What the evidence actually says
Because markets rise more often than they fall, investing a lump sum immediately has historically beaten drip-feeding it in roughly two out of three periods. Money out of the market has a cost, and on average that cost exceeds the protection that waiting provides. That is the statistical answer: if you have it, invest it.
But the average hides the distribution. The times lump-sum loses are exactly the times that hurt most: invest everything the month before a crash and you will feel it for years, statistically fine and emotionally scarring.
What averaging is actually for
Spreading a windfall over, say, six to twelve months is not really a returns strategy; it is a regret-minimisation strategy. It guarantees you are never the person who invested everything at the top, in exchange for usually earning slightly less. That trade can be completely rational, because the investor who panics and sells after bad luck loses far more than averaging ever costs.
The honest question is not which is optimal but which you will survive: if a 20% fall the month after investing a lump sum would make you sell, averaging is cheap insurance against yourself.
The part everyone misses
For most people the debate is academic: money arrives monthly from a salary, so you are dollar-cost averaging by default, and the right move is simply to automate it and never interrupt it. The lump-sum question only exists for windfalls: inheritances, bonuses, house sales.
A sensible windfall rule: decide a schedule while calm (half now and the rest over six months is a common compromise), write it down, automate it, and refuse to renegotiate with yourself based on headlines.