1. variance

Could someone give me a working definition of variance? From what I understand, standard deviation is the data's average amount of dispersal around the mean...so then we square that and get variance but I don't understand what variance tells me. Any help would be appreciated.

jjmclell

2. Originally Posted by jjmclell
Could someone give me a working definition of variance? From what I understand, standard deviation is the data's average amount of dispersal around the mean...so then we square that and get variance but I don't understand what variance tells me. Any help would be appreciated.

jjmclell
Well you don't understand what standard deviation is. It is the root mean
square dispersion (if you will) about the mean. The variance is the mean

RonL

3. Ok, thanks. Can you tell me though why we square the deviation in the variance calc?...Is it just to get rid of negative deviations? It is hard for me to understand why we would want to express the average squared deviation.

4. Originally Posted by jjmclell
Ok, thanks. Can you tell me though why we square the deviation in the variance calc?...Is it just to get rid of negative deviations? It is hard for me to understand why we would want to express the average squared deviation.
When finding some sort of measure of the "deviation on average" we need to average something positive (otherwise for a symmetric distribution at least we would get zero. So to keep things simple we have the family of measures of average deviation:

$\root n \of {Average(|x-\bar{x}|^n})$

To keep this smooth we want $n$ to be even, then the simplest case is for $n=2$ which is the standard deviation.

(The average absolute deviation is used sometimes, but is not as generally useful as the standard deviation.)

RonL

5. jjmclell,

Just to add to CaptainBlack's remarks, another reason why we like to work with variance is that variances "add up": that is, if X and Y are independent random variables, then the variance of X+Y is the variance of X plus the variance of Y:

$var(X+Y) = var(X) + var(Y)$.

This is not true of standard deviations.

Standard deviations, on the other hand, have a more natural dimension, the same as the original random variable. For example, if your sample data is measured in feet (people's heights, for example), then the standard deviation is also in feet. The variance, on the other hand, will be measured in square feet. So the standard deviation may have a more intuitive "feel" about it.