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What does an optimal grading distribution look like?
Researchers on education must have thought about this, so I hope to find some directions here. Suppose that I am grading students on a scale from 1 to 10. If I have no prior knowledge of the ability of the students but have obtained their test scores (let's say from 1 to 100).
Suppose these test scores have only ordinal meaning. A student with a higher score can be said to have achieved better mastery of the course but having double the score does not mean that one has achieved double as much as another student. We may also assume that a (statistically speaking) large number of students representative of the student population took the exam.
What should be the optimal grading distribution?
We may also assume that the grading distribution/scale should achieve two goals: a) it should be informative about student's grasp of the material, b) it should incentivize students to study the material.
Regarding a) from an information theory perspective, we may want to maximize the information (entropy) of the grade distribution. Thus, we would choose a scale which yields a uniform distribution. However, in practice most teachers implement distributions which are peaked. What is the motivation behind this?

Hopefully everybody learned everything perfectly, and grades bunch up in the 8-10 range (yes, people make mistakes).

The optimal grade distribution is the empty distribution --- in the optimal world, people are not motivated by grades, and they do not deceive themselves or others about their knowledge or abilities. Thus no grades are necessary.

@Boris Even in the optimal world you describe, grades would still be useful. Students aren't necessarily able to measure their own level of mastery, grades are one way for instructors to provide feedback to students on what they've achieved

Since some comments seemed somewhat beside the point of the question, I (hopefully) clarified it.

The optimal distribution is whatever honestly reflects student mastery of skills for the course. This assumes that there exists a grading rubric which accurately measures relative mastery of desired skills -- which it should, and is a far higher priority than any other concern of the OP's. Mangling the data to fit some preconceived distribution resembles a type of research fraud.

A uniform distribution would not generally be informative about students' grasp of the material, because people do not behave that way. In the absense of any other information, I would think a normal distribution would be the most obvious guess.

@JessicaB A normal distribution is defined on the reals, not a discrete number of grades. I am also not sure what you mean by "people do not behave that way". Why do they "behave" more like one distribution than another?

If really "test scores have only ordinal meaning" then I'd say that this test is not appropriate to be used in grading at all, since it cannot even discriminate if students have achieved the basic competencies or knowledge required to complete the course, much less grade them on how well they have done so. A test must be able to distinguish on individual level if some absolute competency level X is achieved or not. Grading is an afterthought which may (if so desired) choose the particular level of competency appropriate for each grade to show relative ratings, but it can't be all relative.

Hopefully everybody learned everything perfectly — Ain't no such thing.

@JessicaB because people do not behave that way — I have actually seen people behave this way.  Many years ago, I gave a 40-point exam to 120 students; the score distribution was statistically uniform.  For every half-integer score between 4.5 and 38, one to three students had that score.  (There were also a couple of score below 4.5, and a couple of scores above 38.)  It was freaky.

You may want to look at construct validity.

Thanks, @StephanKolassa this seems to be the first suggestion with reference to some scientific literature. I am surprised how many of the answers so far are personal opinions without a scientific foundation/reference. This is academia.SE after all!

@ff524 Arguably, a list of learning goals with those checked off the student achieved would be much more useful than a flat, all-averaging grade.

I've added some references to education research to my answer. Enjoy :)

"A student with a higher score can be said to have achieved better mastery of the course but having double the score does not mean that one has achieved double as much as another student." I would think a measure double that of another would have much more evidential weight than one being slightly higher (assuming not close to zero).

Assigning grades to fit some "optimal distribution" is misguided.
We don't want to maximize the entropy of the grades in a particular course. It's not a very useful measure of a "good" set of grades. To quote from an answer by Anonymous Mathematician:

Strictly speaking, Shannon entropy pays no attention to the distance between scores, just to whether they are exactly equal. I.e., you can have high entropy if every student gets a slightly different score, even if the scores are all very near to each other and thus not useful for distinguishing students.

(Note that this was in answer to a question that asked about using the entropy of exam scores as an indication of how good an exam is at distinguishing between different levels of mastery. It wasn't suggesting that grades should be curved a posteriori to maximize entropy.)
What we actually want is for grades to signal as closely as possible the students' mastery of course material. If every student in the course has achieved truly excellent mastery of the course material, they should all get high scores. This grade then seems to carry very little information. But in reality, it is much more useful to say that all students in this particular class achieved excellence and deserve a 10/10, than it would be to maximize the "information" carried by the grade and give some students a 1/10 because their performance was slightly less excellent than the highest level of excellence achieved by a student that year. This scenario (where all students achieve excellent or very good grades) is not even that unusual, as Michael Covington points out:

In advanced courses, it can be quite proper for all students to get A’s and B’s, because weak students would not take the course in the first place.

For an individual student, the grade should depend on that student's demonstrated mastery of course material, and hopefully not at all (or as little as possible) on the other students in the class.
If you insist on thinking about it from an information theoretic perspective, what we really want is to minimize the Kullback–Leibler divergence between the distribution of students' achievements and distribution of students' grades.
If you have no information about the exam and what it measures, you cannot assign meaningful grades based on that exam score. If you know about the exam, you can assign meaningful grades based on exam scores, but not according to any optimal distribution - you would assign scores based on how much of the exam students are expected to know to demonstrate various levels of mastery, not by mathematically shaping the grades into some predetermined "optimal distribution."

Edit: Suppose these test scores have only ordinal meaning. A student with a higher score can be said to have achieved better mastery of the course but having double the score does not mean that one has achieved double as much as another student.

Your edits are not going to change the answer; there's still not going to be an optimal distribution. If I have many excellent students in the class and I do an exceptionally good job teaching them, I'll give out many excellent grades, no matter how they rank with respect to one another. If all of my students are terrible and do poorly, they'll all get low grades even if one manages to get in a few more points than another (although in that case, I'll also take a closer look at the class to see why students are doing so poorly.) If half of my class excels and the other half fails to meet minimum standards, I'll give out 50% top grades and 50% failing grades. If their abilities happen to be normally distributed, their grades will be as well. You get the idea.
The distribution of student grades should follow the distribution of demonstrated achievements. Any grade distribution that doesn't is definitely not optimal.

The idea of grading students to some predetermined distribution (of any shape) is known as "norm-referenced grading."  For more information about alternatives to norm-referenced grading, see:

Sadler, D. Royce. "Interpretations of criteria‐based assessment and grading in higher education." Assessment & Evaluation in Higher Education 30.2 (2005): 175-194.
Aviles, Christopher B. "Grading with norm-referenced or criterion-referenced measurements: to curve or not to curve, that is the question." Social work education 20.5 (2001): 603-608.
Rose, Leslie. "Norm-referenced grading in the age of carnegie: why criteria-referenced grading is more consistent with current trends in legal education and how legal writing can lead the way." Journal of the Legal Writing Institute 17 (2011): 123.
Guskey, Thomas R. "Grading policies that work against standards... and how to fix them." NASSP Bulletin 84.620 (2000): 20-29.

These go into some more detail about problems with norm-referenced grading. You asked for a grade distribution that is "informative about student's grasp of the material." The literature I have cited explains that pure norm-referenced grading is not informative about that; it can only inform about student's relative grasp of the material, compared to others in the same group who have taken the same exam. In other words (emphasis mine):

It can be useful for selective purposes (e.g. for the distribution of a scholarship to the 5 best students, or extra tuition to the 5 which are struggling most), but gives little information about the actual abilities of the candidates.

Source: McAlpine, Mhairi. Principles of assessment. CAA Centre, University of Luton, 2002.
In the middle of the 1990s, the Swedish secondary school grading system was changed from a norm-referenced to a criterion-referenced system. Thus it became possible to compare (for the same population) the ability of a norm-referenced grading system and a criterion-referenced grading system to predict academic success.
The first paper looking at this Swedish data set is not written in English. (Cliffordson, C. (2004). De målrelaterade gymnasiebetygens prognosförmåga. [The Predictive Validity of Goal-Related Grades from Upper Secondary School]. Pedagogisk Forskning i Sverige, 9(2), 129-140.) However, in a later paper, the author describes those results as follows:

Cliffordson (2004b) showed in a study of 1st-year achievement in the Master of Science programs in Engineering that the predictive validity of CRIT-GPA was somewhat higher than it was for NORM-GPA.

In that later study, Cliffordson found (consistent with the earlier study)

a somewhat higher prognostic validity for CRIT-GPA
  than for NORM-GPA.

and that across a variety of disciplines,

Despite differences in both design and purpose, the predictive efficacy of criterion-referenced grading is at least as good, or indeed is somewhat better, than that of norm-referenced grades.

For details, see:
Cliffordson, Christina. "Differential prediction of study success across academic programs in the Swedish context: The validity of grades and tests as selection instruments for higher education." Educational Assessment 13.1 (2008): 56-75.

Thanks for the useful link to the other question. If I understand it correctly, then given that student's grades are similarly near to each other the entropy gives at least some answer to a). I of course agree with you that the KL divergence between scores and achievements is more meaningful. However, it is assumed that student's actual achievements are unobservable. I noticed now that the answer to this depends a lot on assumptions of the exam properties.

@HRSE Without knowing about the exam and what it measures, you can't assign meaningful grades. If you know about the exam, you can assign meaningful grades based on exam scores, but not according to any optimal distribution - that is, you would assign scores based on how much of the exam students are expected to know to demonstrate various levels of mastery, not by mathematical tricks.

Great! Many thanks for all the caveats. The question was not meaningful without further assumptions on the exam. However, I am not willing to go as far as saying that a teacher can "read" levels of mastery directly from the exam.

@HRSE Yet you are willing to go so far as to say that the levels of achievement of any group of students, at any kind of institution of higher education, in any kind of class, should be identically distributed to that of any other group? That seems like a much bigger leap to me :)

By "representative of the student population" I of course do not mean "all students across the globe", otherwise I would have written that. All that is precluded is that in a particular sample of exams I do NOT get only "excellent" students (relative to the population under consideration) and in another sample I get only "terrible" students.

@HRSE You could make my examples less "extreme" and they'd still apply. It's just easier to write "all excellent students" and "all poor students" than to list hundreds of hypothetical students and their abilities. The point is that different classes are different, different groups of students are different, and there's no reason to expect any two groups to have the same distribution.

@HRSE if you assume that a particular sample of exams matches the overall distribution of "excellent" vs "terrible" students, it still doesn't allow you to conclude that in those circumstances the grading can be the same. Exam grades and pass rates are correlated not only with the attributes of students, but also e.g. course difficulty. The system must be able to grade situations where 100% of students taking a particular exam will get a "good" grade and also situations where some part of students in some exam get a "failing" grade - two obviously different, incompatible grade distributions.

In advanced courses, it can be quite proper for all students to get A’s and B’s, because weak students would not take the course in the first place. — I strongly disagree.  There is no absolute scale of mastery to which scores must (or can) be calibrated.  Every class is necessarily calibrated to the skills and background of the students taking it.  It is entirely proper to teach advanced classes so that some students get Cs and Ds, and so that the weaker students who aren't taking it would have absolutely no chance of passing.

Also, I don't read the original question as "What distribution should I use to assign grades?" but rather "After I've assigned grades, what does the resulting score distribution tell me about the effectiveness/accuracy of the exam/grading rubric?"

@JeffE I think the point was that it's feasible for a course to have all A's and B's and that's one example of a scenario where this would not be unlikely, not that that's always going to be the grade distribution in an advanced class.

+1 for "The distribution of student grades should follow the distribution of demonstrated achievements. Any grade distribution that doesn't is definitely not optimal."

From the literature you provided I can now see the distinction between norm-referenced grading and criterion-referenced grading and that criterion-referenced grading may be better in case that the exam gives us the information to perform criterion-referenced grading. The question assumes the exam does not fulfill this. Thus, the question remains: if we need to use norm-referenced grading, what are desirable properties of the grade distribution? Saying that criterion-referenced grading (though impossible) is superior clearly dodges the question.

@HRSE you asked for a grade distribution that is "informative about student's grasp of the material." The literature I have cited explains that pure norm-referenced grading is not informative about that; it can only inform about student's relative grasp of the material, compared to others in the same group who have taken the same exam. Grades referenced to a distribution can't give you what you asked for in the situation you describe; you're asking for something that doesn't exist.

@HRSE I think your edit makes it a completely different question. If you want to know about assessing students' relative abilities, maybe you should ask a new question in a new post. (It would also help if you specify the purpose for which you want to know students' relative abilities; different grading systems are likely to be "optimal" for different purposes.)

well, since the original question did not include "a)" but "a)" was added after you suggested this, i think it is O.K. to clarify that relative abilities are meant.

@HRSE It's generally considered poor etiquette to edit a question in a way that invalidates the existing answers.

Let us continue this discussion in chat.

Without trying to evade your question, it is extremely difficult to quantify the impact of different grading distributions. This makes it not possible to find an optimal solution. You state 2 goals which makes optimization difficult. Further, your goals are not well defined.
On the surface, the goal of motivating students seems laudable, but we need to be careful about what we are motivating. We do not want to motivate students to strive for higher grades (i.e., grade grubbing). Nor do we want to promote a cutthroat environment where students attempt to increase their grade, by sabotaging fellow students. We want to motivate students, at both the individual and group levels, to increase their understanding. Education research is pretty clear that external motivators, like grades, do not promote the type of learning teachers strive for.
Your second goal is for the grades to be informative. But you do not define whom you are trying to inform. Do we want grades to inform students about their level of understanding, or do we want the grades to inform potential employers. From my understanding of education theory, summative assessments are not part of the learning process while formative assessments are. Further, it is not clear if letter/numerical grades are useful for formative assessments.

My (anecdotal) experience is that having all marks high does not provide a helpful incentive, because it makes the cost of errors very high. There needs to be space for students to learn things over time and from the assessment process (except of course those assessments at the very end of the course) and benefit from doing so.

In addition to "who," it also matters "what" information you are trying to convey. Do you expect people to use the grade to estimate the student's ability to do a job, to estimate the student's ability to do well in the next course in the sequence, to select the 10 best-performing students for an award, to identify which students should be eligible for TA jobs the next time the course is offered, etc.

No offense but I have no idea how this even comes close to answering the question.  Yes it is great for a teacher to use feedback from students.  Are you saying that teachers should teach without grades?  The answer is so vague that your answer could mean anything.  Just wondering what you answer is and why the OP gave this the checkmark - I feel like I am missing something.

@blankip I am saying that the problem is ill posed and hence there is no single optimal solution.

I think there probably is a baseline method for getting an optimal grading system in almost all cases.   I have done LMSs for 15 years and I have seen some really really good grading systems for very complex things.  The good ones are a bit fluid and sometimes complex.  I have seen step tests where they have people answer questions at a certain level of proficiency that is incredibly accurate but really time intensive to set up.  Also there is nothing better than fear to motivate a human.

The optimal grading distribution looks exactly like the distribution of capability in the field that the members of your class have.
Now, that's not really something you can easily or perfectly measure, but that's your target.
If you could follow your students and grade their performance in later life at a few milestones, you could conceivably estimate the information snr in your grading. That's where you'd apply those concepts meaningfully.
That has nothing to do with the shape of the distribution of grades, however.
You really can't assume any specific shape as valid or not unless you know the distribution of the population.

Oh, I agree. I should reword that.

+1. Although the very last sentence is, sorry, simply wrong. There is no reason whatsoever why a large sample should from an unspecified distribution should turn into a normal distribution. If you are thinking of standard statistical results that imply asymptotic normality (Central Limit Theorems etc.), these involve averages of large numbers of data points, so they are not applicable here.

No, you're right. I missed the wording badly, I just have not taken the time to fix it.

I dropped my mangled attempt. I was trying to explain that you need knowledge of the population to make claims about any expected sampled distribution. Better off without it, really. Maybe I'll come up with an improvement and put it back.

Years ago I was involved in the grading of entrance exams for a well-known university. Our goal, in grading, was to provide the maximum differentiation: as the purpose of the exam was to find the top candidates, whilst recognizing that for some the physics portion of the exam would only carry limited weight, I was asked to ensure that my grading was sufficiently lenient (this was a VERY hard exam) that the median would be 20/40; this required giving partial credit to partially correct answers, and did in fact produce scores from 2 to 39. Yes, that one paper scoring 39/40 on a really hard exam is something I remember 30 years later...
My point is this: "grades" can mean different things to different target groups. Questions one might ask:

Is the student sufficiently prepared to take the "next" course?
Does the student merit special recognition (scholarships etc)?
Does the student qualify for some "limited access" position (a job, entrance into a graduate program, etc...)

The first point should be "objective", without regard for distribution. That is, an exam should be constructed to test the necessary knowledge, and whether the person has it or not should be independent of the grades of all other students. Studying the distribution is possible helpful to the instructor (are you making "good" exams?) - but ultimately asking 10 hard questions, getting on average 4 good answers, and deciding to set the pass grade at 4+ does not guarantee that people who pass the exam have mastered the material.
The second point relies less on a distribution, and more on a "cutoff". Perhaps there are 5 scholarships: you award them to the top 5 scoring individuals. If you aggregate over multiple exams, you take their rankings on all exams (perhaps eliminating N outliers) to come up with a final ranking.
The final point is the only one where a distribution might be helpful.
Unfortunately, most grading schemes don't follow anything like the common sense approach that I describe...

+1 this is exactly right. The OP asked about a grading system that is "informative about student's grasp of the material," which seems like an absolute (not relative) metric. Given the scenario the OP describes, it doesn't seem that a grading system referenced to any kind of distribution can convey that information.

This is a great observation that different testing objectives can have different goals for differentiation. I bet many of us read a question like this assuming our own customary use-case. The OP should clarify the answer by (among other things) explicating their use-case.

I agree with other answers: there is little sense to change your grading paradigm a posteriori until you get some distribution.
However, it can be useful to test your assumptions by formulating your expectations
before inspecting the actual statistics.
What should the overall distribution look like?
What distribution should which individual problems exhibit?
Should a certain subset of the problems be accessible to almost all students?
How many of the problems do you expect a strong, average, weak student to be
able to attempt in the alloted time?
Do results on some problems correlate more strongly than they should,
i.e. do they really test different skills?
Then you can check if the data fit your expectations.
If they do not, you did something wrong.
Here are two examples.
Multimodal Distributions
Experience suggests that you probably get something like a Bell curve if there
is nothing out of the ordinary going on.
In other words, if you do not get such a curve further inquiry may be warranted¹.
As a specific example, if you get

you want to look for a criterion that separates the participants into two (or more)
groups that explain the two bumps:

Possible criteria (which you can likely check) are course of study, participation
in exercises (or some other course-related activity), gender, and so on.
You can then take measures to ensure your course works equally well for everybody.
Hardness of a problem is off
What you expect to see is this:

That is, most students solve (very) easy problems well (these should form the
baseline for passing the exam, if that), average problems distribute normally
(these determine everything between passing and B),
and few students solve the hard problems (those that do probably get an A).
If you see big derivations from your expectations, you may have missed the mark
when formulating a problem (which can influence your grading decisions) and/or
your course did not promote the necessary skills as you thought it would.

The other direction does not work, i.e. there may be issues you do not see in a histogram.

May I ask an irrelevant question: how were these plots generated? I like them very much.

@LeeDavidChungLin IIRC, drawn on my Galaxy Note 10.1 with Squid.

oh how did I never learn about this great app before, ha. Thank you.

Edwards Deming, a guru in the quality movement, thought that there should be only three outcomes: "Student masters the material", which is the objective of the course and the main responsibility of the instructor, "Student shows exceptional proficiency" (those he regarded as outliers and would hold out as exceptional achievements), and "Student did not master the material" (outliers at the other end). To students in the first group he would give an A (that would be a large majority), students in the second group would get an A+, and students in the third group would get an F.

One problem with that (among others) is that if these are the only official records (3 distinct points), statistical after-analysis is weakened for any purpose. For example: http://andrewgelman.com/2012/11/12/thinking-like-a-statistician-continuously-rather-than-like-a-civilian-discretely/

What kind of statistical analysis are you after, other than percentage of students passing, etc? The original question was putting the cart before the horse, assuming that there is an optimal grade distribution in the first place. It is not clear to me (or, it seems, to ff524 or StrongBad) that there is. The OP's question reminds me a bit of the optimal length of a man's legs. Abe Lincoln's answer? "Long enough to reach the ground".

For example: Regressions, correlations, and predictions between prerequisite courses, post-requisite courses, special assistance programs and tutoring, program exit qualifications, follow-up results like graduate or job placement, etc., etc. Anything possibly investigated in program assessment or scholarship of teaching and learning.

Your question is really impossible to answer because you don't give us the measurable goals of your grading distributions.
Some sort of Bell Curve (example #1)
This is clearly flawed because it poses no distinction of the mastery of a course nor does it show the success or flaws of the professor.  If you were a really good teacher and had really good students a "bell curve" approach would mean that students that actually had a really good grasp of the subject material could get an F.
Let's just say it is a rather simple class and you are an incredible teacher.  The grades on a percentage basis are 88-100%.  Do the 88% students get Fs, the 90% students get Ds, and so on?  And if so why would anyone with any brains take a class like this?  I would have to rely on others being dumber, to get a better grade?  I would think what if everyone got a 100% except for me and I got a 97%?  It makes no sense.  (And I know I am being arbitrary using %s as a lot of classes aren't graded like this but just using this for an easy example.
Some sort of Bell Curve (example #2)
Well same situation but this time the class isn't a "normal" class.  It is at a university that has 50 openings into their engineering school with 1000 kids taking classes to get in and they take a set number of classes.  Well then we might want to grade using some sort of bell distribution because it is comparing the students, not their mastery of the course.  After doing this for a few courses you could probably weed out to around 50 students.
Other distribution models
The problem with stating that given a large population that you should have a certain amount of certain grades is really - as mentioned above - about comparing the students to each other and has very little to do with their knowledge of the material of the course.
Even with a simple curve model where you give the best student a 100% and grade from there you could run into some major issues while having a poor rendering of grades.  If you were a terrible teacher and students had poor mastery the top student might have a 80%.  Do you just bump everyone by 20% and feel good about it?
Also when doing things like that how is this reflected in your grading strategy?  For instance what if the first half of your tests are relatively easy and every performs at A level.  Then the next half is very very hard.  The person getting an 80%, gets bumped to 100%.  But then the person who got the first half all right but then missed everything on the harder material... they still get a C?
What do you do?
If your course is part of a core piece of learning and taken by most of the student population you want to work with your school on what they expect their students to get.  Possibly look at the historical records for like classes.  I am sure an Intro to English class has different scores at upper-tier universities compared to state colleges.
From there set objectives for the students and let them know what is expected.  If everyone does what is expected and masters the material, they all get As.  If no one shows up for your classes and learns nothing, they all get Fs.  The hard part is adjusting the class if you feel that the grading scale is too easy or too hard.  But there shouldn't be a fixed distribution.  If we are talking English 101 at Harvard we are seeing at least 70-80% As.
For a upper level class a lot of the distribution goes out the window.  If you are giving a science class in a specific area you must clearly grade according to how well the student grasped the concepts and materials.  There grade should be indicative of how well they can expect to perform if they go to the next level.  If you were teaching Quantum Physics and had a class of 10 that were absolutely below average would just give some As and Bs out?  Then when they take Quantum Physics II the next professor is like "what the hell, these students aren't ready for this class."

Great answer, and I totally agree. The only thing is, I fear that your examples of lunatic-grading-procedures are in fact widely applied, perhaps even in the majority of cases, and in fact highly recommended in some literature.

The right distribution to use is the one that the students are used to.
I would dispute your assumption that more entropy is always better than less.
Lets imagine a student who typically receives a B grade, with grades going from A through D, and with F as well and various pluses or minuses, and there is a bell curve concentrating the class's results around a C+
They are content with a B, they are happy with a B+ and so on. They know broadly where they sit relative to their cohort.
Then, Professor Entropy decides to give letters all the way through W, with F still being a fail (F is the lowest grade with W-- being the 2nd lowest) and +/- still being used. The bell curving has been replaced with a flat distribution. The student receives a H+ for their work. This has a lot more entropy, but tells the student almost nothing about how well they did compared to how well they should have done, or how they did in their other subjects (C+, B- and A- using the old system). If the student wants to compare their mark to other marks they have received, they will have to convert the mark you have given them into the same distribution they are used to.
You want to compare the students to each other. They want to compare themselves to a theoretical self that studied a bit harder, or DID make time to go to that party, or moved back to subject ABC instead of XYZ where they recently switched to.
Are you the target audience for the grades, or are the students??
Apologies for lack of citation, but this is as much disputing your premise/assumptions as it is an answer.

I dispute the OP's assumptions too, but I also disagree with this answer. Presumably if Professor Entropy explains to the students how the grades are computed, then they will be as useful to students as they are to anyone else. (Whether they are useful to anyone, students or not, is of course an entirely different matter.) It's also not clear that helping the student "to compare their mark to other marks they have received" is an intended use of grades - I don't think that always needs to be a consideration in determining what is a "good" grading system.

My undergraduate course tutor's research specialism was statistical mechanics. He wrote the "grade normalisation" software for my university course which was Physics. It would take students raw trades and spit out the final UK grade (1st, Upper 2nd, Lower 2nd, 3rd, Fail). He told me that the idealised distribution should be a normal distribution; students would be randomised around a mean with more or less ability, applying more or less effort, with more or less distractions of being undergrads (aka over partying), and as those factors were random [*].
My course tutor lamented that the problem was that the actual grade distributions on typical exams was more like an inverted normal distribution; bi-modal with a group of folks flunking, a group of folks acing the exams, and very few people doing "average". This was largely because exams are not perfect and hard to pitch correctly; too hard and folks flunk and too easy and too many folks ace them. So the purpose of the software he wrote was to try to renormalise the individual exams against the student population sitting them. Rather than perfect exams grading a student population the imperfect exams can be normalised against the student population. He was a very smart scientist who had been an educator for decades so I suspect that his intuition and approach was sound.
[*] Of the two brightest guys that I still know two decades later with one got a triple 1st and a college medal, the other flunked out due to over-partying; the friend who flunked his undergraduates Physics recently took a professorial role at a respectable university with a research specialism of international development.

Regardless of how bright this tutor was in mechanics, as an education statistics professional I would call this attitude totally malformed. The goal of (STEM) courses should be to master a certain set of skills, and the bimodal distribution is shining a bright and consistent spotlight on the simple fact that some have passed and some have failed. That's not because "tests are hard to write"; that's incoherent BS. Mangling the actual data to fit some preconceived hypothesis is a kind research fraud, and he should know better.

If your typical score distribution is bimodal, then this logically cannot be due to the test being "too easy" or "too hard" - either of these would simply shift the distribution, not create multiple modes. And why "should" grades be normally distributed? There is no law of large numbers involved that I could discern.

My philosophy professor's grading philosophy was that by very definition, most people are average (i.e. C)
Students asked if he graded on a curve and his response: Yes - if my grades don't fall on a bell curve with the bulk of the class making C, several D's and B's and maybe a couple of A's and F's then I've either made my tests too easy (grades skewed to the A) or too hard (grades skewed to F).
I suppose it depends on if you're trying to push a few of your students to succeed, or if you're trying to help as many students as you can get good grades.

"then I've either made my tests too easy (grades skewed to the A) or too hard (grades skewed to F)." - Or my teaching was more/less effective this semester, or the admissions committee has changed their standards and are admitting students who are more/less prepared, or a newly available set of online lecture videos now enables students to learn the material despite my poor teaching, or the instructor of the previous course in the sequence covered a lot of "my" material, or the instructor of the previous course in the sequence introduced a misconception that I haven't corrected, or...

I actually found that I agreed with his approach - but that's probably why I was one of the 50% that didn't drop his class. I mostly made B's on my tests. I feel like I earned that grade - and I'm a lot more proud of the A that I made in that class (he offered a full letter grade of extra credit) because of the effort I put into it. I also learned a lot more about writing and arguing. The delicious irony is that I got terrible grades on my multiple choice tests in a history class. Usually because she'd ask dumb questions like, "What's the most/best/...."

I could've taken the lectures and reading and argued that any of the answers were the most or best, unless there was something that was just plain wrong. Like Abraham Lincolon as a general in the Athenian military or something to that effect.

You feel like you earned your grade because of the effort you put in, but keep in mind that with his grading strategy, you could have earned the same A if you hadn't put in effort, but the rest of the class put in even less.

This is a philosophy professor who (like many in many disciplines) doesn't understand statistics worth crap. I wonder what the response would be to point out that students are actively incentivized to sabotage other students in this system, because then the scaling will push their own grades higher for the same results? I would claim that this system is unethical in a variety of ways.

authors: НЕЛИ НИКОЛОВА


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What is the difference between a Panel Study and a Cohort Study?
I am having a special subject named "Methodologies of Scientific Research" in which there is a reference on Panel Studies and Cohort Studies for conducting a longitudinal survey. I am confused to what is the difference between Panel Study and Cohort Study ? In Wikipedia they look the same, that is Panel Study redirects to Cohort Study.
Edit: My field is related to Computer Science and more specifically Information Technology in an undergraduate level. The subject "Methodologies of Scientific Research" though is not constrained on applications on IT, rather it is more general and we have research examples from other sciences like Sociology, Biology, Economics etc.

Hi, and welcome to Academia.SE! Can you please provide a little bit more context for your question?  It sounds like this is in a biomedical field, but I'm not sure, and terminology may differ between fields.

I'm voting to close this question as off-topic because it does not appear to be about academia as defined in the Help Center.

I'm voting to close this as it's likely a better fit for CrossValidated.

The terminology for longitudinal research designs differs somewhat between disciplines. However, to my knowledge most discipline refer to a "cohort" as an entity that can be distinguished by a certain event as for example the year of birth or graduation from high school. The general idea is that there is an observable variable for the cohort membership and that individuals from the same "cohort" have something in common, e.g. the risk to die from lung cancer due to a common time of exposure. Thus, a "cohort study" looks at one cohort or compares different cohorts. In most cases, cohort studies use longitudinal research designs, but there are also cross-sectional cohort studies.
The use of the term "panel study" seems to me more vague but a "panel study" is in most cases a longitudinal study which has at least observations from two points in time. Panel studies do not need to focus on cohorts.
Hence, cohort and panel studies both look specifically at the timing of certain events and variables.
For further information you could for example have a look at the following articles in the International Encyclopedia of the Social & Behavioral Sciences:

Mason, W.M. and Wolfinger, N.H. 2001: Cohort Analysis, in:
International Encyclopedia of the Social & Behavioral Sciences,
edited by N.J. Smelser and P.B. Baltes, Pergamon, Oxford, pp.
2189-2194.
Duncan, G.J. 2001: Panel Surveys: Uses and Applications,
in: International Encyclopedia of the Social & Behavioral Sciences,
edited by N.J. Smelser and P.B. Baltes, Pergamon, Oxford, pp.
11009-11015.

To the person who flagged for moving - the first criteria for moving a question is for it to be off-topic on the current site. This fits here; no migration is necessary. (I neglected to add this comment before clearing the flag; my fault.)

Note that while many cohort studies follow this logic, many others don't. In epidemiology, cohort is just an identifiable group. It sometimes has a specific rooting in place or time, but not always.

In research,panel and cohort studies are both longitudinal studies.in cohort studies the researcher makes repeated measurements on a group of people,elements or characters which or who share the same experiences. I tend to think that in cohort study,for example a researcher carries out research on class 8 cohorts of 2014 then after the cohort leaves he or she continues carrying research with the class 8 cohort of 2015. The researcher in this case is interested with the class 8 cohort because probably he is doing a research on let's say , "why do class 8 pupils perform poorly in mathematics"
In panel studies,data are collected from the same set of people at specific intervals over a long period of time. The key feature in panel studies is that they collect repeated measurements from the same sample at different points in time.A researcher might for example follow a set of people from when they are in kindergarten to secondary then university and so on.
If you want to read more,look for a book by Bernard Russel on research.
Just my thoughts,I stand corrected.

Cohorts and Panels are techniques employed for selecting a sample frame for a particular study. They do not specify the time frame for data collection. So, therefore, cohorts and Panels can be formed to collect data for both cross-sectional (one-time data collection) and longitudinal studies (repeated data collection over a specified period of time).

What does this add that the prior answer does not?

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Is it possible to attend my "backup" school but reapply to my top choice and go there if accepted?
I applied to several graduate programs and was accepted to my back-up choice. The professor at one of my top choice schools contacted me personally to say that he was interested in my application but had no room for me this year; he suggested applying next year, as "re-applicants" have a good chance of getting accepted on their second attempt.
Though pleased, I mentally dismissed this. I had a tangible acceptance to a school I did want to attend (albeit last choice), and I enrolled in classes. I immersed myself in my new environment, took on a healthy course load with a teaching assistantship, and have received lots of praise for my work and work ethic from my professors and colleagues.
However, with each week I am here, I hear more and more people--students, professors, the head of the department--admit that my program is lacking in many ways. The school offers very few courses in my actual field (a fact they omitted on the school website that boasted what the program had to offer its students). There are two professors in my field, and one is always away. One of them has also been known to cancel classes frequently, a fact that had been addressed to and dismissed by the department. This means that the few opportunities I have to study in my field have decreased. The program is very heavily based on teaching--that is, the teaching as part of my assistantship. Two of the required classes I am taking this semester are teaching-oriented, and while I am happy to teach now, it is not a long-term career goal of mine, and I expressed this to my advisor. It seems, though, that despite what I was made to understand before coming to school, that this program ultimately tailors its candidates for a teaching career.
Ordinarily I would just accept it, as each program has its problems. Yet, I cannot shake the fact that, especially given the last fact, I am going to be geared toward a career I specifically do not want. Add to that the invitation from the professor of one of my top-choice schools, and I find myself wondering if it is unprecedented for a graduate student to switch. I would not be losing any time; if anything, a switch would save time, as the program I am currently in is unusually long, whereas the program I would be entering is comparatively shorter and more compact. Plus, as the assistantship the other school offers is not a teaching-based one, I will be able to take courses in the field of my choice. But before I keep considering, I would like to know if it is even possible.

This seems somewhat unfair towards the "backup" school, especially if you receive a stipend. Further, you should take into account that it is very well possible that you will be starting pretty much fresh next year anyway.

@Curious, You need to provide more information to your question. What is the subject of your graduate work? Did you major in the same or a similar field as an undergraduate or are you switching to a totally new field of study? Is this a professional graduate program or an academic one? Do the schools offer you similar opportunities to work on the project that you are interested in, or is it the specialty of your top choice but not the other school?

@xLeitix If OP is only going to be taking level one courses and not participating in research rotations, then it is possible that the school that they reapply to will accept transfer of courses, and may even accept results of a first years cumulative exams. They won't know unless they ask the other institution about their policies in that regard. Also, if they are entering into a different discipline, then it could be beneficial for them to do that first year's coursework twice, as they will gain a much better understanding the second time around.

The one word of caution would be that poor performance and grades could damage your chances as a reapplicant. Undertaking a graduate course of study is extremely demanding. If you are just doing it because you have nothing better to do until you reapply to your top school, then you may not invest the effort you will need to absorb the information and "master" it. There is a substantial difference between even upper level Undergraduate courses and Graduate course. Make sure you know exactly what you are getting yourself into. Unless you have visa issues, then it may not be the best choice.

Am important question is what you would actually be doing in each program. If the first couple of years are based on course work, you'd want to see if courses from the one school could be transferred to the other (or maybe only specific ones could). It's likely you'd still end up a little behind. If the backup program is one where you jump straight into research with an advisor, then if you transfer, you'll probably have to start over with a new project, and you'll have completely wasted a year.

Related: http://academia.stackexchange.com/questions/40425/reapplying-to-a-phd-program-while-attending-another-one?rq=1

Common scenario: a professor is recruited to go work at a different university; s/he takes his or her students along.  No one bats an eyelash.
The quicker you can make the change the better; you need not feel at all guilty.  It's just not a good fit.  Your current school would be a great fit for someone who wants to teach.  But it's not a good fit for you.

Yes, it is possible.
However, consider what others have commented.
I would like to add that in future, the switching of schools might be seen as a red-flag.

in future, the switching of schools might be seen as a red-flag — Really?

Yes JeffE. For example, if the person is applying for a postdoc position, then the switching of schools during doctoral studies might be seen as a red-flag.

Anything might be seen as a red flag. Do you have evidence that switching schools is actually seen as a red flag by anyone? I've never heard of such a thing.  (I switched schools two years into my PhD program.)

No evidence. Only anecdotal.

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How do I email my professor about assigning me to a group?
I have a class project coming up, the professor asked us to form groups of 6. Everyone in class seems to know each other and have already started forming groups. Unfortunately, I don’t know anyone in that class. I would like to know how can I email my professor about assigning me to a group? I am an international student so I am not really sure how to address this.

Keep it simple:

Dear Prof. X,
I'm Hannah Y (ID n. 12345), an international student in your class
  XYZ. Since I don't know anyone in the class, could you please assign
  me to one of the project groups?
Thank you,
Hannah

I have international students in my classes and such emails are not uncommon. The important thing is to provide all the pieces of information needed to identify you and the course (professors usually have several).

This is the main issue we had and we now have a « random » choice system - all the names are put in a box and groups of 5 or 6 pulled out - depending on the nunber we may have to do a few groups of 4 that get completed late. But, this has reduced the number of later group changes that we need to deal with : many times a group of friends start a project and, at the end, are no longer all friends : lack of input, too much time with a partner etc.

While I agree about including identifying information, I've had students going to the other extreme. One was so thorough about identifying themselves that even the lecturers for other subjects knew who they were.

how can I email my professor about assigning me to a group?

Don't seek complicated solutions for a simple problem.

Find the professor's e-mail address online.
Open your e-mail client and write the letter to him/her along the following lines: "Dear Prof. XXX,  I could not get into a group myself; please assign me to a group. Sincerely yours, Hannah ____"
Press the "send" button.

A repeat of the prior existing answer that adds nothing of note.


text:
Should I keep my achievements to myself?
I'm a PhD student and I have published several good papers and achieved some awards and leadership position early in my career. I always celebrate my work but not boastfully since I poured in a lot of effort and hard work to create it.
But what really makes me glad about it is that many students in my University and young professionals become inspired and motivated by my perseverance and are encouraged to publish their own. Sometimes I even offer help and simple advice (e.g., how to deal with reviewers or prepare the manuscript for submission) to my fellows.
But recently, a group of my fellow students and my friends started to call me out, saying that I give them and others pressure to publish and I am inconsiderate about how they feel on their circumstances. They become resentful and there are days discredits my work. I did not mind it at first, but it is becoming frequent and I have become anxious and guilty about myself and my achievements.
So, I was wondering if ever I achieve something should I keep it on myself and only share to few people? Should I be felt guilty of achieving something on my work?

Neither should I keep [my achievements to] myself and only share to few people? nor [s]hould I [feel] guilty of achieving something...? seem like questions that are specific to academia.

Should you feel guilty for achieving something? Surely you know the answer to that. This feels more like a rant than a question.

Perhaps you can just stop pressuring them. It isn't your achievements, but your actions.

@Spark I apologise  if you see this as a rant but this is more of asking others about their perspectives on what I am going through and how should it be handled.

@Buffy I actually don't know how it pressures them since I know it is our duty as PhD's to produce findings relevant to our fields. So it is suggested to just keep it by.myself?

You shouldn’t apologize for doing well. You should probably watch that what you perceive as being helpful is not seen as patronizing or condescending by others.

I have a hard time understanding the specific behavior you're being criticized for. The only specific behavior I see is offering (unsolicited) advice - but why is the title of your question then about hiding your accomplishments, and not about offering unsolicited advice?

What does "celebrate my work" mean exactly?

Everybody needs at least a few good personal allies to celebrate with---your advisor is hopefully one of those, and some of your close personal friends are likely to be as well.
Your peers in a graduate program, however, are likely to be in a more complex place emotionally. It's also easy to cross the line between sharing and bragging, and easy to come across as condescending (especially regarding "routine" tasks like preparing a manuscript or responding to reviewers).
Don't feel guilty about success, and don't hide your achievements.  But I would also recommend holding back and waiting for your peers to ask you before sharing your news or offering to help.

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Do "filter courses" really filter out "weak" students from subsequent courses?
My department (of mathematical sciences in a US university) is considering some curriculum changes for our majors and the topic of our "mathematical filter course" came up.
For clarity, I'll define a mathematical filter course as a course that is treated as a pre-requisite for other courses but whose content isn't really necessary for those subsequent courses (though in a general sense does make sense to include in the major). The intention being that such a course would help to filter out weaker students who might be interested in those subsequent courses (and potentially one of our majors), but lack either the drive or skill in mathematical thinking to really do well in those courses (and the major). In the past, we used calculus as such a filter, but for our Applied Data Science major we are considering using an applied linear algebra course instead.
While on the surface, the idea makes sense to me (though I don't have a sense of how common such a "filter" course comes up in other schools/programs/fields of study), the conversation left me wondering:
Do filter courses really do what they set out to do and distill the population of students going on to the next course and the rest of the program (or more specifically, do they distill enough to be worthwhile)? or would including such a course in a major's curriculum potentially be a waste of valuable course time that could be better utilized elsewhere in the program?

Comments are not for extended discussion; this conversation has been moved to chat.

Thanks to pedroelpanda, Squary94, and Ben for wonderful answers; my understanding on filter courses is greatly enriched by your responses. It is clear that the idea of a filter course is far more layered than I first imagined, but there are some clear takeaways for me, 1) arbitrarily hard courses are a bad idea, 2) filter courses should be motivated by protecting students from going too far down a path that ultimately is not a good fit for them,  and 3) a filter should still be relevant even if not necessarily direct content-wise for subsequent courses, giving a clear sense of future courses

I assume you mean "filter" courses in the way that some in the US also use the phrase "weed-out" courses, i.e., an artificially difficult introductory course which is designed to decrease the size of the cohort, by scaring away, or failing out, a certain percentage of students.
Generally speaking, "filter" or "weed-out" courses have some benefits and detriments. The benefits are obvious for the department, as they in some way reduce the enrollment in the department's courses, and ensure that those who remain are in some way capable, given that they have already proved themselves in the past.
The detriments can be less obvious. Particularly in mathematics, "filter" courses can foster a toxic environment, and can force students from less traditional backgrounds out of the major. Realistically, students in this linear algebra class who went to private schools or better-funded public schools in the US may have already taken linear algebra in the past. Is everyone on an even playing field? And will this course impact anybody disproportionately? Will a large course with, say, a strict grading curve, foster a competitive environment in which some students will feel discouraged to collaborate with their friends? Are you really selecting for the most talented people, or for people who thrive in a certain environment?
You seem to suggest that some of these "filter" courses are completely auxiliary for the major as a whole, but your example of linear algebra in a data science major says otherwise. As everyone in the comments was quick to point out, linear algebra sounds more like a vital pre-requisite, which may not be directly necessary for most courses in your department, but which most students should know. In my experience, most "filter" courses fulfill a similar role. Generally speaking, if your department forces students to take an artificially difficult class which is entirely useless, then the department should not be surprised when a large amount of students do run away from the major.
Anectodally speaking, my mathematics department in undergrad got rid of an intentional weed-out course a few years before I arrived. The overall enrollment in the department went up, to no one's surprise, but so did the percentages of female students, and those belonging to racial minorities (I do have numbers to back up this claim, but once again, only in the context of my department). You may take that as you will, but personally, I would suggest that this is not because such students were weaker, but rather because something about a competitive weed-out course affected them more than it would affect others.

I have witnessed 4 variations of filter courses in Germany and would like to go into each variation of it from a student's perspective.
The first variation I witnessed was artificial difficulty introduced by intentional obfuscation. The materials used for the course were simply read out by the professor and not really explained. Questions were brushed off and not really explained, which discouraged further questions very quickly. The provided materials were lacking which "encouraged" the student to look into other sources, mainly those listed by the professor at the start of the semester. This bred tremendous amounts of ill will towards the subject and professor.  Participation rates were low and it did force quite a few students into different majors all in order to circumvent the course. It did indeed filter out weaker students from studying but also prevented quite a few others that simply decided that this is not something worth putting up with.
The second variation I witnessed introduced many assignments to pass as a prerequisite to enter the final exam. This felt like a ton of busy work but overall was not too bad. It was really painful for those that worked a second (or third) job on the side since any time taken for those assignments  was time that you could have spent on other subjects. It didn't really filter out much of anyone and only led to some people requiring an extra semester of attendance.
The third variation was a participation requirement. The course was held at 8 am and the materials necessary for passing were not online and only shown/explained for those present. Taking notes and paying attention during the entirety of the course was 100% necessary for passing. It instilled good habits and filtered out those that didn't take things seriously. A lot of the faces in that course vanished over the coming months since many found it difficult to uphold a constant attendance with good levels of concentration.
The fourth was simply a very challenging final exam. It filtered out a lot of students but it also led to the occasional student that simply skipped it for later and had 145 out of 150 credit points with the last 5 held hostage by some basic course held in the first semester.
I did meet and know of many students that have been filtered out by one or the other so it did work but it also always ended up either wasting time or largely inconvenient to other able students.

It's not clear to me that that first variation of a filter course was actually a filter course. It might very well just have been a bad teacher.

I don't know that Germany is a good comparison, since Germany already does a lot of pre-filtering of unprepared students that the US doesn't do. I started to say "Doesn't Germany start filtering at age 14-16?" but then I found a description of the German educational system which seems to indicate that the filtering starts at age 9, determining which sort of secondary school a student attends, which filters out those who won't even go to university.

This is a very helpful answer (+1).  It is important for lecturers to ensure that a filter course is still a course of good quality and substance --- the filtering effect should come from the inherent challenge of the material, not any artificially introduced deficiencies in teaching.

@shoover It's true that there is a whole lot of filtering going on depending on which schools you visited in your younger years but universities still have filter courses in their first semesters. That is mainly because no matter what school you ended up in your younger years, if you wish to attend a university, the means to do so are provided for you. Because of that, it's not uncommon to see students that are 30+ years old on campus.

@Squary94 Good to know. I don't know whether you have any experience with US institutions, but if you don't, you might be surprised at the lack of preparedness of some of the admitted students, accompanied in many cases by a lack of self-awareness regarding their unpreparedness. Also, US institutions vary widely in their selectivity, with some admitting all comers and others only the top 5% of applicants.   Hence the need for filter/weeder courses.

As pointed out in the comments, your premise that a course in linear algebra would act primarily as a filter ---as opposed to being something that is substantively important in data science--- seems dubious to me.  In any case, there are probably a number of general mathematics courses that could act simultaneously as a valuable substantive course and also a "filter" on progression to later stages of the program.
One of the things you will notice in any university program, and which can easily be confirmed by the empirical data on student grades within the program, is that the grades of each student are highly correlated across different courses in the same program (i.e., a student who does well in one course is more likely to also do well in another course, and vice versa).  Thus, failure/success in a filter course may indeed be a reasonable predictor of results in later courses.  For any university department considering this issue, it ought to be possible to get a historical record of student grades in their courses in order to undertake empirical assessment of the extent to which success in one course is predictive of success in another (or of overall success in the program).
A secondary thing to note here is the "gatekeeping" function is not limited solely to an initial filter course.  The entire process of grading students and passing/failing them acts as a filtering function throughout the entire program.  When course lecturers are too lenient in their grading for a course, this has the averse effect of allowing weak students to progress to higher-level courses, which in turn causes difficulties and frustrations for the lecturers of those later courses.  Consequently, appropriate attention to prerequisites and marking standards is something that should be applied throughout the entire program, not just by setting up a filter course at the start.

Do filter courses really do what they set out to do and produce stronger majors at the end of the program?  Or would including such a course in a major's curriculum potentially be a waste of valuable course time that could be better utilized elsewhere in the program?

Since course marks are correlated for by student, any prerequisite course at the start of a program is likely to succeed in filtering out students who are likely to do poorly on later courses.  Some students may come into a program as weak students (e.g., ill-prepared, poor aptitude in the subject matter, etc.).  Many of these may be able to develop into stronger students if they have additional time in their degree program, which is something that sometimes occurs when a student initially fails a filter course and then repeats the course.  I disagree with the other answer here that raises alleged equity problems with this function --- so long as it sets a single objective standard for all students, it is not problematic that some students come into the program less prepared than others, and are therefore less likely to pass through the filter.  Indeed, the entire function of the filter course is to ensure that students meet some minimum objective standard in the course before proceeding to the remainder of the program.
Here it is notable that a filter course can also provide remedial education, particularly if you use a general mathematics course as your filter course.  Consequently, the course can operate both as a filter and as a means of upskilling students who are insufficiently prepared to pass in the first instance.  (In a previous academic role I worked in, we gave incoming students a short mathematics test that tested material from their pre-university schooling courses; students who failed this test were required to take remedial sessions to assist them prior to the start of the regular semester.  This was not a "filter" per se, but it had a similar effect in identifying weak students and giving them some preliminary remedial attention.)  The Department might wish to allow students to attempt the filter course multiple times, to give poorly-prepared students the opportunity to use it as remedial education and then proceed past the filter once they meet the minimum requirements.  Whether or not a filter course of this kind is beneficial enough to warrant the opportunity cost of displacing another course depends on the extent to which the Department is having difficulties with the quality of students in the program.


text:
What are the advantages and disadvantages of Preprints.org compared with arXiv?
Preprints.org is a new site for sharing preprints of papers.
arXiv is one of the longest standing, stable and mature places for many people.
I'm wondering where to upload a paper (which is already open access on GitHub, but not very visible).
One downside of arXiv: Google Scholar seems to direct people to the arXiv and not the latest versions.
One upside of arXiv is that it is open, stable and mature.
In summary, where should I put my preprint for maximum benefit?
Update
Thanks to the first link in the 'correct' answer below, I think I've discovered the OSF affiliated preprint services. See here for more info: Preprint services other than arXiv (for other fields)
Check it out (and potentially submit your preprints) here: https://osf.io/preprints/

"One downside of arXiv: Google Scholar seems to direct people to the arXiv and not the latest versions." What do you mean by this? In the cases I've checked google scholar provided a link to both the published and arxiv versions, with the arxiv version being the latest revision.

@mmeent Scholar definitely gets it wrong sometimes, and treats the arxiv version as the canonical version.

Scholar will get better over time.

arxiv and search engines: to me that's a non-issue because arXiv allows you to give the journal reference and the DOI (which auto-links to the published paper) once the paper is published. You can also update the preprint with the info about the "official" paper and again a link.

I never heard of preprints.org. Just opened it on my field of expertise (algebra and number theory), and I keep seeing proofs of the Riemann hypothesis and the Collatz conjecture (which means that it is full of papers by cranks).

You should generally use the service that most people in your field uses. In my subfield of physics people follow the daily postings on arXiv, so that's the natural choice for me. I imagine a preprint would be effectively invisible if posted elsewhere, except to someone who happens to be browsing my Google Scholar profile, ResearchGate profile or website. One could perhaps post to both arXiv and Preprints.org for wider dissemination, but that might confuse e.g. Google Scholar's citation counts so I'm not sure if it's a net benefit.
Advantages of Preprints.org

Much wider field coverage.
Has a comment system. (The closest thing on arXiv would be emailing the authors.)

Disadvantages of Preprints.org

No flexibility with license. All Preprints.org preprints are posted under a Creative Commons CC-BY 4.0 license. This makes some sense as Preprints.org is designed with open access journals in mind, but really limits the set of journals the manuscript can be published in.
It's not clear to me how Preprints.org would handle misleading comments. One might prefer not to have those on the same page as the preprint.
It's run by MDPI, a publisher that's been rather controversial in the past. Although they claim that Preprints.org is run on a not-for-profit basis, it remains fully funded by MDPI, which may or may not be a sustainable and lasting investment. (Compare e.g. the discontinued Nature Precedings.) In contrast, arXiv is operated by Cornell University (which at least I consider a more respectable entity), and has proven to have lasting power.

Unknowns

One can volunteer to screen preprints on Preprints.org. Is that more or less robust than the screening by arXiv moderators?

The first disadvantage is a really big one IMO. Even if your first choice of journal would be fine with it, until the paper is accepted it's risky to post there and that defeats the whole purpose of a preprint server.

@EspeciallyLime Indeed. In my field (biomedicine) a lot of journals are pre-print skeptical. Adding a "By the way, this already has a license..." is a big problem.

If the use of arXiv in your field of science is common, that is probably the better option for you personally. For example, subject specific abstract indexing services (like inSPIRE or ADS for high energy physics and astronomy) will automatically link their entries for the arXiv version to the journal version. Consequently, you will easily be able to obtain combined citation counts.
However, there are some limits to arXiv. One of the foremost is that they accept pre-prints only for a limited number of subject areas (essentially physics+astronomy+mathematics+(some) computer science with maybe some cross-over into other fields). Consequently, for some fields arXiv simply is not an option. (There also some subfields for which arXiv would accept pre-prints, but the use of arXiv is not so common). In this case, alternatives could be interesting.
Personally, I have never heard of preprints.org. One potential worry I would have is that it is owned by a (commercial) publisher. So, even though it is currently run as a non-profit with free access, I am not sure what safeguards there are against them changing their usage policy in the future. I would certainly try to find out before submitting anything.

"I would certainly try to find out before submitting anything." This question is aiming to find out! Regarding your point about arXiv, I think in practice they do accept papers from any field. See this for example, which is outside the usual arXiv area - I submitted this 3 years ago to a (now accepted) transport planning journal: https://arxiv.org/abs/1509.04425 - the CS (computers and society) category is very, very wide.

@RobinLovelace: Your paper did at least deal with an application of computers to society.  I'm pretty sure if you try submitting, say, a paper on Roman history, arxiv will reject it.


text:
When to leave a postdoc as foreigner that doesn't have enough courage because of current global situation?
so i joined a postdoc in US six months back. Had to move from India. Even after six months have no clue what im working on how to proceed with work. I make suggestions to the PI but he shoots them down. He is making me work but nothing productive which can translate into research paper in the next year. He is always early in the lab and leaves in the evening and expects us to be there during the time he is in the lab. Should i leave or stick it out?

You haven't given any reason to stay and several to leave. What are you asking?

Is it okay to leave so soon? Am i being paranoid or shld these things not bother me?

It seems normal to be bothered, actually. I hope you have some other options.

No currently i dont. Havent even started looking yet cz im new in this country and dont have the courage to make the move again

Would moving cause any problems with your visa?

Very generally speaking, it's often not a good idea to leave a position until you have another secured. There are exceptions when it is necessary to get out of a situation where serious harm is ongoing or imminent, but that doesn't describe your situation.

Though I empathise with your situation, I don't gather from your description what you have been actively doing to make the situation better. You said you don't have the courage to move, so I understand that you might also not have been speaking up.
Do not do that. Do speak up, while being polite, and do try to improve your situation. Have you, for instance, had a conversation yet with your boss? Does he allow you to change research direction if you would think that would be helpful? Can you bring up your own ideas that you would be more excited about?
Though I don't think that one should stay endlessly in a position which does not offer a perspective and there are certainly places worth leaving, be aware that you will encounter issues during every postdoc. You do have to be able to develop the tools to cope with these and carve out your own path successfully.
Do not worry too much about others opinions when deciding to leave, it happens and people know that. Be smart about it, however, and do not talk negatively about this situation to a potential future lab.


text:
Postdoc offer requires response before common deadline
I recently got a postdoc offer from the mathematics department of University X. Many mathematics departments in the US agree not to require responses to postdoc offers before a certain date. University X appears in the list on the American Mathematical Society (AMS) website as one of the universities that has agreed the the common deadline.
Quoting from the website of the AMS,

The following departments in the U.S. have formally adopted an agreement to coordinate their earliest deadline for responding to postdoctoral job offers for those jobs that begin in the fall of 2023. This agreement specifically excludes tenure-track offers, and it applies only to candidates who are less than or equal to two years past the receipt of the Ph.D. The departments listed below have agreed not to require responses to postdoctoral job offers before Monday, February 6, 2023.

My application and the associated offer fit the description in this paragraph, and yet University X has required a response significantly earlier than February 6th in their offer letter.
The offer from University X is good, but I am interested in other schools whose offers may come later (between University X's deadline and Feb. 6). I also applied for the National Science Foundation postdoc which will be announced after University X's deadline (and which is a reason for the common deadline).
Given that University X agreed the the Feb. 6 deadline, but now gives an earlier deadline, what should I do? I don't want to make waves at University X, but I also don't want to pass up other opportunities.
Edit: I did reach out to some other schools to which I have applied. Not all will be able to make a decision early enough for the University X deadline.

Here is what I would write to the math department at University X:
"Dear Professor/Chair/...
First of all, I am excited to receive an offer of a postdoctoral position from your department! I noticed that the deadline for my decision on this offer is January **. At the same time, according to the information that I found on the AMS site [...], Mathematics Department at the University X participates in the Coordination of Postdoctoral Job Offer Deadlines Program, with the common deadline of February 6. Therefore, I would like to inquire if the deadline of January ** from my offer letter is the result of a clerical error. If not, would it be possible to extend this deadline to the common deadline of February 6?"
 One more thing: In the US postdocs salaries are fixed (and frequently are even listed in job ads). Thus, avoid salary discussion (as well as the teaching load). But you can try to negotiate other things if you first accept a postdoctoral position at X but then receive a postdoctoral offer from Y which is, say, a research institute (IAS, MSRI, MPIM, IHES, etc.) with no teaching duties: You can ask if the department would agree to let you go to Y for a year before coming to X. (I know some instances when the answer was positive.)

Since when are postdoc salaries fixed in the US? I did mine in the early 2000nds (in physics) and back then everyone negotiated their salary up. Usually not by much, e.g. $1000-$2000/year, but it definitely wasn't fixed.

@Marianne013 I was talking about math. Was your postdoc in a lab? Those are paid for by soft money and the game is quite different from the one in math.

Even departmental postdocs in math departments do not necessarily have fixed salaries or teaching loads. I negotiated +$5000 on an offer last cycle and a reduced teaching load on another. Regardless, I cannot really see a situation in which it is not advantageous to ask, so even if the chance is low, people should not be discouraged from trying to negotiate a salary increase.

@MoisheKohan No, we were all employed by universities.

I removed the paragraph regarding salary/teaching/etc as irrelevant.

@AnonymousM: Feel free to post your own answer. I used this language because it gives the department an easy way out in case the early deadline is the result of a somewhat embarrassing (but understandable) screw-up: For instance, the departmental Chair is new, hired from a different institution and simply did not know  that University X agreed to the common deadline... The language "clerical error" provides a way for the department to save face in this situation. (I saw this used often enough during my work in academia.)

Anything you do or don't do carries risk here. One simple possibility is that they just made a mistake. Here is an option (though not without risk)...
Reply to them that you are (very) interested but won't be able to make a firm decision until the common deadline that they seem to have agreed to adhere to.
To assure yourself of some position, you could accept it, of course, which only risks the possibility of a better offer.
You could also accept it (fingers crossed behind your back) and then back out if you get a better offer, though you have to weigh the consequences (legal, ethical, practical, ...) of that. That wouldn't be my recommendation, though.
Good luck. Hopefully they aren't being unethical themselves, but if they are, then rejecting it might be wise.
Fraught.

this is a great answer. thanks for posting this @Buffy!

You can say to them:

Would you please extend the deadline to the date recommended by the AMS, which is February 6?

Ask for higher pay, or something like that, too.  Negotiating sometimes gets you what you request.  It sometimes creates useful delays.  I once allowed a postdoc job offer to expire and then had the offer reissued after my questions were answered.
I think it would be very appropriate to criticize them publicly for their dishonesty.  You can wait until after the hiring cycle is complete if that feels more comfortable to you.
The AMS agreement probably has no enforcement mechanism.  It might be interesting to think about what the law says about agreements not to compete.  Maybe your offer comes from a department that has decided to compete.  Maybe it comes from a department that expects to have any unspent budget taken away on February 5th.


text:
My co-author has not responded my email
I am a postdoc in mathematics. I read a paper of another mathematician some time ago and I found out that I could extend it. Therefore, I dropped an email to him, and then he told me if I could do it, it would be an interesting result. He also told me "I would be happy to know if you have anything to share with me," I thought that meant he would like to work with me. So, I wrote an email to him that contains all of my ideas. Now, I found out that my ideas work and everything correct.
My problem is he hasn't replied to my email after two weeks and I might be pessimistic but I am scared he steals my ideas and whatever I explained to him.
Even though I am a postdoc, he is a well-known mathematician. Moreover, we both use university emails to contact each other. What should I do now? Thanks in advance.
I have been writing it, but what can I do if he publishes it before me?

Publish what you have written...? At least in a preprint, then you can easily add stuff and authors later if your "co-author" provides input (I don't see where he suggested to work together on a paper, just that he was interested in hearing what you have done)

Just go ahead and publish. The other mathematician has not contributed to your work in any way other than saying that your results would be interesting. In fact, in my eyes it would be entirely inappropriate for this person to be your coauthor.
I would suggest though to also discuss this with your PI to make sure she/he agrees.

Since you said the senior is a big-named professor, one might speculate that he is too busy to work with you. He is more likely to be too busy to respond and very unlikely that he is trying to steal your work.
Since he is already tenured, the cost of stealing is very very high, and the reward for him will be ultra-low. Any rational mathematician won't steal you ideas as you have emails on hand as proofs.
It is also highly probable that your email is stacked in the huge of piles of email and he missed it, because a senior professor receives tons of emails each day.
Furthermore, seniors are far more likely to respond to a result that is highly polished and well-written with detailed proofs and no grammar mistakes, while most preliminary drafts contain some (minor) mistakes.
So I would send one additional email include a polished version of that paper and if he does not respond again, I will just submit the paper.
Of course, consult your PI (if you have one) before submission.

I don't entirely agree from the 3rd paragraph onwards.  Also I don't know why both answers suggest consulting with one's PI.  Math postdocs don't typically have PIs.

@Kimball Department funded postdocs don't have PI, while postdocs funded by one professor or the professor's research center have a de facto PI.

@Kimball Hi Kimball I would be very grateful if you could tell me which specific part do you disagree with, as I am still learning. Do you disagree with "seniors prefer to read a well-written, highly polished result without grammar mistakes"?


text:
Is this plagiarism? Can I publish my work
I happen to have worked on a Mathematical topic. The story is complicated.
It was not my original idea. My professor had already worked on it for a long time, intermittently. In fact, when we started out, I did not understand the concepts that well. Over a period of time, I picked up and managed to implement the algorithm myself; it proved to be even better than professor's implementation and speed was 10% faster.
We were to publish this paper, but it took a different turn. The professor has some different thought process, and wants to prove a substantial finding. He has linked it to a thought that it can be better further. We have a complete discord of thoughts on this. I do not deny what he is saying is true, but I think it should be the next phase. Our work till now produces best results, of course - under certain conditions.
I do not mind working with him further, and bringing it together to the research community. But I am going for a PhD. And I will not be physically available. I suggested an option of remote interaction, but the professor does not agree. He explained why meetings are important and why remote work cannot be carried out efficiently. After the discord, it is more of an ego issue, he did not directly indicate it. But I agree with most of his points on remote work being difficult.
But I have worked hard on the problem, sometimes 10 hours a day, for 5 complete months. And now I have to walk away without anything on my name. The nature of a professor is extremely dominating, and he will NOT mention my name in the paper once it comes to publishing after following his pathway. This is not my analysis, he clearly said this: "If you are not working on it any further, your name will not be included in the paper" He will probably find another student (MS level) and get it through.
Now my questions are:

Can I publish work till now, as it is also a major finding? I firmly believe I completely understand what is being proposed, under what scenarios finding is valid, simply put, I own the work, even though original idea is not mine. I indirectly talked about this with professor, he nullified that this is even possible, indicating I should not.
Will this be termed as plagiarism? I am ready to mention professor's name, but he does not even agree to idea of publishing it at this stage. So I have two options:
a. Mentioning his name as an author without his consent.
b. Mentioning him only in the acknowledgements, and also mentioning it was his idea and we worked together, but he does not agree this should be published without future additions.
Will this come back to me in a bad way, possibly unimaginable to me at this point of time. I am just starting out my research career and do not want it to be hampered because of my behavior/attitude.
Could you suggest any better way of resolving the situation? Talking to the professor is not really helping, yesterday we spent five hours discussing it. He is acting stubborn, uncooperative and because of that I have a feeling of wasting my six months. Of course, I learned lot in the process, my programming is much better now, my research methodology has improved. But unless I publish something, it remains only with me. I genuinely want to demonstrate my work, my abilities through a publication.

Pl suggest.
Edit_1_after_~20_months
I ended up not attempting to write a paper. At that time, I had a feeling that I did SO_MUCH work but have gained nothing, and wanted some output. I was wrong though, I learned how to approach a problem, how to carry out research iteratively, my programming improved a lot. The professor is still in touch. I have moved to different country and working on my PhD. Not having that work on my name does not matter a bit, but learnings do make a positive difference. I thank people who put out their thinking and suggestion. It was a vile effort from a bu**hurt person and that thinking does not make sense now.

I only read your question to the first bullet 1.Can I publish work till now, as it is also a major finding? I don't think you can. The original idea is his. He should be one of the authors because he has significant contribution. How do you publish the paper without his consent?

How do you own the work if the idea wasn't yours??

A 10% speed up is nothing. Did you mean 10 times faster?

You can not copyright an idea, although it is unethical to take one and cal it only yours in my opinion. The algorithm however, is the protected work. If you used substantial information from the professor to construct the algorithm, that was not already common knowledge, you can not publish the work individually without his consent.

@TheMathemagician For some problems, 10% would be a perfectly acceptable improvement. OP also said that it's 10% faster than his prof's original algorithm and we don't know how much faster it was than the state-of-the-art solutions.

@mkennedy - well, it really depends on the professor's implementation - getting a 10% speed up by changing compiler flags (much less languages) is not uncommon. Without a lot more detail (and work by the student and professor), I would not brag about 10%.

The story doesn't quite make sense to me.  The OP is some kind of student but not a PhD student, but spent 10 hours a day working on one problem for five months.  I don't understand how that could be possible for anything but a PhD student completing their thesis -- students have other things to do.   (Also ten hours a day thinking about one problem sounds like unhealthily much time, in my personal opinion.)

@PeteL.Clark: "sometimes" 10 hours a day. Also, admitted in the beginning that initially things were difficult to grasp, so working on fundamentals of the problem actually makes one spend more amount of time. Anyways, not everything everyone asks needs to make sense to everyone.

Adorn, if an experienced person such as @PeteL.Clark says it doesn't make sense to him, then there is a problem. There are many fewer genuine mysteries than people may claim.

@paulgarrett, pl see an update

There are a number of other questions on this site relating to these issues, although I'm not going to search for them now.
There are different issues going on here:
Firstly, you want to publish the bit of research you have done so far, but the PI thinks it should instead be turned into something more substantial before publishing. In this he is almost certainly right. It is not considered good practice to slice work into small pieces to get more papers.
You are considering publishing work with someone else against their wishes. Don't. Especially if you want to stay within the research community.
I take it you are concerned about having material for your PhD. Unless you are somewhere where publishing is a requirement for graduating, this is actually irrelevant. A PhD thesis should contain what you have done, and the normal progression of research is considered when assessing it.
You want to work remotely. I am confused as to who your PhD supervisor is. If he is your supervisor then you should expect to see him in person. If you are doing a PhD elsewhere, then how do you expect to have time to work on this project while also working full-time on your PhD topic?
Note also that writing a faster algorithm does not in itself really constitute new research in maths. Is there a particular theoretical change that made it faster? Or is it just that you coded a bit better? Does it actually make a difference in practical computational time, such that it allowed you to test some new examples that gave you the key insight for a theorem (or at least conjecture, or counter-example)?
The professor has said your name won't go on the final paper if you do no further work. It's possible that the professor is being unreasonable, and you can look up other questions along those lines. But the more likely situation is that your name won't go on the paper because what you have done doesn't count for authorship. Generally in mathematics the authors are only those who have contributed to what appears in the paper. It sounds unlikely that that would be true of the work you have done.
An unconventional solution, but one that would perhaps be more to your liking, might be to agree to release your write-up of your algorithm alongside the final paper, either as an appendix or as a separate paper. It probably won't do that much for you career-wise, but it might make you feel better and preserve your relationship with the professor. Also, by the time that happens you may be in a better position to admit that he's right. I know for myself it's hard to admit that your work isn't really that valuable, because it matters a lot to you. Over time you will become a little less emotionally involved.
Also, remember that this is someone you will want a reference from, if you are to stay in academia.

While I agree mostly, I think the refusal to list the OP as a co-author is punitive. I worked with co-authors who insisted in including me in a paper where my only contribution was to point out an additional "theorem" in the draft I read was false. I appreciated their generosity.

@AndrewLazarus (Delayed response I realise) By the standards of many subjects, what your co-authors did was wrong. There is at least one instance of a reasonably precise description of what is required for authorship (it's on here somewhere, but I'm not looking right now), and what you describe is nowhere near enough to pass.

It is more difficult now that there seems to be arguments between your advisor and you. You would need to improve your communication with him before making progress. Five months is not enough to have the perspective your advisor has. If you submit a paper with him, it will be his name that will get it accepted, not yours.

If you submit a paper, none of the names on the paper get it accepted. The content of the paper does that.

@TobiasKildetoft On the other hand, having the wrong names, or the right names without permission to submit, is sufficient to prevent the paper being published.

The only possible way is asking him to publish your name as one of the co-authour when the paper comes to pubish. It takes time to get publish, but you should wait as you beive it is a significant finding. meanwhile, start applying same mathedology to some other algo for furthur improvements. As you abilities in programming are enhanced, you can achive this fastly.