to weigh or measure something against a fixed standard: to mete out a quantity, gauge a person's true worth, or come out right when tested
The picture behind the word is a set of scales or a measuring cup. At its plainest it means to hand out an exact amount: in Isaiah's hymn of creation God "metes out" the waters, marks off the heavens, and fixes the weight of the wind. Isaiah sets it beside מדד, the verb for measuring length with a line, so that together they sweep across every kind of measuring.
From weighing quantities the word moves to weighing people. Proverbs pictures the LORD lifting human hearts and motives onto the scales, gauging what a person is truly worth beneath what they claim. To be "weighed" by God is to have one's real character assessed against a standard that nothing can fool.
The same image turns sharp in Ezekiel. The people protest that the way of the Lord is not balanced, which older translations give as "not equal" and newer ones as "not fair." Their charge is that God's dealings do not come out even. God throws it back: it is their own conduct, not his, that fails to weigh true.
So one verb reaches from apportioning a measured amount, to weighing a heart, to a life "coming out right," held together by the idea of a fixed standard that something is measured against.