significance test for average distances? I have many measurements of distances between places. Say, distances between a school and selected houses in different cities.
I want to compare the distribution of distances across cities.
I want to answer the question "Are the distances distributed roughly the same?".
Which test would be the most appropriate?
Is the fact that we're dealing with distances make it any different from a "normal" statistical significance test?
 A: Comment. Depending on the random mechanism of scattering your 'selected' houses
and the geographical shapes of your areas of interest (cities, school districts?), the distance
data may be skewed, and the two samples may be skewed by different amounts. In that case, as commented by @whuber,
descriptive statistics and graphical displays may give you the best view of any differences in distances.
Random data below, sampled in R, show one possible scenario
for two samples, each of $n = 100$ houses.
set.seed(302)
x = sqrt(rchisq(100,2))
y = 1.2*sqrt(rchisq(100,2))

summary(x); sd(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.1775  0.8279  1.1892  1.2154  1.6448  2.9068 
[1] 0.564254  # sample standard deviation
summary(y); sd(y)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.1374  0.8643  1.3449  1.4824  1.9572  4.3618 
[1] 0.844284  # sample standard deviation

boxplot(x,y, col="skyblue2", pch=20)


A Welch two-sample t test will 'correct' for different sample variances,
but not for skewness. It does find a significant difference according to
the small P-value shown, but the distribution upon which that P-value is
based may not be correct.
t.test(x,y)$p.val
[1] 0.009322874

A nonparametric two-sample Wilcoxon rank sum test expects two samples
of about the same shape (somewhat questionable here) in order to be
interpreted as a straightforward test of differences in population medians.
This tests also gives a P-value below 5%, but its interpretation is not
obvious.
wilcox.test(x,y)$p.val
[1] 0.04630706

Even with samples of 100, a Kolmogorov-Smirnov test of the "goodness-of-fit" between the two samples
does not show a significant difference. (This test has notoriously low power, so its non-significant P-value is not a surprise.)
ks.test(x,y)

        Two-sample Kolmogorov-Smirnov test

data:  x and y
D = 0.16, p-value = 0.1545
alternative hypothesis: two-sided

Returning to graphical descriptions, it seems clear than distances in
sample y tend to be larger than distances in x. The ECDF of the second
sample (red) plots mostly to the right of the first---hence below.
[The K-S test statistic $D = 0.16$ is the maximum vertical distance between
the two ECDFs.]
plot.ecdf(x, ylab="ECDF", main="Sample Y (red) Dominates")
 plot.ecdf(y, add=T, col="red")


