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A really good paper on something most stats folks know but most everyone else forgets: Percentages applied in order are not themselves additive. You have to “do the math” since the base changes after each percentage increase or decrease.

So, the classic case: You have a 20% off coupon, and the item is marked off 25% from regular price. What’s your actual savings at the register? Most people immediately add 20% and 25% and arrive at 45% off, immediately running to the register.

But if the price was $100, your final price will be $60. That’s only 40% off. You thought it was going to be a better deal, didn’t you?

The paper mentions an even more insidious error in its intro. “A 60% decrease followed by a 70% increase on the standardized test scores in the state of California seemed to cheer up a lot of people”, but they follow with “though this actually results in a net *decrease* of 32%”.

Percentages are very ambiguous things. Sometimes, it makes sense to explain an event with an understanding of its context, and that’s something that percentages can help with (10,000 may be a lot, but not when its out of 10,000,000,000… 0.0001%). However, big and small bases can cause vast errors of judgment (if I get a 66% click rate on a message, that’s amazing. Its less amazing if its 2 people out of 3 mailed, or 2 clicks on 3 impressions.) I always advocate setting the stage by warning people if bases are small or large.

And now we have a reminder that combining percentages is more devious than you expect. Don’t be fooled! Do the math, or do the time. (Ok, that last wasn’t quite warranted).

BTW, if you really, really want to quickly add the percentages in the above example: 25% off is the same as multiplying by 0.75. 20% off is the same as multiplying by .80. When expressed this way, you can multiply (NOT add) them together. So, .75 * .80 =0.6 (the % of the original price remaining); 1-0.6 is the discount (0.40, or 40%). If you are augmenting (adding 25%), then just do 1.25. I know, this is simple math, but as the paper points out, most people forget what they learned in 9th grade so many years ago.

(Thanks to Dr. Dobb’s editor Jonathan Erickson for mentioning this paper.)

PS: Ok, for extra credit:

100 + 20% + 25% + 60% = 240

Now, remember what we said? Each percentage has to be calced in order since the base changes each time?

So, what do you expect from this, reversed order, with a large base as the first calculation:

Is 100 + 60% + 25% + 20% bigger or smaller than the first one?

Right, trick question. I didn’t say if the sparrow was African or European. And yes, both calcs wind up with the same answer.

* * *

The really scary part of the test scores example is that it would take 250% increase just to get back to status quo after the 60% decrease. That’s why we should always do things right the first time.

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