Calculation of quadratic mean diameter

Given, you have estimates of:

  1. Basal area per acre (BAPA, ft2/ac)
  2. Trees per acre (TPA, #/ac)

Avery and Burkhart (1994) give a formula for calculating the quadratic mean diameter of a stand based on BAPA and TPA (see middle of page 290).

The derivation of this formula is give below.

The average basal area per tree (BA, ft2)

BA = BAPA / TPA

Using the formula BA = p D2 / 4, then the diameter of the tree of average basal area (a.k.a., quadratic mean diameter, Dq) is:

Dq = [4 BA / p ]0.5
= [4 (BAPA/TPA) / p ]0.5

BUT, we want Dq in inches, therefore we need to include a conversion from feet (for BA) to inches (for Dq).

THEREFORE,

Dq = [4 (BAPA/TPA) / p ]0.5 * 12
= [ (BAPA/TPA) / {(p / 4) / 144} ] 0.5
= [ (BAPA/TPA) / {0.005454} ] 0.5
= same as pg. 290

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