Weapon weight increases the amount of time it takes for you to ADS (aim down sights) or "scope".
This is explained in a loading screen tool-tip.
For example having a lighter scoped weapon will allow you to ADS quicker and reduces the fade-out time of the black overlay.
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All of the weapon modifications will highlight what they're changing.
Having a "lighter" weapon in itself will not directly change anything, but the modification may impact another statistic.
For example; An advanced receiver increases damage and gives you a better rate of fire A full stock gives you better recoil A quick eject drum magazine gives you better ammo capacity and improved reload speeds Additionally, "heavy" weapons will effect your movement speed while they're equipped.
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A loading screen tip for the PC version specifically says that weapon weight inversely affects draw speed.
I'll screenshot it if I see it again for support, but I can safely say that yes, weapon weight matters (though it might not matter much).
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Most of the mods that increase weapon weight also increase VATS cost.
This includes the heavy receiver (light reduces cost, but anything with "sensitive trigger is better) longer barrels (stub is cheapest), scopes (Reflex is best, focus seems to be keyword there), and stocks as opposed to pistol grips.
Muzzles don't matter though.
Someone also said that weapon weight also affects bash damage.
I haven't really tested this
but I have read that Minigun + Shredder + basher perk brutalizes enemies.
gaming.stackexchange.com/users/132286/John Pinkerton
I asked this question myself recently, and found this posting.
After performing my own experiment, I have determined that although heavier types of weapons (gauss rifle v pipe rifle) have slower drawing and aiming speeds, there is no discernable difference between a heavier version of the same type of weapon (fully upgraded pipe rifle v light pipe rifle).
I also determined that there is similarly no difference between a pistol and a rifle version of the same weapon type.
In conclusion, the drawing and aiming speed of a weapon appears to only be tied to the type of weapon, and not it's actual weight.
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Due to recent experience with games like Destiny and how significantly things have changed since Fallout 3 and New Vegas..I've been wondering if there are any implicit effects induced by higher/lower weapon weight. For instance: would a 10mm pistol that's heavier than a stock version of itself have reduced recoil but slower handling (draw/stow speed)? Perhaps slower reload speed?
Does weapon weight affect anything other than carry weight?
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<fallout-4>
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Weapon weight increases the amount of time. A loading screen tip for the PC version specifically says that weapon weight inversely affects draw speed.  Most of the mods that increase weapon weight also increase VATS cost.   Weight supposedly affects the time it takes to aim with a weapon, but few other things might be also affected.
People state that weight affects the time it takes to scope. While few others mention that weight might affect other things too.