Is an antistatic wrist strap necessary?

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Yes grounding on the chassis does work, but it's not a continuous contact, all you need to do is walk a few paces or, even worse, roll your chair around if it's on roller's and wher-hay 15kv of static suddenly appears. If you're really conscientious it'd be ok, but I'm not. Also requires a grounded supply to your case (some aren't)

Look at the size of that thing!
 
Lead has been know to become brittle, and reduce it's electroconductivity quite a bit. Probably from other materials seeping into the cracks and insulating.

No, you just need one ground strap. You're whole body is made of ions and water, which conduct static potential very well. A beer will help ground you even better.
 
My father has a degree as an electrical engineer and he also taught it at a university. He highly recommends using a static mat and wrist strap. In addition he also had this to say..."its crazy to leave the power plugged in while having one of those on and operating on it". Given you may not necessarily being messing with the power supply, things can happen and quite simply your risking your life depending on whether you have it plugged in or not. So....How much guts do you have? If youve gotten shocked before and it didnt hurt much consider yourself lucky because DC power flucuates greatly, it can go on levels from almost painless to fatal. Personally however ive followed his advice and id rather be safe than sorry and havent had a single problem since because last time I wasnt using one even though I was touching the chassis pretty much at all times, long story short...screwed a few components.

Good luck,
_RK_