# Thread: what is standard deviation?

1. ## what is standard deviation?

I wonder if someone can help me understand Standard Deviation? All explanations I can find show a calculation, and then a result at the end of it.

For instance, on example I have seen is 100 coin flips will have a standard deviation of "5". But what does a standard deviation of "5" mean, or represent?

2. st deviation is a measure of how spread out your data or distribution is.
It's the square root of the variance and the smaller these are the more concentrated your values will be.

3. Originally Posted by matheagle
st deviation is a measure of how spread out your data or distribution is.
It's the square root of the variance and the smaller these are the more concentrated your values will be.

Right OK....so i take it then that you would need to work out your variance before you were able to work out your standard deviation?

(these questions are in relation to probability by the way)

4. Originally Posted by Layman
Right OK....so i take it then that you would need to work out your variance before you were able to work out your standard deviation?
Yes. Variance = second moment - mean^2.

Originally Posted by Layman
(these questions are in relation to probability by the way)
As opposed to being about... Statistics?

5. Could you give me an example of how to calculate variance? Perhaps using the 100 coin flip example?

6. Originally Posted by Layman
Could you give me an example of how to calculate variance? Perhaps using the 100 coin flip example?
How about $2$ coin flips...

Define $X$ as the number of heads that appear in $2$ coin flips

$E\{X\} = 0P(X=0)+ 1P(X=1) + 2P(X=2) = 0(1-1/2)^2 + 1[1/2(1-1/2)] + 2(1/2)^2$

$E\{X^2\} = 0P(X=0)^2+ 1P(X=1)^2 + 2P(X=2)^2 =$ $0[(1-1/2)^2]^2 + 1[1/2(1-1/2)]^2 + 2[(1/2)^2]^2$

$Var(X) = E\{X^2\} - (E\{x\})^2$

$StandDev(X)= \sqrt{Var(x)}$

7. I have no idea what all of that means.....where do i start with that?

8. $E(X)$ means "expectation" or mean... You know, average.

Like whats the mean of $8,3,6,1?$

Its $\frac{8+3+6+1}{4}$

How about $8,8,3,3,3,6,1?$

Its $\frac{(2)8+(3)3+6+1}{7} = \frac{2}{7}8 + \frac{3}{7}3 + \frac{1}{7}6 + \frac{1}{7}1$

Take, $\frac{2}{7}8$

Think of $\frac{2}{7}$ as the probability that $"8"$ occurs. Right cause we have $7$ possible numbers and $2$ out of the $7$ are $"8"$s. Follow?

Now let $X$ be a random variable that takes on one of the values, $8,8,3,3,3,6,1.$ So $x=8$ with probability $2/7;$ $x=3$ with probability $3/7....$

We define expectation it the theoretical sense as

$E(X) =$ possible value that $x$ can take on $\times$ probability of this value $+$ possible value that $x$ can take on $\times$ probability of this value + ...

$E(X) = iP(x=i) + jP(x=j) +....$

$E(X) = \sum_{i} x P(x=i)$