You have ordinal categorical data, so it may not be appropriate to do a two-sample t test. which requires nearly-normal data. You have described sample distributions that seem far from normal. Possibly your data are not even really numerical. (Assigning numbers as category labels does not necessarily lead to truly numerical data.)
With no hint clue about the appearance of your summarized data or the number of subjects, the only ethical response is to show
you artificial data for which a two-sample Wilcoxon (rank sum)
test are appropriate.
Then you can judge whether your data might
be sufficiently similar to use the same test.
set.seed(119)
x1 = sample(1:5, 100, rep=T, p = c(1,1,2,3,1))
table(x1)
x1
1 2 3 4 5
13 16 25 34 12
x2 = sample(1:5, 100, rep=T, p = c(1,1,2,3,4))
table(x2)
x2
1 2 3 4 5
2 8 18 34 38
From the tables it seems that sample x2 has fewer low scores
and more high scores than x1.
Nonoverlapping notches in the sides of boxes in two boxplots, is often a sign of significant differences.
boxplot(x1, x2, horizontal=T, col=c("wheat","skyblue2"), notch=T)
Will a Wilcoxon test find that
the higher scores in x2 are statistically significant?
The answer is Yes, with a very small P-value.
wilcox.test(x1,x2)
Wilcoxon rank sum test with continuity correction
data: x1 and x2
W = 3086, p-value = 1.296e-06
alternative hypothesis:
true location shift is not equal to 0
When using the Wilcoxon signed-rank test to compare two samples
of rather different shape, it may be unclear whether it is as
simple as saying one sample has a significantly higher median.
However, empirical CDF (ECDF) plots of the two samples clearly show that
x2 tends to 'stochastically dominate' (have larger values) then x1 The dashed blue ECDF is to the right of the brown one (thus mainly below.)
plot(ecdf(x1), col="brown", main="ECDFs of x1 (brown) and x2")
lines(ecdf(x2), col="blue", lty="dashed", lwd=2, pch="o")
