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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 $<x>_i$ ?

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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.

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  • $\begingroup$ Indeed, quantile mean would work well for the average of $x$. But for $y$ it does not, and since quantile suggests ordered according the variable itself, I would prefer to stay away from using just "quantile". Perhaps I could say something like "the averages of $y$ for the quantiles of $x$". However, this is somewhere in between both sides, i.e., not completely clear without explanation and not really short (shorter than full explanation, though). $\endgroup$ – Lu Kas Jan 26 '17 at 12:37
  • $\begingroup$ The regressogram is interesting, I didn't know this. The suggestion of quantile regressogram is really good, indeed, though it only works for a figure (perhaps all I need). The only problem is that regressogram is not well know, I think. Or it is just me? But I would indeed define it once first anyway. I have to think about whether I'll accept this as the answer, but I'll definitely upvote your answer. $\endgroup$ – Lu Kas Jan 26 '17 at 12:43
  • $\begingroup$ If it is of any interest, since there doesn't seem to be an existing term, I asked at latin.stackexchange if they would have some ideas for a neologism. Someone suggested, in analogy with domain and codomain, "co-quantile" (average or mean) for this specific type of average of $y$. Seems quite fitting, doesn't it? $\endgroup$ – Lu Kas Jan 27 '17 at 11:37
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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.

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    $\begingroup$ It is indeed close to a moving average, like you show. But the problem is that it is a moving average of the data sorted according to $x$, whereas the original data is not sorted. And also it is moving over the indices of $x$, not over the values of $x$. The amount of measurements in a given interval of $x_0$ to $x_0 + \Delta x$ is not necessarily the same as in say $x_0 + \Delta x$ to $x_0 + 2 \Delta x$. So moving average might confuse people. $\endgroup$ – Lu Kas Jan 25 '17 at 17:27

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