The idea is to give more emphasis in some examples of data as compared to
others by giving more weight. For example, we could give lower weights to
the outliers. The motivation to write this function is to compute percentiles
for Monte Carlo simulations where some simulations are very bad (in terms of
goodness of fit between simulated and actual value) than the others and to
give the lower weights based on some goodness of fit criteria.
USAGE:
y = WPRCTILE(X,p) % This is same as PRCTILE
y = WPRCTILE(X,p,w)
y = WPRCTILE(X,p,w,type)
INPUT:
X  vector or matrix of the sample data
p  scalar or a vector of percent values between 0 and 100
w  positive weight vector for the sample data. Length of w must be equal to either number of rows or columns of X. If the weights are equal, then WPRCTILE is same as PRCTILE.
type  an integer between 4 and 9 selecting one of the 6 quantile algorithms.
OUTPUT:
y  percentiles of the values in X
When X is a vector, y is the same size as p, and y(i) contains the
P(i)th percentile.
When X is a matrix, WPRCTILE calculates percentiles along dimension DIM which is based on: if size(X,1) == length(w), DIM = 1; elseif size(X,2) == length(w), DIM = 2;
EXAMPLES:
x = randn(1000,1);
w = rand(1000,1);
y = wprctile(x,[2.5 25 50 75],w,7)
Durga Lal Shrestha (2020). Returns weighted percentiles of a sample (https://www.mathworks.com/matlabcentral/fileexchange/16920returnsweightedpercentilesofasample), MATLAB Central File Exchange. Retrieved .
1.0.0.0  Added option with different 5 algorithm to compute the quantile 

Change of the screenshot file as it was very big. 

Change of Screenshot as wrong y tick marks 
Inspired: Quantile calculation
Create scripts with code, output, and formatted text in a single executable document.
Felipe G. Nievinski (view profile)
For those looking for a weighted median: wmedian = @(X) wprctile(X,50);
Lorenz (view profile)
I claim that this can be implemented in expected linear time. As you are using sorting, you have at least O(n log(n)), assuming Matlab uses comparisonbased sorting (which is proven to need at least n log(n)  O(n) element comparisons in average).
Thanks for pointing it. Indeed I have used the the formula pk = k/n (type = 4 in R package). What you suggested is type 5 (p(k) = (k  0.5)/n)which is used in MATLAB.
I have updated the code using 6 different algorithm to compute the quantile.
It's all right except that the coordinates fed into the interp1q function is incorrect. However, it's a 2line fix:
After the line "cumW = cumsum(sortedX(:,2));", it should read
coord = (cumW  sortedX(:,2)/2)./(sum(sortedX(:,2)));
q = [0;coord;1];
instead of q = [0;cumW;1];
To A P
Please mention what is incorrect. Do you mean underestimate the median? But if you see the figure WPRCTILE overestimates the median than by PRCTILE. However this depends on the weight vector.
INCORRECT! This understates the median.