归一化与”把食谱按一人份缩小”、或用”每百公里油耗”而非”总耗油升数”表达油耗，本质上是同一个思路。
- Extensive measure – grows with size – favours large institutions.
- Intensive measure – independent of size – allows like-for-like comparison.
- Normalised measure – expressed per unit of input – is standard practice in economics, physics and bibliometrics.
- 广延量——随规模增长——有利于大型机构。
- 强度量——与规模无关——可实现同类比较。
- 归一化量——以”每单位投入”表示——是经济学、物理学与文献计量学的通行做法。
4. The Denominator: Who Counts as “Faculty”? | 分母：谁才算”教员”
The denominator is where most disagreements arise, because “faculty” has no single legal or statistical definition. Rankings and audits normally use full-time equivalent (FTE) academic staff, meaning part-time staff are counted proportionally: two staff each contracted at 50 percent contribute 1.0 to F.
分母是争议最多的地方，因为”教员”并没有唯一的法律或统计定义。排名机构与审计机构通常采用”全职当量”（full-time equivalent, FTE）学术人员，即兼职人员按比例折算：两位各承担 50% 合同的教员合计贡献 F 值 1.0。
Typical inclusions and exclusions in a research-oriented count are set out below. The exact rules differ between data providers, so an institution’s ratio can legitimately change if the denominator definition changes, with no change in research output at all.
下表列出以科研为导向的统计中常见的计入与排除项。不同数据方的具体规则并不相同，因此仅仅改变分母的定义（科研产出完全不变），一所机构的比值也可能合理变化。
| Usually included in F | Usually excluded from F |
| Professors, readers, lecturers on research-and-teaching contracts | Purely administrative and clerical staff |
| Research fellows and postdoctoral researchers (research-only contracts) | Students, including PhD candidates |
| Part-time academic staff, converted to FTE | Honorary or visiting staff with no substantive contract |
| Clinical academics with a research component | Technicians, librarians, estates and IT staff |
The strategic implication is clear: an institution that employs many teaching-only staff has a large denominator but a numerator that those staff barely add to, so its ratio falls. Conversely, an institution that concentrates its staff in research-active roles raises its ratio without publishing a single extra paper.
其策略含义十分清楚：如果一所机构雇用了大量”仅教学”的教员，分母被撑大，而这些人几乎不贡献分子，比值就会下降。反过来，一所机构把人员集中在科研活跃岗位上，即使一篇论文都不多写，比值也会上升。
5. The Numerator: Which Citations Count? | 分子：哪些引用算数
The numerator is not “all citations ever received”. Data providers impose a window and a source. A common design is a five-year window of publications, with citations counted over a longer period (often six years including the publication years), drawn from a single index such as Scopus or the Web of Science Core Collection.
分子并不是”有史以来获得的所有引用”。数据方会设定时间窗口与数据来源。常见设计是选取五年的出版物窗口，在更长的时间范围内统计引用（通常为六年，包含出版年份），并且只取自同一索引数据库，如 Scopus 或 Web of Science 核心合集。
- Window: only papers published in the defined years are eligible to be cited.
- Source: citations recorded in the chosen index; books, patents and non-indexed journals may be missed.
- Attribution: a paper is credited to an institution if at least one author lists that institution.
- Self-citations: sometimes removed, because a group citing itself inflates its own numerator.
- Counting rule: one citing paper normally contributes one citation to each cited paper, regardless of how many times it cites it.
- 时间窗口：只有规定年份内发表的论文才有资格被计入被引。
- 数据来源：只统计所选索引中记录的引用；专著、专利与未被索引的期刊可能被遗漏。
- 归属规则：只要有一位作者署名该机构，论文即计入该机构。
- 自引：有时会被剔除，因为研究团队引用自己会人为抬高分子。
- 计数规则：一篇引用文献通常只对每篇被引论文贡献一次引用，无论它引用了多少次。
6. Field Normalisation: Citations Differ by Discipline | 学科归一化：不同学科的引用差异
Raw citation counts are not comparable across disciplines. Molecular biology papers routinely attract hundreds of citations; pure mathematics papers often attract fewer than ten. A medical school therefore raises an institution’s citations per faculty almost automatically, while a conservatoire or a law faculty depresses it.
原始引用数在不同学科之间不可直接比较。分子生物学论文常常获得数百次引用，而纯数学论文往往不足十次。因此，一所医学院几乎会自动抬高学校的师均引用率，而音乐学院或法学院则会把这一指标压低。
The standard correction is field normalisation. Instead of counting citations, you compare actual citations with the world average for papers of the same field and the same publication year. The resulting quantity is often called the field-weighted citation impact.
标准做法是”学科归一化”。不直接统计引用次数，而是把实际引用数与同一学科、同一出版年份论文的全球平均水平作比较。所得量通常称为”学科加权引用影响力”。
FWCI = (Citations received) ÷ (Expected citations for that field and year)
A value of 1.00 means the work performed exactly at world average; 2.00 means twice the world average; 0.50 means half. This dimensionless ratio is then aggregated before being divided by faculty numbers.
数值为 1.00 表示成果恰好处于世界平均水平；2.00 表示世界平均水平的两倍；0.50 表示一半。这个无量纲比值先被汇总，再除以教员人数。
7. A Worked Example | 计算示例
Consider four institutions with the citation totals and FTE staff shown below. Work through each division carefully: the institution with the largest citation total is not the one with the highest ratio.
设有下列四所机构，引用总数与全职当量教员数如下表。请逐步完成每一处除法：引用总数最大的机构，并不是比值最高的机构。
| Institution | C (citations) | F (FTE faculty) | C ÷ F |
| A – large comprehensive | 300,000 | 5,000 | 60.0 |
| B – mid-size technical | 90,000 | 1,000 | 90.0 |
| C – small specialist | 15,000 | 200 | 75.0 |
| D – very large open university | 120,000 | 6,000 | 20.0 |
Institution A produces 3.3 times as many citations as B, yet B’s ratio is 50 percent higher. Institution D has four times A’s citation efficiency gap reversed: it has 40 percent of A’s citations but a ratio one third of A’s, because its denominator is larger. This is exactly the distortion the division is designed to expose.
机构 A 的引用数是 B 的 3.3 倍，但 B 的比值却高出 50%。机构 D 的情况同样说明问题：它的引用数只有 A 的 40%，比值却只有 A 的三分之一，原因就在于分母更大。这正是”做除法”所要揭示的偏差。
8. Small Institutions and the Ratio Effect | 小规模机构与比值效应
Dividing by a small denominator amplifies every fluctuation. If F = 200 and one newly recruited professor brings 3,000 citations with her, the numerator rises by 3,000/15,000 = 20 percent, so the ratio jumps from 75.0 to 90.0 overnight. In an institution with F = 6,000, the same individual changes the ratio by 3,000/120,000 = 2.5 percent, from 20.0 to 20.5.
除以一个很小的分母，会把任何波动放大。若 F = 200，一位新引进的教授带来 3,000 次引用，分子上升 3,000/15,000 = 20%，比值便一夜之间从 75.0 跳到 90.0。而在 F = 6,000 的机构里，同一位学者只使比值变化 3,000/120,000 = 2.5%，即从 20.0 变为 20.5。
Consequently, small and specialised institutions show the highest year-on-year volatility in this indicator. Rankings often respond by applying smoothing, by grouping institutions into bands, or by capping the indicator’s influence. None of these fixes removes the underlying arithmetic.
因此，小型与专业性机构在该指标上的年度波动最大。排名机构常见的应对办法包括平滑处理、把机构划分成区间，或对该指标的影响力设上限。但这些修补都无法消除底层的算术效应。
9. Mean versus Median: The Skew Problem | 平均值与中位数：偏态问题
Strictly, C ÷ F is a mean: total citations shared out equally among all faculty. But citation distributions are strongly right-skewed. A small number of researchers attract a very large share of citations, while many attract none at all. The mean therefore describes almost nobody.
严格来说，C ÷ F 是一个平均值：把引用总量在所有教员之间平均分配。但引用分布具有强烈的右偏特征。少数研究者吸引了极大比例的引用，而许多人一篇被引都没有。因此这个平均值几乎不能描述任何一个具体的人。
Suppose five academics in a department receive 0, 0, 5, 20 and 500 citations. The mean is 525 ÷ 5 = 105, which exceeds four of the five individuals’ totals. The median, by contrast, is 5. Both are correct; they describe different things.
假设某系五位学者的被引次数分别为 0、0、5、20 和 500。平均值为 525 ÷ 5 = 105，超过五人中四个人的实际数值；而中位数是 5。两者都正确，只是描述的对象不同。
Mean = Σx ÷ n | Median = middle value when n values are ordered
Practical indicators built on the mean are sensitive to outliers and to a handful of “star” researchers. This is one reason why serious bibliometric analyses report the median, the proportion of highly cited papers, or the share of publications in the world’s top 10 percent by citations, rather than relying on a single averaged quotient.
建立在平均值之上的实用指标，对离群值和少数”明星”学者高度敏感。这也是严肃的文献计量分析会同时报告中位数、高被引论文比例，或”全球被引前 10% 论文占比”，而不只依赖一个平均商值的原因之一。
10. How Rankings Actually Use It | 排名机构如何使用该指标
Two major international rankings publish a citation-per-faculty style measure, with different weights and data sources. The figures below are approximate and change between editions; always check the current methodology document.
两个主要的国际排名都会公布师均引用类指标，但权重与数据来源不同。下列数值为近似值，且各版本会调整，使用时应查阅最新方法论文件。
| Feature | THE World University Rankings | QS World University Rankings |
| Indicator name | Citations per faculty (research pillar) | Citations per faculty |
| Approximate weight | About 30 percent | About 20 percent |
| Data source | Scopus | Scopus |
| Normalisation | By subject field and publication year | By subject field, with staff FTE denominator |
Notice the two-layer structure. Layer one: normalise citations by field and year. Layer two: divide by faculty FTE. The second layer exists precisely because the first layer, on its own, still leaves an institution’s total score driven by staff numbers.
注意这里的两层结构。第一层：按学科与年份对引用做归一化。第二层：除以教员全职当量。第二层之所以存在，正是因为只有第一层时，机构总分仍被人员规模所驱动。
11. Practice Questions for A-Level Students | A-Level 学生练习题
These questions test compound-measure reasoning, percentage change and standard form – all of which appear in A-Level Mathematics and in any EPQ or personal statement that discusses research metrics.
下列题目考查复合度量推理、百分比变化与科学计数法——这些内容既出现在 A-Level 数学中，也出现在讨论科研指标的 EPQ 或个人陈述里。
Q1. An institution has 1,850 FTE academic staff and receives 111,000 citations. Calculate the citations per faculty ratio to one decimal place.
第 1 题. 某机构有 1,850 名全职当量教员，获得 111,000 次引用。计算师均引用率，保留一位小数。
Q2. Its citations then rise by 18 percent while staff numbers rise by 12 percent. Show that the new ratio is higher, and find the percentage change in the ratio.
第 2 题. 随后引用数上升 18%，教员人数上升 12%。证明新的比值更高，并求出比值的百分比变化。
Q3. In part 2 the correct answer is 1.18 ÷ 1.12 = 1.0536, i.e. a 5.4 percent increase. Explain in words why the ratio change is not 18 − 12 = 6 percent.
第 3 题. 第 2 题的正确解法是 1.18 ÷ 1.12 = 1.0536，即上升 5.4%。请用文字解释为什么比值的变化不是 18 − 12 = 6 个百分点。
Q4. An institution wants its ratio to double while its citation total rises by only 40 percent. By what factor must it reduce its reported FTE faculty count?
第 4 题. 某机构希望比值翻倍，但引用总数只上升 40%。它的全职当量教员数需要变为原来的多少倍？
Q5. Two departments of equal staff size are merged. Department X has a ratio of 120 and Department Y has a ratio of 30. Show that the merged ratio is 75, the arithmetic mean, and comment on why this could be considered unfair to Department X.
第 5 题. 两个规模相同的系合并。X 系比值为 120，Y 系比值为 30。证明合并后的比值为 75（即算术平均），并评论为什么这对 X 系可能并不公平。
12. Key Takeaways | 要点总结
The whole topic reduces to a small number of defensible claims. Keep them in mind whenever you meet a research metric in a prospectus or an application form.
整个话题可以归结为几条站得住脚的结论。当你在招生简章或申请表中再遇到科研指标时，请记住它们。
- Citations per faculty = total citations ÷ full-time equivalent academic staff.
- The division exists to remove the size advantage of large institutions, converting an extensive quantity into an intensive rate.
- The denominator definition matters as much as the numerator; part-time staff are counted as FTE.
- Field normalisation must happen before, or alongside, the division by faculty, because citation rates differ enormously between disciplines.
- The measure is a mean, and citation data are heavily right-skewed, so it can be dominated by a few highly cited individuals.
- Small institutions show high volatility; large institutions show stability. Neither fact says anything about research quality by itself.
- Always ask which staff were counted, which index was used, and which years were included before comparing two numbers.
- 师均引用率 = 引用总数 ÷ 全职当量学术人员数。
- 做这个除法是为了消除大型机构的规模优势，把一个广延量转化为强度比率。
- 分母的定义与分子同样重要；兼职人员按全职当量折算计入。
- 学科归一化必须在”除以教员”之前或同时进行，因为不同学科的引用率差异极大。
- 该指标是平均值，而引用数据高度右偏，因此可能被少数高被引个体主导。
- 小机构波动大，大机构更稳定；这两点本身都不能说明科研质量高低。
- 比较两个数字前，先问清：统计了哪些人员、用了哪个数据库、包含哪些年份。
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