Stack size plays a very important role in the game of no limit Texas Hold'em.
At any point during the game there is the possibility that all of your chips can be put on the line.
Hence the size of your stack has a significant impact on how you play a hand.
But it is important that your are not only aware of your own stack size, but the size of your opponents' stacks also.

Nevertheless, a \$200 stack may sound like a lot on its own, but if the blinds are \$5/\$10 then this is a very small amount to be playing with.
That's why we will look at the stack size in relation to the blinds of that particular game.
If a player has a stack of \$1250 and the Blinds are \$5/\$10 then he has a stack size of 125 Big Blinds (BB).
The following table will give you a rough categorization of what is considered to be short, small, medium, big and deep stack:
 Short: 20 big blinds or less. Small: 20 - 40 big blinds. Medium: 41 - 90 big blinds. Big: 91 - 200 big blinds. Deep: 200 big blinds or more.

This categorization is not definitive and one can argue about it's bounds. Frankly it's not that important. The point is that the amount of chips in front of you and the amount in front of the players around you influence a number of aspects in online poker, ranging from the psychological to the mathematical.

Most poker rooms have limits on how much or how little you can bring to the table from the start.
There is typically a 10BB minimum and a 100BB maximum for the majority of limits.
This means that players have the option of buying in as a short, small, medium or big stack from the off.

As you can see there is also a "deep stack" condition if you have 200 big blinds or more.
This is common in cash games if a player has bought in for the maximum amount (usually 100BBs) and has either doubled up through an all-in confrontation against another player of equal stack size, or has managed to grind there way up into the realm of the "deep stack".

This article takes a closer look at the average stack size in poker at different stakes with regards to the number of players at the table.
We choose to compare NL10, NL100 and NL1000 without any particular reason, and analyzed more than 9 million hands in total.
A detailed breakdown of the hands we analyzed can be found at the end of the article.

The result of our analysis can be seen in the chart below:

NL10 (Blinds \$0.05/\$0.10)

Starting with NL10 (blinds: \$0.05/\$0.10) our computations shows the average stack size a player has at the beginning of a hands is roundabout 123.5 BB.
That's ~2% above average.

The average of 123.5 BB is ~20% above the default (and maximum) buy-in the PokerStars client offers - which makes sense.
People tend to rebuy to the maximum of 100 BB once they fall below, because they want to maximize their winings in case they double up by an all-in.
And players that are above the 100 BB maximum can't take money off the table without leaving the table.

NL100 (Blinds \$0.50/\$1.00)

Looking at NL100 we can find similarities to NL10.
The average stack size is ~122.5 BB. That's more of less the same as NL10.
In the heads-up games the average stack size of ~128.43 BB is ~5% above average for 3-6 player games.

NL1000 (Blinds \$5/\$10)

Looking at NL1000 the tendency for heads-up games to involve more money per player becomes evident.
In a 3-6 player game the average stack size is 129.27 BB. The average stack in a heads-up game is 167.49 BB.
That's a plus of ~30%.

for the sake of completeness, here's the chart showing the number of hands we analyzed for each stake and number of player:

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