When I was in school, pupils were given numerical grades, or the equivalent of numerical grades but disguised as words, on their performance in various school subjects and also behaviour. A key feature of these is that it allows the pupils to compare their performance with their peers.
In a newspaper story, not recent anymore, a child was asked if they would be happy with receiving only information about their strengths and weakness, with no numerical grade. Their response was that it is okay, as long as they still get to know how good they are when compared to others.
One might imagine that good students might get complacent or poor students might give up when such comparisons are possible, but one might also imagine good students getting self-confidence and poor students seeing that something is wrong and maybe deciding to fix the matter. One might assume that the negative effects outweight the positive ones; an answer to a question about grading assignments at a university certainly points at such a conclusion: https://matheducators.stackexchange.com/a/2267/2083.
I am interested in any research concerning what effect quantitative, or quantifiable, feedback has on learning, particular when it comes to mathematics and to younger children. This would be compared to feedback which tells about the strengths and weaknesses of the pupil, but does not give an explicit means of comparing oneself to others.
But as this is quite specific, I am open to research where one or both of the conditions are loosened; the subject might not be mathematics and the students need not be young. I am particularly interested in the effect of evaluation that happens at the end or midpoint of school year, rather than the results of every exam or smaller test.
Some related studies
Giving final grades is a common form of summative assessment. Finding studies about its effects on student learning seems tricky.
For example, in Toimivatko päättöarvioinnin kriteerit? : Oppilaiden saamat arvosanat ja Opetushallituksen oppimistulosten seuranta-arviointi koulujen välisten osaamiserojen mittareina the author simply declares final grades as summative. This is reasonable for the article, which aims to investigate if the given grades are comparable in different schools.
In the article Osaaminen ja hyvinvointi yläkoulusta toiselle asteelle - Tutkimus metropolialueen nuorista, in second chapter, it is taken for granted that finishing grades of comprehensive school have an effect on future studies (in gymnasium or professional schools), but few references are given. There is nothing on comparison of giving or not giving such grades and the focus is on the transition from comprehensive school to the next stage in one's life.
The developmental project Arvioinnin vaikutus merkitykselliseen oppimiseen opiskelijoiden ja opettajien näkökulmasta gets as close as I can find. They asked teachers and students in professional high school (university of applied sciences) and university about different ways of assessing in courses. There, both students and teachers found numerical assessment to be quite good and motivating. However, the study did not actually compare the effects on learning on different assessment methods.
