How are Percentages Used and Abused?

Percentages are used everywhere. From discounts in shops to stories about the efficacy of a new drug. It is perhaps this widespread usage that makes us so susceptible to misinterpreting them. Of course retailers want you to think you're getting a better deal than you are. But it is always a good idea to think carefully about any percentage presented to you and work out what it means.

Let's be curious and ask, 'how are percentages used?'

Per cent

Firstly, it is probably wise to think about what % really means. Per cent is Latin for 'by the hundred'. In other words 50% means 50 per hundred. So if there are twenty items and 50% of them are defective then that means half, and so ten items are defective.

Absolute or relative?

A higher percentage does not mean that the value is higher. If I have two envelopes with cash and was to offer you 75% of envelope A or 6% of envelope B, which offer would you take? Instinctively you might go for the first offer. However, without knowing how much cash is in each individual envelope you can't work out which is the better deal. If envelope A had £100 then you would get £75. But envelope B might have £10,000 and 6% of that is £600 - a far better deal.

In this case the absolute value is more important than the relative (percent) value. A real life scenario of this would be deals in a supermarket. Two similar products are on offer. The more expensive one (say £15) has 25% off. The cheaper one (say £10) has no offer. Panic buying may lead you to pick up the one with 25% off, even though it is £1.25 more expensive.

Sometimes the relative value is more important. If I want to compare two new drugs to see which is the best I might look at the percentages (of course this is greatly simplified as there are many more factors in drug development) of those cured.

Drug A was tested on 7000 people. Of those, 4500 were cured after taking the drug. Drug B was tested on 10,000 people and 6000 were cured. Which is the better drug? Well the absolute values would point to drug B, but the relative values disagree. Drug B cured 60%, whereas drug A cured 64% of people.

It doesn't add up!

There's a sale on. All items in the shop have 50% off. You have a voucher that gets you 20% off. The best part is that you can use both offer in conjunction with each other! That's a whopping 70% off.

Only it isn't. The voucher will take 20% off the half-price. In other words if you have an item that costs £80, the 50% off will take it down to £40. Your 20% off voucher will take 20% off of £40. (40x0.2=8) So you get an additional £8 off taking the price down to £32. That's actually 60% off the starting price. Not 70%.

Base value

New start-up company XYZ ltd is positively booming! Figures show that in the last 6 months employment by the company is up 500% and their customer base has grown by 1000%. Compare that to big company ABC Inc. which has seen employment increase by just 0.5% and the customer growth slow to just 2%.

Wow! XYZ ltd is schooling ABC Inc. in how to run a business, aren't they? Well, are they? Let's look at the facts. XYZ ltd is a new start-up company. That means it will likely employ less people than a big company and also have a smaller customer base.

If we say XYZ was completely ran by its founder for the first months of its life, then a 500% increase is actually just an additional 5 people bringing the total up to 6 employees. ABC on the other hand may have employed 20,000 people. An increase of 0.5% is a gain of 100 employees. Sure, XYZ's rate of employment is higher, but ABC is now employing 20,100 employees - significantly more than XYZ. The statistics for the customer base are the same. If XYZ had 2 customers then they now have 22 customers, an increase of 20. ABC may have 500,000 customers, to a 2% rise is actually an increase of 10,000 customers.

Percentages are useful to compare figures whose base values are different. The drug example shows that. But, it is vital not to be misled by percentages as they are not always appropriate.

100

If you are ever shown a break down of opinion in the form of percentages the first thing you should do is check the numbers add up to a hundred. If they don't, then take an conclusions made with a pinch of salt. It is worth noting that sometimes ~99 or ~101 may be the total because of rounding.

Fox News, America's Newsroom, 12/8/09, via Media Matters

Those percentages add up to 120%. Media Matters showed this as Rasmussen's poll data:

• 35 percent responded "Very Likely" 

• 24 percent responded "Somewhat Likely" 

• 21 percent responded "Not Very Likely" 

• 5 percent responded "Not Likely At All" 

• 15 percent were unsure.

That adds up to 98% most likely due to rounding. It turns out Fox News' 'somewhat likely' category contains both 'very likely' and 'somewhat likely' responses. Then 'very likely' repeats in its own group. The two not likely categories got put together. Hey presto! 120% and figures that match the story (rather than a story to match the figures).

Of course, it should be noted that I do not support the result of the survey. I cannot tell who Rasmussen asked. It could be they asked the general public. But asking the general public if they think scientific data was falsified is about as useful as asking if the colour blue is nicer than the colour green.

What do think about percentages? Let me know in the comments below. You can share this post with the links to the left, or you can follow It Is All Science with the links to the right.

Remember, it is all science. Let's be curious!