The PDI comes from the Cumulants analysis and is a measure of the width of a DLS measurement. Standard deviation is normally used to specify the closeness of a consecutive group of measurements rather than the width of the distribution.
You might want to read: Controlled Production of Poly (3-Hydroxybutyrate-co-3-Hydroxyhexanoate) (PHBHHx) Nanoparticles for Targeted and Sustained Drug Delivery
The standard deviation is showed in the Zeta sizer.
If you do multiple measurements and select them all, at the bottom it show the standard deviation of your particle size (ake 5 measurements for example). However, usually DLS underestimate the std. I would use NTA which is better for calculation the std.