Less is Not the Same as More

The title of this posting is clearly a self-evident statement, but despite this, there is a clear tendency for some, perhaps many, to forget its truth and significance. A few years ago, I noted a research organization had adopted an approach in writing its headlines on particular research reports with claims that the reports described how to achieve desirable outcomes such as: "Reduce sales costs by six times"; "Cut overhead by 4x", etc.

While such outcomes are desirable, they are also mathematically impossible! While it is a relatively simple thing, mathematically speaking, to increasecosts by as much as two, four, six or any number of times, it is wholly different with respect to reducing costs or any other typical performance measure. Once you reduce costs as much as onetime, you have eliminated them entirely, and the only way to get past zero in such a case would be to translate what were once costs into revenue. One times anything always yields the same number you started with, so nothing real can be reduced to below zero.

More recently, in fact, just last night, I listened while a highly respected newscaster on ABC News noted that the dosage of one airport full-body scan involved as little as 2,000 times less radiation than a chest x-ray, and 200,000% less than a CT scan. ["Pilot Rebellion: Pilots Refusing to Use Full Body Scanners or Submit to Patdown" ABC News with Diane Sawyer, Nov 10, 2010 (abcnews.go.com)] Again, while it is a simple matter for the dosage of one application to involve 200,000 times as i.e. 200,000 times moreas another, it is impossible for it to involves 200,000 times less.

The math is pretty simple. If a given treatment involves 1 unit of radiation or some other toxic or dangerous content, and another involves 200,000 units of the same risk, then that which involves the higher amount does create 200,000% or 200,000 times more risk. But if the desire is to describe how much lower the low-risk number is than the high-risk counterpart, the best that can be said is that the risk of the low-risk option is only 1/200,000 as great, not 200,000% or 200,000 times less.

When you start with the higher figure and wish to describe how the lower relates to it, you can never get below a 100% or one times reduction, because everything below that is in negative space. 200,000 divided by one equals 200,000, but one divided by 200,000 equals .000005, meaning that one is 99.9995% less than 200,000, not 200,000% or 200,000 times less.

To get from 200,000 to one, it is necessary to get rid of 199,999, or 99.9995% of 200,000. To get rid of 200,000% of 200,000 would require eliminating 200,000 times 200,000 or 40,000,000, and where would all that extra "stuff" come from? It may be understandable for news media and marketers to look for the most eye- or ear-catching way of describing differences, but in the case of more vs. less, the rules of mathematics severely limit what is true regarding differences involving "less". 100% less or one times less is usually the absolute limit.