As you say, there are several different ways to define quantiles.
For your small sample: Your sorted data are as follows (from R):
x = c(49,33,28,33,21.1,9,0)
sort(x)
[1] 0.0 9.0 21.1 28.0 33.0 33.0 49.0
With this small sample of $n = 7$ observations, there is no one obvious
choice for the 90th percentile. Various statistical software programs
make different compromises.
The documentation of R's quantile procedure lists nine different rules
for such compromises accessible with the parameter type. Exact formulas and rationales for most of them are shown there. (R provides the opportunity to use different types of quantiles for compatibility with other software programs.)
Type 7, the default in R, gives one of the values for the 90th percentile that you mentioned.
quantile(x, .9) # usually, no 'type' specified, R uses Type 7
90%
39.4
quantile(x, .9, type=7)
90%
39.4
Your other value results from using Type 9:
quantile(x, .9, type=9)
90%
47.4
Because of the large gap between observations 33 and 49, a large
variety of other values are possible with different types, including
33 (from Type 3) and 49 (from Types 1 and 2).
Larger samples: In practice, quantiles are mostly used for samples of moderate and large
size, for which the different rules don't make substantial differences.
y = sort(rnorm(1000, 50, 7))
quantile(y, .9)
90%
59.01486
quantile(y, .9, type=1)
90%
59.01104
quantile(y, .9, type=3)
90%
59.01104
y[898:902]
[1] 59.00132 59.01043 59.01104 59.04926 59.09998
Consequences of rounding. Rounding a sample to integers can induce many ties, and thus fewer
differences among the rules:
z = sort(round(rnorm(1000, 50, 7)))
quantile(z, .9)
90%
59
quantile(z, .9, type=1)
90%
59
quantile(z, .9, type=3)
90%
59
z[898:902]
[1] 59 59 59 59 60
When you are taking a basic statistics course, use whatever rule is given
in your text or recommended by your instructor. But be aware that if you
use software to find answers, you won't necessarily get exactly the same answer as
an answer book.
In practice, you will probably use a particular kind of statistical software,
and be happy enough with whatever rule the software uses as its default.
Addendum per comments about application: If your samples are coming from a normal population, it may be best to use the fact that the sample mean $\bar X$ and sample SD $S_X$ are based on sufficient statistics, so
tend to provide better information about the normal population than do quantiles. Then the procedure would be to find the 90th percentile of the normal distribution $\mathsf{Norm}(\bar X, S_X).$
In the simulation below, I compare the actual 90th quartiles q.9 with estimates q.e obtained under the assumption of normality. The latter estimates do a little better. If the population distribution is $\mathsf{Norm}(\mu = 25. \sigma = 5),$ then its 90th percentile is 13.41. Also, in the simulation I tried Type 9 quantiles which are said
to give unbiased approximations of population quantiles of normal distributions.
qnorm(.9, 25, 5)
[1] 31.40776
set.seed(808)
m = 10^5; n=40; mu=25; sg=5
q.9 = q.e = numeric(m)
for(i in 1:m) {
x = rnorm(n, mu, sg)
q.e[i] = qnorm(.9, mean(x), sd(x))
q.9[i] = quantile(x, .9, type=9) }
mean(q.e); sd(q.e)
[1] 31.36575 # aprx 31.41
[1] 1.072285 # less variable than below
mean(q.9); sd(q.9)
[1] 31.4376 # aprx 31.41
[1] 1.302801 # more variable than above
Both methods tend to give nearly the right 90th percentile on average,
but the method using the normal distribution is more consistently
close to the correct answer. Root mean square error (RMSE) is one way to measure 'goodness' of an estimator, low values are better. The RMSE
using quantiles is a little larger, but this method does not assume normality. (RMSE from Type 7 quantiles is about 2.87.)
sqrt(mean((q.e-31.41)^2))
[1] 1.073193 # smaller RMSE
sqrt(mean((q.9-31.41)^2))
[1] 1.303086 # larger
Something similar might be done for sampling from an exponential population, but if you have no idea what kind of distribution the
population has (or even whether it is stable), then using sample
quantiles to estimate population quantiles may be a safer course of
action.
