Measuring siginificant difference between Likert-scale value? I have two tables, each containing 64 5-point Likert-scale scores. I would like to investigate if there is a significant difference between the two tables.
Which test would be best to use in this situation?
 A: The answer depends on what you mean by 'different'.

*

*Likert scores are ordinal.
Do you want to know whether one set of scores is generally higher?

*Likert scores are categorical. Do you want to know whether the sample distributions into the five categories differ significantly?

Higher or Lower? Suppose scores are integers 1 through 5, with 5 indicating the
most favorable view on a topic. Sample a tends to have higher (more favorable) scores than sample b. [Sampling and computation in R.]
set.seed(719)
a = sample(1:5, 64, rep=T, pr=c(1,1,1,6,6)/15)
b = sample(1:5, 64, rep=T, pr=c(3,3,3,3,3)/15)
table(a)
a
 1  2  3  4  5 
 7  4  9 21 23 
table(b)
b
 1  2  3  4  5 
13 13 13 12 13 

We do a two-sample Wilcoxon (rank sum) test on vectors a and b.
wilcox.test(a,b)

        Wilcoxon rank sum test with continuity correction

data:  a and b
W = 2683.5, p-value = 0.001898
alternative hypothesis: 
   true location shift is not equal to 0

Looking at empirical CDF (ECDF) plots of the two groups of Likert scores, we see that
scores a dominate (are generally higher than) scores b. That is the ECDF
for a (blue) lies to the right of the ECDF for b (orange). [To make an ECDF plot, sort $n=64$ scores from smallest to largest, starting at $0$ at the left, jump up at each score by $k/n$ if there are $k$ observations tied at a value, ending at $1$ at the right.]
plot(ecdf(a), col="blue", lwd=3)
 lines(ecdf(b), col="orange", lwd=2, lty="dotted", pch="o")


More or Less Controversial? Sample a has more extreme scores (strongly opposed or in favor) then does sample b.
set.seed(2020)
a = sample(1:5, 64, rep=T, pr=c(5,2,1,2,5)/15)
b = sample(1:5, 64, rep=T, pr=c(2,3,5,3,2)/15)
A = tabulate(a);  B = tabulate(b)

TBL = rbind(A,B);  TBL
   [,1] [,2] [,3] [,4] [,5]
A   26    7    7    5   19
B    7   14   20   15    8

We do a chi-squared test of homogeneity, using a table TBL of scores a and b.
chisq.test(TBL)

    Pearson's Chi-squared test

data:  TBL
X-squared = 29.013, df = 4, p-value = 7.768e-06

We strongly reject the null hypothesis (P-value near $0)$ that the profiles of favor and disfavor
are the same for groups with scores a and b.
Because scores are not generally higher for one group than for the other, a Wilcoxon signed rank test would have found no significant difference between the groups.
wilcox.test(a,b)$p.val
[1] 0.2067968

Notes: (1) We could have done a chi-squared test of homogeneity in the first part and it might have rejected the null hypothesis, but that would not have
answered the question whether one group tends to have higher scores than the other.
(2) It is not necessary for the vector pr of relative frequencies
used in the sample procedure to sum to unity. R automatically adjusts
the vector to do so.
