What's the right way of finding the median of the lower and upper quartiles of a box and whisker plot? Let's say I have a set of test scores from 20 applicants
$5, 15, 16, 20, 21, 25, 26, 27, 30, 30, 31, 32, 32, 34, 35, 38, 38, 41, 43, 66$
Obviously, the median of this set would be $30$ and $31$, so I would just get the average, though that's not what I'm looking for.
What I'm really looking for is the median of the lower and upper quartiles.
I have found 2 ways, both yielding different answers.
1st way: Finding the lower quartile, the range would be $5$ to $30$, and the median of the set would be $21$. This is based on a video that I watched.
2nd way: Same problem, but instead of just finding the median, I use a formula.
$x_{kth}=(\frac{k}{4})n=\frac{1}{4}(20)=5$
This means the median for the lower quartile is the 5th spot, which is $21$. Sounds good, except that I have to do an extra step.
Get the average of the values $x_k$ and $x_{k+1}$ positions.
This means I have to average the 5th and 6th positions.
$Q_1=\frac{5_{th}+6_{th}}{2}=\frac{21+25}{2}=23$
This is entirely different from $21$. This method came from a textbook.
Which way is correct?
 A: Different textbooks and brands of software have various methods of
finding quartiles (and other quantiles). 
If you have 40 distinct
observations, sorted in order, then it is clear that the lower
quartile should be between the 10th and 11th values. But if the
sample size is not evenly divisible by 4, or if there are ties,
the compromises must be made, and there is no general agreement
exactly how to do that. The different methods may make a noticeable
difference in small samples, but usually not in large samples.

In R statistical software, the help screen accessible by ? quantile
  gives formulas for nine different types in common use, including rationales for
  some of them. (By default, R uses type 7 in the list.) In addition, traditional boxplots--as originally specified
  by John Tukey--use 'fourths' instead of 'quartiles'; they are about the
  same as quartiles, and some people feel they are easier to find by hand
  in small datasets.

The criteria for 'identifying' outliers are based on the quartiles, so
moving from one software package to another, you may see differences in
which observations are so identified.
Quantiles for your data, of types 7, 3, and 5 from R, are shown below.
The last item shows Tukey's 'five number summary', which agrees with
type 5 for your data. (Your data have no ties in the vicinity of
quartiles, so some major disagreements among types are not illustrated.)
x = c(5,15,16,20,21,25,26,27,30,30,31,32,32,34,35,38,38,41,43,66)
quantile(x)
   0%   25%   50%   75%  100%  # defult for 'quantile' is min, quartiles, max
 5.00 24.00 30.50 35.75 66.00 
quantile(x, type=3)
  0%  25%  50%  75% 100% 
   5   21   30   35   66 
quantile(x, type=5)
  0%  25%  50%  75% 100% 
 5.0 23.0 30.5 36.5 66.0 
fivenum(x)
 5.0 23.0 30.5 36.5 66.0

