Percent change is the relative difference between an old value and a new value. Positive values represent an increase over time, while negative numbers indicate a reduction.

For example, if the price of a candy bar changes from $1 to $1.10, it’s a 10% increase.

In this post, learn the percent change formula and the step-by-step calculation process. I’ll work through examples and explain a surprising characteristic of increases and decreases at the end!

## Percent Change Formula

Use this formula to calculate the percent change:

Here is the step-by-step process for the calculation:

- Take the new value and subtract the old value to find the change.
- Divide that difference by the old value.
- Multiply the result by 100 to convert the decimal to a percentage.

In the percent change formula, notice how the subtraction produces a positive numerator when the newer value is greater than the older value and a negative numerator when the newer value is smaller. That allows the formula to produce a positive percent change for an increase and a negative value for a decrease.

The denominator uses the old value because that is the baseline or starting value. You want to determine the shift relative to the initial value.

## Examples

Let’s work through two examples.

Suppose a nature reserve has an initial population of 128 elephants and a year later it has 142. What is the percent change?

Here’s the process:

- Calculate the difference between the new and old values: 142 – 128 = 14
- Divide that difference by the old value: 14 / 128 = 0.109
- Multiply the proportion by 100 to obtain the percentage: 0.099 * 100 = 10.9%

The elephant population in the nature reserve grew by 10.9% over a year.

What does it mean exactly? We know that the elephant population increased by 14. Those 14 additional elephants represent 10.9% of the baseline value of 128 elephants.

Next, imagine Fred has $12,000 in his savings account. Unfortunately, he spends too much money, and next year he has only $9,000.

What is the percent change?

- 9000 – 12000 = -3000
- -3000 / 12000 = -0.25
- -0.25 X 100 = -25%

The negative percent change (-25%) indicates that Fred’s savings account balance decreased by 25%.

In this example, the shift in value over a year is a decrease of $3000 (-3000). That -3000 is -25% of the baseline value of 12,000.

## Caution with Positive and Negative Percent Changes

You might be surprised to learn that if you have an X% percent increase followed by a -X% decrease, you don’t end up at the original value! Let’s see this in action and understand why.

We start with $100 in our account. Over a year, we save and end up with $125, producing a 25% change.

For the next year, we start with $125 and spend $31.25, leaving us with $93.75 at the year’s end. Let’s plug those numbers in the percent change formula:

In this example, we start with a 25% increase followed by a 25% decrease. It seems like our account balance should end up at the original starting value of $100. Instead, it’s down to $93.75! How did that happen?

The key reason is that the baseline value in the denominator is different.

In the example, we have the same percentage changes but in opposite directions, 25% and -25%. However, they don’t cancel each other out equally because the changes occur in the context of different baseline values in the denominator: $100 vs. $125. Consequently, the same percent change with a larger baseline value produces a larger absolute difference:

%Change X Baseline Value = Absolute Change

If you hold the percent change constant but increase the baseline value, you end up with a larger absolute change.

In our example, the absolute change associated with a -25% change and a baseline of $125 is greater than +25% change with a baseline of $100. Hence, our account balance declined overall.

Learn more about percentages in my posts, Relative Frequencies and Their Distributions and Percent Error.

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