Your ranked placements explained:

Lets use an example.

For the example we will say League has 1000 players in Ranked.

At the end of season there are 27 divisions. 5x Bronze, 5x Silver, 5x Gold, 5x Plat, 5x Diamond, Master and Challenger.

30 of the 1000 players (3%) ended the season in the top 7 divisions. 5x Diamond, Master and Challenger. 70 of the 1000 players (7%) ended in Plat. 200 of the 1000 players (20%) ended in Gold. 400 of the 1000 players (40%) ended in Silver. 300 of the 1000 players (30%) ended in Bronze.

At the beginning of the season there are 20 divisions. 5x Bronze, 5x Silver, 5x Gold, 5x Plat. We lose 7 (26%) of the divisions.

30 (3%) of the players are pushed down to Plat.

So it now looks something like:

50 of the 1000 players (5%) start in Plat. 150 of the 1000 players (15%) start in Gold. 300 of the 1000 players (30%) start in Silver. 500 of the 1000 players (50%) start in Bronze.

Bare in mind there is some fluctuation due to players over or under performing in their placements.

There is a number that is attached to each player called MMR. This is why you will never see a Bronze player from the previous season go 10-0 in placements and get placed in Plat division in the new season, unless, Riot resets the MMR of all players to 0 (It would be so fun!) at the beginning of the new season.

When the season ends we can say for our example the top player ended with a 5000 MMR and the guy that never wins a game and is at the very bottom of Bronze V ends with an MMR of 0.

It looks something like this:

Diamond to Challenger - 4000 and 5000 MMR Platinum - 3000 MMR Gold - 2000 MMR Silver - 1000 MMR Bronze - under 1000MMR

At the start of the new season it looks something like this:

Platinum - 4000 and 5000 MMR Gold - 3000 MMR Silver - 2000 MMR Bronze - under 2000 MMR

As the season progresses that 5000 MMR that started the season in Plat works its way back up to Challenger. The plat division gets its 3000 MMR players back, Gold its 2000 MMR players back, etc.

If you wait until the higher divisions fill up (Diamond to Challenger) before doing your placements you will likely place closer to or even higher than you ended the season.