just take 10 points : 1,2,3,4,5,6,7,8,9,10 (discrete samples)
the mean is: 5.5 and you may take variance to be approx. = 3
this standardizes the data to -1.5,-1.167,-0.83,-0.5,-0.1667,0.1667,0.5,0.833,1.1667,1.5
(now imagine the same thing for a continuous domain)
so if you observe the plot, the PDF will be a curve closer to 1 at E(X) which is well, the most probable value of X and that after standardization is 0. So, the curve is denser around X=0 and not so dense as you move to sides which is why you observe a bell curve post standardization.
but if you look at the original data; the variance = 3(approx.) > 1. If the data were continuous; the mean being 5.5 would imply that the curve peaks at X=5.5(well, not exactly X=5.5 because then f(x) = 0, lets say in an interval epsilon around 5.5) yes; however the variance being larger means the next value is at a greater distance. Notice how 2-1 = 1, 3-2 = 1 but; -0.83-(-1.167) = 0.337 < 1. think of balls scattered in a playground vs. gathered in one place. the density is higher in that one place but negligent in other parts of the playground. in the first case, the density is well, not that high except at some place where you have 2 balls or something.
this also helps answer the question: why would you want to collect them all in one place? well because then it's easier to find.