Percentage Change Calculator

Table of contents:
A percentage change calculator is useful for determining the difference in two values measured in percentage.

This is an expression that comes up all the time in various fields. We've all seen news stories citing a statistic like "Unemployment is down 7%," or a commercial claim that a brand of toothpaste is "75% more effective at fighting cavities than the leading brand." However, percentages are not always intuitively grasped.

There are several ways that percentage statistics can be reported which lend themselves to logical fallacies. Working with percentage change formulas is an important exercise in visualizing increase and decrease rates over time.

The Percentage Change Algorithm

The formula used is simply:

(((X - Y) / |Y| ) * 100)

Where X is the new value, and Y is the original value. Note the pipe ("|") symbol around the second Y; that denotes unsigned integer value. That's in case the value of Y is negative. In plain English, we describe it as "Take the difference of the new value and the original value, then divide it by the original value (ignoring the minus sign if it's negative), then multiply it times 100."

For example, if we start with 50 and end up with 100, we subtract 50 from 100 to leave 50, divide it by 50 to get 1, then multiply it times 100 to get 100%.

For a real-world application of working with negative values, imagine you were on a game show where you win and lose money based on your answers to trivia questions. You had a score of $2000, but after several wrong answers, you're in the hole with -$200. Your percentage change was that you lost 110% of your prior score. (( -200 - 2000 ) / 2000 ) * 100 ). You can see that 100% of 2000 is 2000, and 10% of 2000 is 200, so the total money you were set back is $2200, 110% of 2000.

Now, if you managed to ring in with several correct answers, imagine your score back up to $2000. The percentage change this time is 1100%. This looks like an awkward operation at first, but we've just added $2200 back to our balance, which happens to have been -$200. Subtract ( -200 ) from 2000, which is the same as adding a positive difference, making 2200. Divide that by the unsigned original value: ( 2200 / 200 ) = 11. Multiply times 100: 1100%. To see this more clearly, mentally add $400 to both values: from $200 to $2400 is also a percentage change of 1100%.

This calculator works both ways, allowing you to see both the increase or decrease in the percentage rate between the two values. You would typically report this as "our productivity is up 10%" or "Chicago's murder rate was down 3% this month."

Real World Example: Population Statistics

World human population and the population of individual countries is a concern on two fronts. You want your resource production to be adequate to feed and house everyone, so you have to watch how quickly the population expands. A population trend that rises too fast is cause for Malthusian alarm: How are we going to feed all these people? On the other hand, a population decrease can also be cause for alarm, because if it keeps going long enough, you become extinct! Poor planning for population increases and decreases has led to many a crisis in government management in the past.

To calculate population growth, we say "X - Y = Z," where X is the current population, Y is the previous population, and Z is the growth rate. If the rate is positive, the population is growing. If it's negative, the population is dwindling. A zero value means a static population. If everyone in China were to be paired off in marriage and each two-parent family in China had two children without fail, that's a static population; also known as "replacement population" as each family's children is just enough to take the place of the parents after they die off.

If we then divide the growth factor by the previous population and multiply that times 100, we get a population growth rate. Note that we do not have to worry about negative signs when it comes to population statistics, as we can never have a number of people below zero. So our population percentage change formula works as:

(((X - Y) / Y ) * 100)

In the year 1800, the world's population was estimated to be 1,000,000,000 - one billion people. By the year 1900, the world population had grown to an estimated 1,600,000,000, or 1.6 billion people. The population growth was 600 million people. The population change was 60%, over a century's time.

In the year 1998, the world population was 5,988,846,103, by which time census techniques and computer technology allowed us to get a more accurate number. In 2018, twenty years later, the world population was 7,632,819,325. The world population change over that two-decade period was 1,643,973,222, round it off to 1.6 billion. The population change percentage from the year 1998 to 2018 was 27.45%. At first that seems like a slower growth rate, but you have to account for the shorter period of time.

To compare using the same time span, chart the world population from 1900 at 1.6 billion to the year 2000 at 6,145,006,989 people. That's a growth rate of 284%!

Now as always, interpreting the data is its own sort of challenge. Are people simply reproducing faster? Likely that's not actually the case. Instead, people are dying at a far lower rate from before, thanks to improvements in modern medicine and general overall standards of living. If anything, recent statistical data points to the opposite trend; our growth rate is slowing as people have fewer children, even if they're still living longer.

Other Applications Of Percentage Change

Percentage change data comes up frequently in such diverse pursuits as finance, industry, biology, and chemistry. It's applicable whenever you need a rate expressed concisely as a comparison. Practice spotting situations where expressing a percentage change between two numbers helps to put their difference in perspective.

Percentage Change Calculator

Percentage Calculator
Percentage Increase Calculator
Percentage Difference Calculator
... more