Let me show results from the Mann-Whitney-Wilcoxon
Signed rank test for some simulated data in R.
First, the data meet your specifications because they
are sampled from gamma distributions, which are right
skewed and so not normal. The respective population means are $\mu_1 = 10,\,
\mu_2 = 20.$ Also, the population variance from which the first
sample was drawn is less than the population variance for
the second sample.
Simulated Data and 'Location'. In a real life situation we would not
have all of this information about the populations. The
point here is to see how well the samples (both of size 20)
reflect the populations from which they were drawn. In
particular we want to see if a 2-sample Mann-Whitney-Wilcoxon test
can detect that the location of the second population is located
above the first.
set.seed(2020)
x1 = rgamma(20, 3, .3)
x2 = rgamma(20, 4, .2)
summary(x1); sd(x1)
Min. 1st Qu. Median Mean 3rd Qu. Max.
2.401 6.846 10.218 10.335 13.861 24.977
[1] 5.370183 # sample SD of x1
summary(x2); sd(x2)
Min. 1st Qu. Median Mean 3rd Qu. Max.
8.202 10.945 16.474 17.362 22.043 34.085
[1] 7.15061 # sample SD of x2
Location shift. Notice that the 1st quartile, median, mean, and 3rd quartiles
for the first sample are all below the corresponding values for
the second sample. There are many ways to express the 'location'
of a sample (or distribution). The median is among them, but
certainly not the only one. (Also, the maximum value in the first sample
lies below the maximum value in the second sample, but in these
right-skewed distributions maximum values are rather unpredictable.)
Thus, if one does not want to use the median as the specific
measure of location, there are other ways to discuss location. A general
term for expressing that two samples (or populations) do not have
the same location, is that there is a "location shift" of one from
the other. You should discuss this with your instructor, but I think
location shift might be the best terminology for you to use.
Boxplots (each based on quartiles, median, min and max)
show that the sample x1 tends to be 'located' below
the sample x2.
boxplot(x1, x2)
Mann-Whitney-Wilcoxon Test. The M-W-W test does show a significant location shift (P-value about 0.002).
Notice the terminology "location shift" in the R output for this procedure.
wilcox.test(x1, x2)
Wilcoxon rank sum test
data: x1 and x2
W = 87, p-value = 0.001767
alternative hypothesis:
true location shift is not equal to 0
ECDFs and 'Stochastic Dominance'. Another technical term for saying that one sample is located above another
is "stochastic dominance/" If you google around the internet you will see
that there are several senses (sometimes called 'orders') of stochastic
dominance. A lot of this may be above your level, but on some pages you'll find some simple gambling examples and see
illustrative graphs. You might use the terminology "stochastic dominance," if you're prepared to explain it.
ECDFs. One of the simpler kinds of stochastic dominance has to do with ECDF plots.
Especially for large samples, the empirical CDF (ECDF) of a sample imitates
the CDF of the population from which the sample was chosen.
To make an ECDF of a sample begin by sorting its values from smallest to
largest. The ECDF is a stairstep function. With a sample of size $n$ it jumps up by distance $1/n$ at each sorted data value. (If $k$ observations are tiec at a value, then the jump there is $k/n.)$ So the ECDF starts at $0$ on the left and rises to $1$ at the right. One sample "dominates" another if its ECDF
is to the right of the ECDF of the other.
The left panel below shows the CDFs of the two populations from which our samples were randomly chosen: blue for the first sample and maroon for the second. At right (same colors) are the ECDF for the samples. [R code below
the graph.]
par(mfrow=c(1,2))
curve(pgamma(x, 3, .3), 0, 40, col="blue", lwd=2, ylab="CDF",
main="Population CDFs")
curve(pgamma(x, 4, .2), add=T, col="maroon", lwd=2)
plot(ecdf(x1), col="blue", main="Sample ECDFs")
lines(ecdf(x2), col="maroon")
par(mfrow=c(1,1))
