"normalizing" std dev?

Question

First of all, I'm not a statistics person but came across this site and figured I'd ask a potentially dumb question:

I'm looking at some P&L data where the line items are things such as Sales, Direct Cost, Advertising Cost, etc. I have 12 months of data and am trying to plot each of the line items on a scatter plot where x-axis is variability across the 12 months and the y-axis is the magnitude of the entire year (e.g. sum of 12 months).

I'm using std. dev for variability and what I end up getting is pretty much a line (bottom left to upper right) since the std dev is a factor of magnitude (if I'm saying that right). Is there a way to "normalize" std. dev. or calculate variability that isn't dependent on size/magnitude?

Thanks!

Answer 1

For non-negative economic quantities like sales and costs where spread might tend to be proportional to mean, it's often sensible to look at coefficient of variation, which is sd/mean.

CV's are dimensionless (it doesn't matter if you measured in dollars or millions of dollars, nothing changes for CV). The above link gives some advantages and disadvantages.

Sums of terms will tend to have lower coefficient of variation (so annual aggregates will tend to have lower coefficient of variation than monthly totals).

Answer 2

You want to use a standard score, also known as a z-score.

The equation is:

$$z = \frac{x - \mu}\sigma $$

Where $x$ is the value you're testing, \mu is the mean of the values and $\sigma$ is the standard deviation.

So, for example, the following sets of numbers of different scales will both return the same $z$-score when $x$ is the same distance from the mean relative to the scale:

[10, 20, 30, 40, 50]
x = 40
z = (40 - 30) / 15.8113883
z = 0.632455532

[1000, 2000, 3000, 4000, 5000]
x = 4000
z = (4000 - 3000) / 1581.13883
z = 0.632455532