Percentages are incredibly useful when talking about how something is a part of something larger: this many percent tax, that many percent of people are unemployed, etc. When percentages are much larger than 100, however, they lose their meaning and their usefulness. Unfortunately, they seem to be increasingly common.

Here are a few examples of more or less recent stories:

*Drug with rage-inducing >5,000% price-hike now has $1/pill competitor**Pharma company boosts the price of an old acne cream 3,900%**An Indian businessman built the world’s largest air-cooler company, giving shareholders 91,000% returns in a decade*- A story about how Domino’s stock price has increased by 2,000% since 2010

But perhaps the silliest example I have ever seen is SpaceX's claim about reducing the cost of sending stuff to Mars:

"We aim to improve the cost per ton to Mars by 5 million percent." https://t.co/kCtBLPbSg8

— SpaceX (@SpaceX) September 27, 2016

Using percent here to make the numbers sound bigger and more impressive is backfiring, I believe, because many people don't understand percentages all that well – and in particular percentages way above 100.

## What Is A Percentage?

Percentages are a simple and incredibly useful concept: divide a number that represents a part by the number that represents the entirety. That gives you a fraction from 0 to 1. We don't like those, so we multiply by 100 for nice whole numbers in a range we can deal with: 0 to 100.

And that is what percent means: *out of one hundred*. When you pay a tax of 12%, that means that for every 100 dollars you pay, 12 go to tax. An unemployment rate of 5% means that five out of every 100 people don't have a job.

A percentage makes numbers comparable and meaningful. If you were told that there were 7.6 million unemployed people in the U.S. at some point in time, that wouldn't be very helpful. Is that a lot? How does that compare to other countries? How does that compare to earlier years, when the population was different?

There is a reason these numbers are reported as percentages. To figure out whether 7.6 million unemployed is a lot, you have to know the size of the labor force: 159.7 million. We're terrible at comparing numbers like that, however, so we divide the unemployed by the labor force and multiply by 100: 4.8% unemployment rate. That is a number we can understand, compare between countries, etc.

## Percentages Over 100

When a value exceeds its normal range, the percentage can go over 100. That is not unreasonable, and when the value is only a little above normal, the percentage is a useful way of expressing that. In particular, when talking about change over time, this can be useful. Stock prices are a classic example here: the stock of X company is up 5% this year. This is a useful way of talking about that because stock prices can be vastly different. A change of a dollar can be large for some and minor for others. Unless you're familiar with that particular stock, the percentage change gives you a much better idea what that means.

Percentages over 100 are also often used to emphasize that a goal has been reached or exceeded. When your sales goal was 85 million dollars and you brought in 97 million, you will be bragging about having reached 114% of your goal. That's fine, that is meaningful.

## Large Percentages, a.k.a. Multiples

In most situations, I don't find it particularly useful to talk about 120% or 150%. Instead, you want to say that something is *20% higher* or *50% higher*. And it gets particularly silly when the percentage reaches 200% and more.

There are words for this purpose. Something that's 200% of something else is also known as *double* that value. 300%? We call that *triple*. A 5,000% price hike? Why, that's otherwise known as 50 times the price (50 times 100% makes 5,000%).

What gets lost in chasing large numbers here is that the multiple is actually much more impressive, even if the number is smaller. I find a 50-fold price increase much more surprising than a 5,000% one. It takes me much longer to parse the percentage difference, and I doubt most people even understand what that means. It's just another large-sounding number.

Another issue is precision: a 3,900% increase suggests that this is the exact amount. That's not even the case, though: this story talks about how an acne cream's price increased from $241 to $9,561 – that's 3,967%. If precision were of any importance here, the percentage would actually be helpful. Since it isn't, and that percentage easily rounds up to 4,000% anyway, it would have made more sense to talk about a 40-fold price increase.

## Crazy Percentages

So how did SpaceX get to the claim that they would reduce costs by *five million percent*? This story clarifies that they're after reducing costs by roughly four and a half orders of magnitude. Four orders of magnitude is a factor of 10,000; multiply by 100, and you have one million percent. And since it’s 4.5 orders of magnitude, multiply by five to get a reduction by five million percent!

To actually express this in percent, the math works a little differently, though. The current costs are 100%. Reducing that down to one 50,000th means we’re diving by that number, which gives us 0.002% (or 0.00002 as a fraction out of 1). Not as sexy perhaps as five million, but at least a meaningful number.

Neither 0.002% nor a 5,000,000% reduction are easy to understand, though. What exactly is wrong with saying: *we're going to reduce costs to one 50,000th*? You can now transport 50,000 times as much for the same price! That's incredibly impressive. Why throw around a meaningless number when you can tell people what they get out of your improvements in terms that make sense?

## Numbers, They Mean Things!

All of my examples here are from communication, either news pieces or press releases. These numbers are meant to tell the audience something. They're also meant to impress, no doubt. But by pushing for the largest number you can make up, you lose people's ability to understand what you're even trying to tell them.

Numbers need to be framed in terms that make sense. Huge percentages don't. Using simple terms like multiples is much easier to understand, and ultimately more impressive, than some huge but meaningless percentage that will be lost on most people.

JG42 says

Some companies want to avoid the percentage problems and use Basis Points instead: http://www.investopedia.com/terms/b/basispoint.asp

richardbrath says

Even small percentages can be confusing to an average user in practice. If a stock gains 25%, you lose that gain if it falls 20%. If a stock loses 33%, you need a gain of 50% to regain where you were at.

In practice, log-based axes on scatterplots and linecharts can correct for the math-perception error. With a log-based axis, the length associated with the percentage gain/loss is consistent rather than using percentages.

Instead of labeling the log base axis in normal units, it may be interesting to label a log axis using a ratio. With such an axis, the interval from 5:1 to 1:1 is the same size as the interval from 1:1 to 1:5. Or, even big ratios can be compared 50:1 to 1:1 will be the same size as 1:1 to 1:50.