The appropriate average for calculating the average percentage increase in population is the Geometric Mean.

Geometric Mean:
The geometric mean is a type of average that is calculated by taking the nth root of the product of n numbers. It is often used when dealing with growth rates, ratios, and percentages. In the case of calculating the average percentage increase in population, the geometric mean is the most appropriate choice because it takes into account the compounding effect of growth over time.

Explanation:
When calculating the average percentage increase in population, it is important to consider the compounding effect of growth. The geometric mean takes into account the fact that population growth is not linear, but rather exponential. It captures the proportional change in population over a given time period, which is essential when analyzing population growth.

Example:
Let's say we have the following population data for a city over a 5-year period:

Year 1: 100,000
Year 2: 120,000
Year 3: 144,000
Year 4: 172,800
Year 5: 207,360

To calculate the average percentage increase in population, we can use the geometric mean. Here's how it's done:

1. Calculate the percentage increase in population for each year by dividing the population of each year by the population of the previous year and subtracting 1:
Year 2: (120,000 - 100,000)/100,000 = 0.2 = 20%
Year 3: (144,000 - 120,000)/120,000 = 0.2 = 20%
Year 4: (172,800 - 144,000)/144,000 = 0.2 = 20%
Year 5: (207,360 - 172,800)/172,800 = 0.2 = 20%

2. Take the product of these percentage increases:
0.2 * 0.2 * 0.2 * 0.2 = 0.0016

3. Take the fifth root of this product to get the average percentage increase:
(0.0016)^(1/5) ≈ 0.2 = 20%

The average percentage increase in population over the 5-year period is approximately 20%.