https://www.linguee.com/japanese-english/translation/学生.html
This references a translation provided by Keio University in 2007....  translated simply as 'competitiveness'
各界に人材を輩出してきた歴史や社会的な影響力を背景に、 学生募集 においては高い難易度と倍率を維持している。 
Backed by its history that has fostered talents in various fields and its influence on society, the university maintains the high level of difficulty and competitiveness. 

https://katekyo.mynavi.jp/juken/6496#1
【1】出願倍率（応募倍率）：出願者数÷募集人数  Number of applicants/desired number of recruits
【2】受験倍率：受験者数÷募集人数   Number who actually took the entrance exam/ desired number of recruits
【3】実質倍率：受験者数÷合格者数   Number who actually took the entrance exam/ actually admitted (accounting for students who pass the test, but choose another school, etc) 

このように、入試の段階が進むにつれ「出願倍率＞受験倍率＞実質倍率」と倍率が下がっていきます。逆に言えば、出願倍率と実質倍率の差がない学校は、それだけ第1志望校にされる人気校といえます。

On other sites, 志願倍率 seems roughly synonymous with 出願倍率

Short answer to your question. Perhaps 
"Competitiveness rating" (for a school or entrance exam, etc).  
("a numerical score roughly defined by the number of applicants divided by the number of available positions or applicants accepted")


Long answer.

The purpose appears to be to either:
1. Indicate to the student how hard it's going to be to get into a University(or private high school/middle school etc), or to pass an entrance exam
2. Indicate how prestigious a school is by showing how many more people apply than are accepted 

"High competitiveness rating" , "low competitiveness rating", "rising/falling competitiveness rating" would all match up.  Ideally there could be a hint at how that is calculated (applicants / accepted), but the details clearly depend on which 倍率 is being discussed.

Roughly speaking, it seems to be a ratio of applicants to accepted applicants, with various qualifiers (see references: 出願倍率 受験倍率 実質倍率)

I followed up with some Japanese college classmates.  I was told the following:
1. 倍率 is a very common, well understood term (if you're applying to schools anyway). 
2. They could produce the simple definition when I asked what it meant(applicants/accepted applicants)
3. Importantly: According to classmates 倍率 is only used for entrance exams or situations where there is a ratio of applicants-to-accepted.  It's not used for, say, final exams.  90% getting a D or better wouldn't create a 倍率 of 10/9 = 1.1, for example.  In other words, it has nothing to do with pass/fail in a grading context.  If this is correct, it seems to disagree with the 2011 definition of "pass rate (of an exam, etc.)". That entry could be struck entirely. Compare, for example "入学試験の倍率" (bunch of google results) with "期末試験の倍率" (no results).

Other than 求人倍率, which hurts to think about, this appears to be fairly consistent for 倍率 entries.  Take this entry:
応札倍率 bid-to-cover ratio
応札=bid, 
https://en.wikipedia.org/wiki/Bid-to-cover_ratio
" the number of bids received divided by the number of bids accepted"
"The higher the ratio, the higher the demand. A ratio above 2.0 indicates a successful auction with aggressive bids."
This is also a measure of competitiveness for the bidder. 

Finally, jmdict also has an entry for 合格率
I'm not sure whether it's worthwhile to  "see also" one to the other. The are roughly, but not quite, the inverse of each other.

Although some of the JEs use ratio it's clearly off the mark as a translation. I wonder about "factor" as an alternative. Any better suggestions?

@@ -27,2 +27,2 @@
-<gloss>acceptance ratio (e.g. university entrance)</gloss>
-<gloss>examination pass ratio</gloss>
+<gloss>acceptance factor (e.g. university entrance)</gloss>
+<gloss>examination pass factor</gloss>

Now that I know what a 倍率 is, the entry no longer effects me of course.  I guess my concern is that any attempt to use the suggested gloss of "acceptance ratio" is likely to be wrong.  "A rising acceptance ratio/倍率","falling acceptance ratio/倍率", "extremely low acceptance ratio/倍率", these are all precisely backwards.  This is the exact mistake I made on a simple exam question "As more and more students continue to want to attend Tokyo University, ....".  I suggested "倍率が下がる一方だ". A confused teacher substituted "上がる", as if it was an obvious blunder.  Some time ago I had added　倍率 as a flashcard to study, based on the jmdict entry, and I thought I knew what it was....

This is why I tried to follow the model of the entry for '求人倍率'.  Those entries are both a bit clunky, but they are not misleading. "applicant-to-acceptance ratio" is not super common in English, but at least someone who sees it will know to be careful with a translation.

I know it is not generally the job of a J-E dictionary to write out definitions, but at least 入学の倍率 and 試験の倍率 have succinct explanations. Literally "number of actual applicants divided by number of accepted applicants". I don't know if it's possible to spell it out. 

For the new note: "25% would be a 倍率 of 4", my concern is, 25% of what? Maybe "An acceptance rate of 25% would be a 倍率 of 4" would be a clearer explanation.  But this suggests reverse meanings for "acceptance rate" and "acceptance ratio"...  I don't know if jmdict ever warns its users explicitly... "Warning: 25% passing would be a 倍率 of 4, 50% would be a 倍率 of 2, etc.". 

Well, I appreciate any change to add clarity, even if that is just the new note.  I've used jmdict (through jisho.org) for the last 5 years of Japanese study, and it's been an invaluable resource.  I'm now studying in Japan, and finally having enough Japanese interactions to encounter small issues like this one.  I've got one other simpler potential-correction on another entry when I get through midterm season.

Thanks again.

There are several of these ratio terms, and they are difficult to gloss succinctly. The JE 
dictionaries duck away. I prefer the previous versions with some tweaks.

@@ -26,2 +26,3 @@
-<gloss>applicant-to-acceptance ratio(for an entrance exam, etc.)</gloss>
-<gloss>exam taker-to-pass ratio (For 50% passing, 倍率 would be 2)</gloss>
+<s_inf>25% would be a 倍率 of 4</s_inf>
+<gloss>acceptance ratio (e.g. university entrance)</gloss>
+<gloss>examination pass ratio</gloss>

https://dictionary.goo.ne.jp/word/倍率/
https://www.eikoh-seminar.com/koukou-juken/magazine/004860/

受験する人の数 ÷ 募集定員 = 倍率

https://www.schoolnetwork.jp/hs/plus/column/2019/vol3.php

https://opencampus-guide.jp/faq/cat05/入試の倍率ってなに？.html

This last site further defines the terms 志願倍率 and 実質倍率

https://kenkou888.com/category13/bairitsu.html

Finally, here is a reference for "applicant-to-acceptance ratio" as an option.
https://pubmed.ncbi.nlm.nih.gov/2764406/

The applicant-to-acceptance ratio in the nation's 127 medical schools has slowly but steadily declined during the last decade to 1.7:1 (60.5% acceptance rate) for the 1987-1988 academic year.

With regards to the "acceptance rate" definition, I recently discovered this is more like the "inverse acceptance rate".  If 100 people take a test, and 50 pass, the 倍率 would be 100/50=2. Confirmed with a professor and numerous Japanese classmates after making a blunder based on this dictionary entry :) The definition on dictionary.goo.ne.jp seems ambiguous to me, but the sample sentence is not: 「倍率の高い名門校」. Here's a random website with some sample calculations: 

https://www.eikoh-seminar.com/koukou-juken/magazine/004860/

Contrast with the English speaking expectation of "Acceptance Rate"
https://blog.collegevine.com/what-do-acceptance-rates-really-mean/ 
"Harvard setting the bar low at just 5.2%. "
or Pass Rate:
https://sciencing.com/calculate-passing-rate-7476998.html
"Divide Pass by Total"

While "inverse acceptance rate" might technically be correct, I feel like this entry requires an explanation.  I'm not sure what the best way to do this is. I have made an attempt, but someone may have a better suggestion.

I suspect, but cannot confirm, that this is related to the other "amplification" meaning of 倍率.  Another (synonymous?) term seems to be 競争倍率. Bigger number = more competitive. 

jmdict also has an entry for 求人倍率 which is written out as "job opening-to-application ratio". That is unambiguous. Oddly, I guess this is from the perspective of the employer trying to fill positions, so a high 求人倍率 means "this is a difficult position for us to fill, we are forced to work hard to fill the position (by lowering our standards)".  Conversely, a low   求人倍率 means the employer can be extremely selective in choosing employees.

@@ -26,2 +26,2 @@
-<gloss>acceptance rate</gloss>
-<gloss>pass rate (of an exam, etc.)</gloss>
+<gloss>applicant-to-acceptance ratio(for an entrance exam, etc.)</gloss>
+<gloss>exam taker-to-pass ratio (For 50% passing, 倍率 would be 2)</gloss>

@@ -27,1 +27,1 @@
-<gloss>(exam) pass rate</gloss>
+<gloss>pass rate (of an exam, etc.)</gloss>

GG5

Add “acceptance rate” sense – saw in article yesterday, seems v. common.

@@ -24,0 +24,5 @@
+<sense>
+<pos>&n;</pos>
+<gloss>acceptance rate</gloss>
+<gloss>(exam) pass rate</gloss>
+</sense>