# How to Calculate an Average Percent Change

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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:

\bigg(\frac{\text{Final } - \text{ initial value}}{ \text{ initial value}}\bigg) × 100

Or in this case,

\bigg(\frac{1500 - 1000}{ 1000}\bigg) × 100 = 0.50 × 100 = 50\%

So the average percent change must be

\frac{50\% }{5 \text{ years}} = +10\% \text{ 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

\bigg(\frac{1100 - 1000}{ 1000}\bigg) × 100 = 10\% \text{ for the first year,} \\ \,\\ \bigg(\frac{1200 - 1100}{ 1100}\bigg) × 100 = 9.09\% \text{ for the second year,} \\ \,\\ \bigg(\frac{1300 - 1200}{ 1200}\bigg) × 100 = 8.33\% \text{ for the third year,} \\ \,\\ \bigg(\frac{1400 - 1300}{ 1300}\bigg) × 100 = 7.69\% \text{ for the fourth year,} \\ \,\\ \bigg(\frac{1500 - 1400}{ 1400}\bigg) × 100 = 7.14\% \text{ 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.

\frac{42.25}{5} = 8.45 \%