Pitfalls of percentages

[C1ay]

Several years ago, a nurse at a Twin Cities hospital temporarily filled in for a head nurse who took a short leave. The nurse was given a 10 percent raise; when she returned to her old job, she took a 10 percent pay cut.

 

lefthttp://hypography.com/gallery/files/9/9/8/PercentageKey_thumb.jpg[/img]If that sounds equitable to you, Akshay Rao, General Mills Professor of Marketing at the Carlson School of Management, wishes to point out a few things. Rao and Haipeng Chen, a Carlson school Ph.D. who is now an assistant professor at the university of Miami, have completed a study showing that people often treat percentages like whole numbers, resulting in systematic calcuation errors. Their paper will be published in an upcoming issue of the Journal of Consumer Research.

 

The nurse is a perfect example of the kind of error they're talking about. Suppose her original salary was $1,000 a week. A 10 percent raise would bump it up by $100, to $1,100. But 10 percent of that salary is $110, not $100. Subtracting $110 from $1,100 would leave her with only $990--a one percent cut from her original salary.

 

"It took a month to convince [the human resources department] that she now earned less than before," says her husband. The error was corrected.

 

In their paper, Rao and Chen report that a common problem with percentages arises when consumers are unaware of how multiple discounts work. For example, if a store offers 25 percent off everything but with "an extra 25 percent" off certain items, that doesn't mean 50 percent off the original price. Instead, the actual discount amounts to about 43 percent. That's because the store will take the second 25 percent off the discounted price. Therefore, on a $100 item, the second discount will be 25 percent of $75, for a final price of $56.25--not $50.

 

Firms can benefit at the expense of consumers who don't realize this. In their study, Rao and Chen tested the impact of offering a 20 percent discount and an additional 25 percent discount versus an equivalent 40 percent discount in a retail store. When double discounts were offered, the store reported more buyers, sales volume, revenue, and profit.

 

In the case of double discounts, it's understandable that consumers would not know how the second discount is computed. After all, the two discounts are applied only once, at the cash register.

 

But in other cases, such as fluctuations in the stock market, time elapses between one percentage change and the next. Even then, some people forget that they must continually adjust base values when computing sequential percentage changes. When a value goes up and then down (or vice versa), things can really get interesting.

 

"Imagine your stock portfolio went up 40 percent last period, and down 30 percent this period," Rao says. "You are not better off by 10 percent. Your portfolio is down two percent."

 

The problem with percentages isn't that we're not smart; it's just that percentages are a relatively new mental exercise. But with effort, we can compensate.

 

"We argue for increased math education because there are some calculations that aren't innate to the mammalian brain," Rao explains. "For example, the brain can detect changes in light intensity. This is a very sophisticated calculation for the brain, but it confers an evolutionary advantage. For instance, a change in light intensity may mean a predator is approaching. But percentages are an artificial construct that requires learning. From a public policy standpoint, we have to teach people how to do this."

 

Sometimes, mistakes can have national ramifications. Rao gives the hypothetical case of the Pentagon asking for the same percentage increase in its budget every year. But as a budget grows, so does the number of dollars represented by that percentage. If Congress tried to increase the appropriation by the same dollar amount rather than the same percentage, the story could be spun as Congress cutting the military budget, says Rao.

 

Or, suppose Detroit improves the mileage of its fleet by, say, two miles per gallon each year. The percentage improvement will be less and less, even though the rate of progress toward a mileage goal remains the same.

 

Rao's advice? "Make sure you know what the base of calculation is, especially for multiple percentage changes," he says. And if you're shopping for sales, bring a calculator.

 

Source: University of Minnesota

[Chacmool]

Very interesting! I wonder how many times I have been fooled...

[Tormod]

Very interesting! I wonder how many times I have been fooled...

 

About 10% of the times.

[C1ay]

About 10% of the times. :D

 

10% of the time or 50% of 20% of the time? :(

[freeztar]

About 10% of the times. :)

 

So if I'm fooled an additional 10% more than Chacmool then I'm being fooled 20% of the time, right? :D :(

 

Although not higher math, this is a very good justification for learning math, at least at the basic level. (But this is for another thread...)

[Mercedes Benzene]

It's funny that many people would simply ignore this and go on with their lives.

 

...just another way that big-business attempts to fool the people.

[Chacmool]

It's funny that many people would simply ignore this and go on with their lives.

 

...just another way that big-business attempts to fool the people.

Big business isn't only attempting, they are succeeding in fooling people.

[Chacmool]

So if I'm fooled an additional 10% more than Chacmool then I'm being fooled 20% of the time, right? :D :(

 

Although not higher math, this is a very good justification for learning math, at least at the basic level. (But this is for another thread...)

Right on. I'm going to pass this on to my sister - who is a math teacher - to demonstrate to her students how important math is in everyday life.

[jungjedi]

I'm being ripped off!:(

[Nootropic]

I don't think it's so much math education as it is consumer awareness. I had no idea that's how stores calculated discounts, and I've recieved a quality public math education.

[freeztar]

I don't think it's so much math education as it is consumer awareness. I had no idea that's how stores calculated discounts, and I've recieved a quality public math education.

 

Yes it is misleading.

Taxes, however, are not (at least here in the USA). You are taxed on your gross income, be it state or federal.

 

But in a consumer situation, knowing the bottomline price is the weapon against such an attack. In most cases it is trivial. Consider a pair of jeans that cost $15. Suppose the store has a 20% discount on those jeans, and you have a store "savings card" (ie regular shopper card) that entitles you to and extra 5% off your purchase.

 

In the case of a store applying the percentage discounts in-line, rather than parallel, you would pay $15*.15=$2.25 $15-$2.25=$12.75 $12.75*.05=.64 $12.75-0.64=$12.11

parallel would be: $15*.20=$3 $15-$3=$12

 

So you spend an extra $0.11 with the in-line case.

 

This is potentially very dangerous with high dollar items, especially with a larger percentage gap.

 

Consider a plasma TV on sale (hypothetical) for $5,000. The store is having a sale of 10% off everything. You have a card that entitles you to a 30% discount on any one item (combinable with other offers, which is most often not the case). So the store takes $500 off the TV and applies your 30% discount card afterwards. So the $4500 becomes $4,500-$1,350=$3,150. Now if the discounts were combined (parallel), then you would only spend $4,500-$2,000=$2,500. :eek2:

 

I'm sure car salesmen love this one.

[Michaelangelica]

Just got a one page promo from the Liberal senator-Minister for Communications"

(bit of a worry) along with everyone else in the state I suspect.

She showed a list of previous 'bad' labor party spending 10 years ago compared with what her 'good' liberal party was spending now

EG

Labor 1997 100 mil

US 2007 (Libs.) 150mil

It looked very impressive.

Of course she used whole numbers and did not factor in the average 4% P.A. inflation over those ten years.

Anyone (math wizard) want to do the sum for me?

Is there a formula I can use?

[freeztar]

Just got a one page promo from the Liberal senator-Minister for Communications"

(bit of a worry) along with everyone else in the state I suspect.

She showed a list of previous 'bad' labor party spending 10 years ago compared with what her 'good' liberal party was spending now

EG

Labor 1997 100 mil

US 2007 (Libs.) 150mil

It looked very impressive.

Of course she used whole numbers and did not factor in the average 4% P.A. inflation over those ten years.

Anyone (math wizard) want to do the sum for me?

Is there a formula I can use?

 

After ten years, starting at $100 mil and increasing at a rate of 4%/yr, you end up with $148 mil.

[C1ay]

Just got a one page promo from the Liberal senator-Minister for Communications"

(bit of a worry) along with everyone else in the state I suspect.

She showed a list of previous 'bad' labor party spending 10 years ago compared with what her 'good' liberal party was spending now

EG

Labor 1997 100 mil

US 2007 (Libs.) 150mil

It looked very impressive.

Of course she used whole numbers and did not factor in the average 4% P.A. inflation over those ten years.

Anyone (math wizard) want to do the sum for me?

Is there a formula I can use?

 

Use the standard compund interest formula for future value where

FV = Future Value

P = Principal

r = Rate (expressed as a decimal)

n = number of periods

 

FV = P(1+r)^n

100,000,000 compounded for 10 periods is 148,024,428.5