Variance and standard deviation, help me out in understanding them

Quote: SilentBob420BMFJ

I understand that "x SD covers y% of results", but I don't understand "It has a SD of x".

Then maybe you don't "completely understand variance". Variance (as EV) is a number describing any random variable. The best description of a random variable is it's full probability distribution, but most often (especially when talking about the "long run") EV and variance is fully sufficient.
EV and variance sums up the most important features of any random variable, namely the center and the width of its underlying probability distribution.

Say you have a game X where you win 40% of the time with even payout. You lose 50% of the time, and you push 10% of the time.
The probability distribution is simply a table: P(X = -1) = .5, P(X = 0) = 0.1, and P(X = 1) = 0.4 (and P(X = ...) = 0 for all others)
Instead, especially if the probability distirbution is more complex, you can also state EV(X) = -0.1, and SD(X) = 0.94
This X is about the game.

Now comes something different. You play the game X mentioned above, say, N=100 times. The result after those 100 plays, is a _different_ random variable, let's call it Y. The probability distribution of this Y is much more complex. If you would want to tabulate it, it would have 201 entries P(Y = -100), P(Y = -99)... P(Y = +99), P(Y = 100), meaning the probability of losing all 100 games, lose all but 1 game (which is a push), up to the probability of winning all 100 games.
If you did play the game 100 times, and you are up +10, then knowing the exact probability P(Y = +10) is of very little practical value. The probability of a +10 result is pretty low - not only it is a losing game, you have 201 different possible results. So P(Y = +10) will be something like less a percent.
What is more important is EV(Y) and SD(Y). You could calculate EV and SD from the P(Y =...) table itself, but since Y is a very complex random variable, this is difficult. However, you can quite easily calculate EV(Y) and SD(Y) from the game X itself:
EV(Y) = N * EV(X) = 100 * -0.1 = -10
and
SD(Y) = sqrt(N) * SD(X) = 10 * 0.94 = 9.4

So your performance Y of the game X, after 100 plays, has a SD of 9.4. Your expected result was EV(Y) = -10, and your actual result was +10.
So you are +20 units up compared to EV. Since the SD of your performance is 9.4, your result is +20/9.4 = 2.1, usually stated as "2.1 SD away".

Now comes the bell curve into play, this is the part you seem to understand. While the result of exact 10 units up is neglectable, a result happening 2.0 SD away (or better) happens with probability of 2.2%.

Quote: MangoJ

Then maybe you don't "completely understand variance". Variance (as EV) is a number describing any random variable. The best description of a random variable is it's full probability distribution, but most often (especially when talking about the "long run") EV and variance is fully sufficient.
EV and variance sums up the most important features of any random variable, namely the center and the width of its underlying probability distribution.

Say you have a game X where you win 40% of the time with even payout. You lose 50% of the time, and you push 10% of the time.
The probability distribution is simply a table: P(X = -1) = .5, P(X = 0) = 0.1, and P(X = 1) = 0.4 (and P(X = ...) = 0 for all others)
Instead, especially if the probability distirbution is more complex, you can also state EV(X) = -0.1, and SD(X) = 0.94
This X is about the game.

Now comes something different. You play the game X mentioned above, say, N=100 times. The result after those 100 plays, is a _different_ random variable, let's call it Y. The probability distribution of this Y is much more complex. If you would want to tabulate it, it would have 201 entries P(Y = -100), P(Y = -99)... P(Y = +99), P(Y = 100), meaning the probability of losing all 100 games, lose all but 1 game (which is a push), up to the probability of winning all 100 games.
If you did play the game 100 times, and you are up +10, then knowing the exact probability P(Y = +10) is of very little practical value. The probability of a +10 result is pretty low - not only it is a losing game, you have 201 different possible results. So P(Y = +10) will be something like less a percent.
What is more important is EV(Y) and SD(Y). You could calculate EV and SD from the P(Y =...) table itself, but since Y is a very complex random variable, this is difficult. However, you can quite easily calculate EV(Y) and SD(Y) from the game X itself:
EV(Y) = N * EV(X) = 100 * -0.1 = -10
and
SD(Y) = sqrt(N) * SD(X) = 10 * 0.94 = 9.4

So your performance Y of the game X, after 100 plays, has a SD of 9.4. Your expected result was EV(Y) = -10, and your actual result was +10.
So you are +20 units up compared to EV. Since the SD of your performance is 9.4, your result is +20/9.4 = 2.1, usually stated as "2.1 SD away".

Now comes the bell curve into play, this is the part you seem to understand. While the result of exact 10 units up is neglectable, a result happening 2.0 SD away (or better) happens with probability of 2.2%.

I never said I completely understand variance, at least not the way you think. Imagine you wanted to learn how a car engine works, or the transmission, etc. Well then what would you say if more than just one person comes along and keeps explaining in different ways that cars have 4 wheels, a gas pedal, a brake, and gears. They tell you cars are a form of transportation. Then they gave you an example of how you start a car and shift it into gear. Wouldn't you say "I know how cars work" in that respect? That's exactly what's going on here.

I'm not, however, referring to you with that example. What you say is too complex for me, as well as that other poster who said he was more confused. But we both know your answers are good if you understand them. I do not have the math knowledge you have, which is probably the issue. I know basic algebra and that's it. Functions and how to place them on a graph, I don't know. And if you're thinking it's gonna be hard to grasp more than the basic stuff of variance without more than basic algebra, you're probably right.

I just figured maybe it was more clear cut than it is. I didn't want to know how to calculate my own variance given xyz, I just wanted to be able to know exactly what a variance of x means in a certain game, but it's just not that simple. Can't just say that it means you'll be up/down 20 units after 100 hands or something like that if the variance were 20.

Quote: SilentBob420BMFJ

I never said I completely understand variance, at least not the way you think. Imagine you wanted to learn how a car engine works, or the transmission, etc. Well then what would you say if more than just one person comes along and keeps explaining in different ways that cars have 4 wheels, a gas pedal, a brake, and gears. They tell you cars are a form of transportation. Then they gave you an example of how you start a car and shift it into gear. Wouldn't you say "I know how cars work" in that respect? That's exactly what's going on here.

If I would want to learn how a car engine "really" works, I would (no joke) first take a course on advanced thermodynamics - before even looking at a car engine sketch.
Being able to draw pressure/temperature diagrams is much more fundamental understanding of the engine, than just "ok here the gas enteres, it get's pressurized there, then is a ignition, and after that this part is moving ..."

If it's either to trivial (all those who give examples) or to complex (all those who give you formulas and ideas), you should define a level of detail you are comfortable with - and try to tell us. The best thing you could do is, try to explain us what variance is in your own words. We will surely notice all gaps you are missing, and can fill you in with details. If you cannot describe what variance is, you probably understand less than you think. Again, waving with "all this I already know, tell me something else" doesn't help you, you should ask yourself "although I know all those points, I cannot figure why it is important to understand variance".

If would like to start from the beginning, you should understand the concept of a random variables. Random variables are not just the outcome of simple dices, they are also more "abstract" random variables, like the performance of your next week's session. Unless you figure out what the outcome of a simple dice has in common with your next week's performance, you won't understand variance. If you would say something like "I know dices, but I don't play craps. So why does it matter for my performance ?" - then you may know about dices, but you definetly don't know about random variables.