Is there a term for an average per equal amount of data? Let's say I have data $y$, which is dependent on $x$. First I sort the data according to $x$, and then I take the average of the first $k$ values of $y$ and $x$, and then the average of the second k values, and so on; i.e.,
$$
<y>_i = \frac{1}{k}\sum_{(i-1)\cdot k+1}^{i*k} y
$$
and 
$$
<x>_i = \frac{1}{k}\sum_{(i-1)\cdot k+1}^{i*k} x
$$
for some value of $k$. 
Is there a term for this type of average? And is there a term for the graph when I plot $<y>_i$ against $<x>_i$ ?
 A: They could be called "quantile means" ... eg. if you divided the data into deciles*, they might be referred to as "decile means", similarly for quartiles, percentiles etc. This works quite well for the $x$-values, however, for the $y$s you'd need to somehow specify that it was means of $y$ based on quantiles of $x$ (because otherwise they  might be seen as means of the $y$-variable within its own quantiles). If you want to refer to the $k$, then if $m=n/k$ they could be called $m$-quantile means.
* confusingly "quantiles" can mean either the $k-1$ dividing lines or the $k$ equal-sized groups into which you divide a variable. Terms for specific quantiles are in the Wikipedia article on quantiles
I'd be somewhat tempted to call those "binned means" -- that term doesn't imply that the bins all have equal number of values (it's more general), but it's probably more readily understood and can be made more specific via an adjective. However, again for the $y$, you'd need to be able to specify that it was the means of $y$ based on bins of $x$.
If you had specified x-intervals (usually equi-spaced) rather than equal-proportions of data, this calculation on $y$ could be called a "regressogram". e.g. see this question. You might add an adjective there to specify that it's equal-proportions rather than equispaced.
On reflection, I'd lean toward the third option as the least ambiguous; you could perhaps just call it a regressogram and then explain the details.
I suppose you could combine the first and third suggestion to supply the adjective and call it a (m-)quantile regressogram (e.g. decile regressogram for ten such bins); the name is sufficiently suggestive that many people would infer the likely meaning without a definition (though I'd still recommend defining it when you first use it, whichever term you use).
So perhaps you would say "quantile means" for $x$ and "quantile regressogram" for $y$ (e.g. "decile means" and "decile regressogram"). I think that would convey the concepts involved.
A: The indexing you're using is a tad confusing, but it seems like you want a moving average? If that's the case, here is some R code to demonstrate:
#Let k be the window length
#let x be the input data, y be the output data, f be a function that takes x and returns y
xmean <- rep(NA, (length(x)-k) )
ymean <- rep(NA, (length(x)-k) )

for (i in 1: (length(x)-k))
{
  xmean[i] <- mean(x[i: (i + k)])
  ymean[i] <- mean( f( x[i: (i + k)]) )
}

I'm not sure about a term other than moving average. 
