# Dictionary of Mathematical Eponymy: Yamartino's Method

There’s a problem, in measuring (for example) wind direction that I hadn’t ever thought about until it was pointed out to me: taking averages is fraught with danger.

For example, if you have three readings: one at a bearing of 010, one at 000 and one at 350, you might quite reasonably say “the average is 0” because the average of 10 degrees east of north, due north and 10 degrees west of north is due north.

However, if you add the bearings and divide them by three, you get 120 - and whatever the average wind direction of three gusts clustered around north is, it’s not pointing to the south-east.

*Yamartino’s method* is one way of solving not just this problem, but estimating the standard deviation too.

### What is Yamartino’s method?

Yamartino’s method, instead of averaging the angles, averages the *unit vectors* in the appropriate directions. Given $n$ readings of angles $\theta_i$, you calculate:

- $s_a = \frac{1}{n}\sum_{i=1}^n \sin(\theta_i)$
- $c_a = \frac{1}{n}\sum_{i=1}^n \cos(\theta_i)$

… and your average wind direction is the value of $\arctan\left(\frac{s_a}{c_a}\right)$ in the approriate quadrant.

Yamartino’s method estimates the standard deviation of the measurements. Taking the value of $\epsilon$ to be $\sqrt{1 - \left(s_a^2 + c_a^2\right)}$, the standard deviation estimate is $\sigma_\theta = \arcsin(\epsilon) \left[ 1 + \left( \frac{2}{\sqrt 3} - 1 \right) \epsilon^3 \right]$.

This gives a standard deviation of 0 for constant wind speed, and values within 15% of the true standard deviation for all possible measurements - in typical conditions, the estimate is typically within 2%.

### Why is it important?

The Yamartino method is the EPA’s preferred method for calculating the standard deviation of wind direction - nicely, it only requires one pass through the data (the ‘obvious’ method requires that you know the mean, so it takes two passes).

I imagine it is also a useful exercise for the EdExcel large data set.

### Who is Robert J. Yamartino?

Robert J. Yamartino has been the principal scientist at Integrals Unlimited in Portland, Maine, since 1984. His first degree was from Tufts University, awarded in 1966, and he graduated with a PhD in High Energy Physics from Stanford in 1972. `