why you can't average APIs

The Asia Society's Room with a View website I mentioned in my last post is excellent. However, they have one indicator of air quality that is a little problematic, and here I'd like to explain why. The indicator I'm referring to is average API (which they call "average pollution").

Mathematically, it doesn't make sense to average APIs. This is because the conversion from pollutant concentration to API is non-linear, as shown in this graph:


Why this complicates averaging is best explained through an example:

Consider three different days with APIs 25, 100, and 250. The average API is 125, which corresponds to a PM10 concentration of 200 ug/m^3. Unfortunately, though, the actual average PM10 concentration for those three days is not 200.

Our three days with APIs 25, 100, and 250 correspond to PM10 concentrations of 25, 150, and 385 ug/m^3, respectively. The average PM10 is 187 ug/m^3, which corresponds to a real "average" API of 118, several points lower than that estimated using the other method.

Because of the non-linear conversion, to get an accurate "average API," we have to convert to PM10, average those, then convert back to API.

The difference isn't huge, and I do think that average APIs may occasionally be useful for snapshot, comparative indicators of the air quality of a given time period (as used by the Asia Society). However, it should be understood that this method usually gives a higher (worse) estimate of air quality than reality, and the average should never be used to convert back to pollutant concentration.

Detailed equations for converting back and forth from API to PM10 may be found at the bottom of this post.