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I want to know if this is a good place to calculate variance in my data,and how to interpret or explain the units of the variance answer.

I have 2 lists of corresponding data. ListA has a starting price, ListB has the ending price of the corresponding item in ListA.

Subtracting the two lists item-wise and looking at the differences in price is not very relevant because the items in the list are not closely related. For example is could be prices of bikes mixed in with prices of cars.

I want to be able to say something like "These 30,000 items typically varied in price (beginning versus ending) by x%"

*some specificity on what typically means may be needed*

It sounds kind of like an average but I'm not sure if that's the bets way to capture what's going on since (again) the items are not closely related even though the "method" by which their prices might change is. For example the could be House prices, sold by a set of the same realtors. The "method" is real-estate sales, but the houses themselves are not related to an extent that we should be compare the prices of the houses. Instead we should compare the mark-up or mark-down of the house prices form their own starting points, as this could later be related to the effectiveness of the realtor to sell.

Is this a variance problem? If not, what calculation will lead me to an outcome where I can make the statement above?

"These 30,000 items typically varied in price by x%"
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Percentage variation is given by the growth rate, i.e. (Closing price-Starting price)/Starting price. Have you considered using that? It seems to fit well to your problem. – johnny May 17 '12 at 22:59

If the starting price (at time zero) of asset $i$ is $x_{i,0}$ and the ending price is $x_{i,t}$ then you can define the return over the period by:

$$r_{i} = \frac{1}{t} \cdot \frac{x_{i,t}-x_{i,0}}{x_{i,0}}$$

The factor of $1/t$ takes into account the fact that the relevant time periods for each asset might be different - an increase of 10% over 1 year is more significant than an increase of 20% over 10 years.

You can then look at the mean return:

$$\bar{r} = \frac{1}{N}\sum_{i=1}^N r_{i}$$

which tells you the average percentage increase for each of the assets, and the standard deviation $\sigma$ of returns:

$$\sigma^2 = \frac{1}{N} \sum_{i=1}^N (r_i - \bar{r})^2$$

which tells you how much variability there is between different assets.

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σ2 is the variance not the standard deviation. Regard ing the units for variance it is the square of the units for the item in the sample. It also sounded like the OP wanted to express variation in terms of the range of increase rather than the variance or the standard deviation. – Michael Chernick May 18 '12 at 5:03

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