I'm trying to install a TV mount that has no instructions. Using my drill gauged plate, 15/64" is the smallest hole these lag screws can thread into.
What size drill bit should I now use for pilot drilling into the wood stud?
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When you're drilling and tapping threads into metal to accept a machine screw, you have to use the precise right size drills and taps. But you don't need to be real precise with lag screws in framing lumber. Without a pilot hole or with a too-small pilot hole the wood may split. If the pilot hole is too large, obviously the fastener won't hold.
Look at the lag screw, observe the diameter inside the threads (R, the "root diameter") in the illustration below) not the thread diameter (T). Match up a bit that's slightly smaller than the root. It sounds like you have 1/4" hardware and a 3/16" bit would be fine.
Based on your description, I took a guess at the bolt size, and assumed 3/8", which should be the diameter of the unthreaded shank. That would get a 15/64" pilot hole. If the shank isn't 3/8", measure it and check this chart: https://www.portlandbolt.com/technical/lag-bolt-pilot-hole-diameters/.
If you don't have drill bits graduated by 1/64". round down and use 7/32". That will be close enough. Rub a little dry bar soap or candle wax on the lag bolt to lubricate it and make it easier to drive in.