Calculating a percentile change in a number is straightforward; calculating the average of a set of numbers is also a familiar task for many people. But what about calculating the *average percent change* of a number that changes more than once?

For example, what about a value that is initially 1,000 and increases to 1,500 over a five-year period in increments of 100? Intuition might lead you to the following:

The overall percent increase is:

[(Final - initial value) ÷ (initial value)] × 100

Or in this case,

[(1,500 - 1,000) ÷ 1,000) × 100] = 0.50 × 100 = 50%.

So the average percent change must be (50% ÷ 5 years) = +10% per year, right?

As these steps show, this is not the case.

## Step 1: Calculate the Individual Percent Changes

For the above example, we have

[(1,100 - 1,000) ÷ (1,000)] × 100 = 10% for the first year,

[(1,200 - 1,100) ÷ (1,100)] × 100 = 9.09% for the second year,

[(1,300 - 1,200) ÷ (1,200)] × 100 = 8.33% for the third year,

[(1,400 - 1,300) ÷ (1,300)] × 100 = 7.69% for the fourth year,

[(1,500 - 1,300) ÷ (1,400)] × 100 = 7.14% for the fifth year.

The trick here is recognizing that the final value after a given calculation becomes the initial value for the next calculation.

## Step 2: Sum the Individual Percentages

10 + 9.09 + 8.33 + 7.69 + 7.14 = 42.25

## Step 3: Divide by the Number of Years, Trials, Etc.

42.25 ÷ 5 = **8.45%**