How should I interpret the median and the interquartile range? I have a 7 point Liker Scale, ranging from 1=Strongly disagree to 7=Strongly Agree that has the purpose to measure if the respondents recognize the logo of a brand. I estimated the median and the interquartile range are, respectively, 5 and 3. How should I interpret this?
 A: Here is a simulated sample with median $5$ and IQR $6-4=2,$
so your scores are somewhat more spread out than mine, according to the IQR.
set.seed(2020)
x = sample(1:7, 200, rep=T, p=c(.5,.5,1,2,3,2,1)/10)
summary(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  1.000   4.000   5.000   4.595   6.000   7.000 

The official interpretation for my data is that not more than 25%
of the sample is below $4$ and not more than 25% is above $6.$
table(x)
x
 1  2  3  4  5  6  7 
11  9 16 50 62 32 20    # counts out of 200

mean(x < 4);  mean(x > 6)
[1] 0.18    # 18% < 25% below Q1
[1] 0.1     # 10% < 25% above Q3

You could also say that my data are sufficiently concentrated in the
center that only IQR + 1 = 3 central Likert categories (4, 5, and 6) include
at least 50% of the observations.
mean(x < 4 | x > 6)
[1] 0.28   # 28% < 50% outside of Quartiles

Your data are a little less concentrated toward the center than mine. You have
IQR + 1 = 3 + 1 = 4 central Likert categories that account for at least
50% of the data.
Caution: (1) It is possible for either Q1 or Q3 (or even both) to correspond to the
same Likert category as the Median. And, even if IQR = 2, either Q1 or Q3 could be the same as the Median.
set.seed(808)
y = sample(1:7, 200, rep=T, c(.5, .5, 1, 4, 1, 1, 2)/10)
summary(y) 
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  1.000   4.000   4.000   4.385   6.000   7.000 

(2) Among various software programs there are almost a dozen slightly
different ways of determining quantiles, including Q1 and Q3 (and thus IQR).
It is possible, but not likely, that two different programs could get different
IQRs from the same data.
A: While the median suggests most respondents are in agreement in their recognition of the brand logo, however the interquartile range indicates a high level of polarity in their judgments.
