Averages & Weighted Averages

6.1 Simple arithmetic mean

Average = Sum ÷ n
Missing value = (Average × n) − sum of known values

Example: six regions average €31 500 GDP per capita. Five total €174 000. Sixth = 31 500 × 6 − 174 000 = €15 000.

6.2 Weighted average

Weighted average = sum of (value × weight) ÷ sum of weights

Example: City A (2M pop, €32 000), City B (5M, €27 000), City C (1M, €41 000). Weighted avg = (64 000M + 135 000M + 41 000M) ÷ 8M = €30 000. Simple average = €33 333 — wrong because it ignores city sizes.

6.3 Population density as a weighted average

Remaining population = Total population − known populations
Remaining area       = Total area − known areas
Density of remainder = Remaining population ÷ Remaining area

6.4 Anchor method (calculation shortcut)

Deviation for group i = value_i − anchor
Weighted avg = anchor + sum of (deviation × weight) ÷ sum of weights

Example: factories P (95 units/worker, 300 staff), Q (110, 700 staff), R (80, 400 staff). Anchor = 95. Weighted avg ≈ 98.2 units/worker.