One thing you have wrong is I'm pretty sure you can't loop through two arrays at the same time for y in Y and x in range(1, 27):
You have to loop through one, you can do this by looping through the indices of the array instead of looping through the array itself.
for y in range(0,26):
x = y + 1
z += Y[y]
...
Another problem you're having, is that you're actually creating a new graph for each time the code in your for loop exectues, each of these graphs only has one point it. These are two of the graphs produced when your for loop starts working correctly, notice how the values at the x and y axis change.

You need to save each z value to another array and then print a single graph using that array after your for loop has finished (The for loop will change a bit, because you don't need both y and x)
import numpy as np
import matplotlib.pyplot as plt
a = np.array([[1,1],[2,0],[3,0],[4,1],[5,4],[6,7],[7,5],[8,1],[9,2],[10,3],
[11,10],[12,9],[13,27],[14,20],[15,39],[16,40],[17,54],[18,69],[19,74],
[20,191],[21,126],[22,102],[23,121],[24,219],[25,204], [26,235]])
X = a[:,0]
Y = a[:,1]
plt.scatter(X,Y)
plt.show()
#""""""""""""""""""""""""THE CHANGED PART"""""""""""""""""""""""""""""""""""
Z = np.zeros(26)
sum = 0
for y in range(0, 26):
sum += Y\[y\]
Z\[y\] = sum
plt.scatter(X,Z)
plt.show()][3]][3]
#""""""""""""""""""""""""THE CHANGED PART"""""""""""""""""""""""""""""""""""
Produces the graph below

pltcome from?