# Measure of distance between two survey responses

I've found some survey data where respondents answer 63 question by giving a response for each question between 0-10 (0 for strong disagreement, 10 for strong agreement). So I can view every respondent as a integer point in $$[0,10]^{63}$$.

Question: is there a "standard" way to measure the distance between two respondents?

Something like the taxicab distance between them seems somewhat reasonable, but likewise so does any $$L^p$$-distance. Though these wouldn't account for questions that contributed the most of least variation in responses, for example.

My intended application is using this to measure distances between points to experiment with persistent homology.

• Sometimes this is called a 'Likert' scale, for which you can find many examples on this site (or with an online search). Depending on the circumstances there can be discussion whether such a scale is 'ordinal categorical' or 'numerical'. Commented Dec 1, 2020 at 20:20

If you have Likert-10 data for two groups, you could compare the two groups using a two-sample Wilcoxon rank sum test. Hypothetical data and test below with 200 subjects in each group.

Data and summary: The p vectors show relative 'popularities' of the score from 0 to 10 in each group.

set.seed(2020)
x1 = sample(1:10, 200, repl=T, p=c(1,2,3,4,5,6,4,3,2,1))
x2 = sample(1:10, 200, repl=T, p=c(1,1,2,3,4,5,6,7,6,5))

table(x1)
x1
1  2  3  4  5  6  7  8  9 10
6 13 21 32 26 37 29 15 12  9
table(x2)
x2
1  2  3  4  5  6  7  8  9 10
7  5  5 19 23 23 29 42 25 22

summary(x1)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
1.00    4.00    6.00    5.48    7.00   10.00
summary(x2)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
1.000   5.000   7.000   6.725   8.000  10.000


Wilcoxon test shows a highly significant difference between the two groups with a P-value near $$0.$$

wilcox.test(x1,x2)

Wilcoxon rank sum test
with continuity correction

data:  x1 and x2
W = 13606, p-value = 2.524e-08
alternative hypothesis:
true location shift is not equal to 0


'Notched' boxplots. Here are boxplots of the two samples of Likert-10 scores. That the 'notches' in the sides of the boxes do not overlap suggests that the two groups differ.

boxplot(x1,x2, notch=T, col="skyblue2")


• This answer introduces 'two groups' that the question does not mention. The OP is asking the distance between (any) two respondents. In this frame, we have a scale between (0,10) with a vector sum of 63 (the number of questions). The Wilcoxon rank test is a significance test, which is not asked for. As for a distance, I'd recommend the Hellinger Distance or even the elementary similarity distance (effectively the histogram overlap). Other distances to consider would include the Jensen-Shannon Distance and the Variation of Information Commented Dec 28, 2022 at 23:35