what is standard deviation?

**Layman** · April 6th 2010, 11:33 PM

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?

**matheagle** · April 6th 2010, 11:36 PM

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.

**Layman** · April 7th 2010, 01:09 AM

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)

**Anonymous1** · April 7th 2010, 04:14 AM

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?

**Layman** · April 7th 2010, 04:19 AM

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

**Anonymous1** · April 7th 2010, 04:27 AM

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

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

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

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

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

**Layman** · April 7th 2010, 06:10 AM

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

**Anonymous1** · April 7th 2010, 06:25 AM

$E(X)$ 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 + ...

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

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

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