deciles in skewed distributions Does it make sense to use deciles for distribution that are skewed. For example, consider the exponential distribution with lambda >= 1.5 or lambda distribution with k = 1 and theta = 2 or any other that is highly overweight. Or for cases like this, are there better/smarter ways to segment the data?
 A: Consider the following random sample of 1000 observations from an exponential distribution with rate 2 (mean 1/2). Various statistical software programs
use slightly different rules for determining quantiles of samples, I used R here. (For samples as large as 1000, differences in quantile rules are usually
not if practical importance.)
set.seed(623)
x = round(rexp(1000, 2), 3)

A boxplot illustrates the right-skewness of this distribution. The many 'outliers' are characteristic of exponential samples.

Here are the deciles of the sample:
q = quantile(x, seq(0, 1, by=.1));  q
    0%    10%    20%    30%    40%    50%    60%    70%    80%    90%   100% 
0.0000 0.0529 0.1108 0.1767 0.2480 0.3225 0.4208 0.5476 0.7280 1.1684 3.5480 

Because of the right skewness of this distribution, the deciles are increasingly far apart as observed values increase.
Ordinarily, it is not a good idea to make histograms with unequal bin widths,
but to illustrate this pattern of deciles, I used them as cutpoints in binning a histogram of this sample. Each bar of the histogram represents about 100 observations. The red curve is the density function of the exponential population.
 
Here is R code for making the histogram.
hist(x, prob=T, br=q, col="skyblue2")
  curve(dexp(x, 2), add=T, col="red", n = 10001)

Here are the deciles of the distribution $\mathsf{Exp}(\text{rate}=2).$
round(qexp(seq(0,1, by=.1), 2), 4)
 [1] 0.0000 0.0527 0.1116 0.1783 0.2554 0.3466
 [7] 0.4581 0.6020 0.8047 1.1513    Inf

Except for the maximum, the sample deciles estimate the population deciles.
