Let’s take a big, nationally representative group of 15,000 or so kindergarteners and follow them through 4th grade. If you look at how well they read in kindergarten, you can predict well but not perfectly how well they read in 4th grade:

That makes sense- some kids (most commonly boys) have a lot of trouble learning to read but catch up, some kids learn the basics of reading fairly easily but have trouble moving on to more complicated texts.

Same with kindergarten math and 4th grade math: if you know how well a kid does in kindergarten, you can predict well but not perfectly how he or she will do in 4th grade:

The average, overall relationship between kindergarten and 4th grade math scores and between kindergarten and 4th grade reading scores is almost identical:

The “Early Childhood Longitudinal Study of 2011,” (ECLS-K: 2011) which this data is from, gave kids a general knowledge test in kindergarten and a science test in 4th grade, and it turns out if you add up a composite of their reading, math, and science scores in kindergarten it does a very good job of predicting the sum of their reading, math, and science scores in 4th grade- a correlation of almost 0.8, quite high given measurement error and all the things that can happen between when a kid is 5 and when he is 9 or 10:

It turns out this is about as well as you can do; the ECLS-K also gave kids a bunch of mental tasks in kindergarten like remembering a string of numbers in backwards order (correlated about 0.45 with 4th grade test scores):

But while all these mental tasks are individually correlated with how kids do in 4th grade, adding them in doesn’t help you much in predicting 4th grade outcomes. Same with knowing the kid’s demographic and background characteristics like race, sex, whether the kid’s parents were married when the kid was born, whether the kids’ parents went to college, are still married, are high income or low, report having trouble buying enough food, are eligible for food stamps, and whether the kid’s teacher reports that the kid has an easy time focusing in class. Each characteristic is correlated with later test scores, but overall, knowing all of them is barely more predictive for 4th grade scores than knowing kindergarten scores by themselves:

The characteristics of kids’ schools make a little difference, especially towards the middle; a kid who scores at the 50th percentile in kindergarten who goes to a high poverty (more than half eligible for free/reduced price lunch) school can expect to score at about the 40th percentile of the national distribution in 4th grade; a kid who scores at the 50th percentile who goes to a low poverty (less than 20 percent eligible for free/reduced price lunch) school can expect to score at about the 60th percentile by 4th grade:

But that difference gets even smaller towards the top of the distribution: kids above the 90th percentile do basically identically, on average, in high and low poverty schools:

Of course, this slight difference could be the result of the schools themselves, or the result of neighborhoods or peers or any number of other things. It is, apparently, not something that simply giving money to poorer schools can easily affect.

In your second last graph, are you plotting composite scores?

In your last graph, I take it that each dot represents a school rather than a student. It took me a while to figure that out, though, so maybe worth tidying up the labels a bit.

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Each dot represents a student. They are classified as in a high poverty school if their school is over 50% FRPL, low poverty if their school is under 20%. This is their composite percentile.

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In that case I don’t understand why your second last graph is different to your last graph. Your last graph really doesn’t look like what would be produced by plotting individual student test scores.

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Last graph is the conditional mean for rounded to the nearest percentile scores E[ScoreGr4|ScoreK, School Poverty], the other is raw data

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