Not much happening around the nest this week to report since it was a short holiday week. While the penguins around here worked most called it a holiday. So this is as good a time to follow up last week with some discussion on averaging as that is another big topic.

# Averaging

Everyone knows how to average. We all learned it in grade school. Now there is even a simple function in spreadsheet programs as well as SQL. But much like rounding it may not be that simple. there are times when a simple average is not appropriate. This it of great importance in forestry. As we don’t have dream sequences here lets break to a example.

Diameter | Trees |

2 | 50 |

4 | 20 |

6 | 10 |

8 | 5 |

10 | 1 |

So the average diameter is?

- 6
- 3.37
- 3.89
- 42

Well 3 of those are correct. While it has been postulated that 42 is the answer to the universe it is not in this case. so that leaves us with three main types of averages.

- Simple Average
- Add the diameters together and divide by 5.

Sum(Diameter)/Count(Diameter classes) - Mathematically correct however completely inappropriate in this case.

- Add the diameters together and divide by 5.
- Weighted Average
- Diameters times trees divided by trees.

Sum(Diameter*Trees)/Sum(Trees) - Another correct answer. This can even be appropriate. However in this example probably not the right answer. But don’t dismiss this as it is the proper way to weight values for an average. So the average volume per unit area might be

Sum(Vol*Area)/Sum(Area)

- Diameters times trees divided by trees.
- Quadratic Mean Diameter (QMD) (in this case)
- the square root of the diameter times the trees divided by the trees.

SQRT(Sum(Diameter^2*Trees)/Sum(Trees)) or - (Sum(Diameter^2*Trees)/Sum(Trees))^(1/2)
- Correct! especially for this case. this will not only give the user a weighted average but it will also be weighted towards the larger trees which tend to be the trees of interest.

- the square root of the diameter times the trees divided by the trees.

WARNINGS

- be cautious. many times people will represent simple average as a Quadratic. with this example because foresters talk about QMD all the time.
- Many routines will ignore NULL or empty values. so if the user is averaging 2 diameter distributions by species then one stand doesn’t have a species. The value may not be divided by the proper count or ‘n’. Some times Null or empty is a proper answer.