Confessions of a Math Cheat
I hate numbers! They’re confusing, hard to keep track of and make my eyes water if I look at them too long. Ugh!
Nevertheless, people insist on throwing them around. They like to talk about them, crunch them, live and die by them, put lots of them in PowerPoint slides and generally find ways to torture me with them.
But I’ve found a way to fight back! I beat them at their own game. How do I do it? I cheat. Hah! There’s really no reason to be oppressed by numbers if you can find simple ways to work around them. In fact, you can win the numbers game if you simply refuse to play it fairly. With that in mind, here’s a few of my favorite math hacks.
Take off a Zero (or Two) to Calculate Percentages
When I first started out in the media business, the company I worked for had a great training program. Great, that is, except for the numbers.
One day they were teaching us to calculating ratings from a certain report and they had an incredibly convoluted formula that they wanted us to learn. For all the ins and outs, it really boiled down to comparing two numbers like these
Population: 811,756; Audience: 51,184
They put it up on the board, had one of the star pupils work through it and then circled the answer (14%). Then they asked if everybody else agreed. I was still young and dumb enough to answer no.
I didn’t use their crazy formula, just noticed that it was a simple percentage problem. The best way to deal with percentages is to simply take off a zero (or two) to give you 10% (or 1%) and then ballpark. A quick look and it was clear that:
10% = 81,175
That’s a whole lot more than 51,184, so it was clear that 14% was out of the question. While they were patiently re-explaining the formula and generally treating me like an idiot, I did two more quick things in my head:
5% = 81,175 / 2 = about 40,000
7.5% = halfway in between 40,000 and 81,175 = about 60,000
So it wasn’t all that hard to see that 51,184 wasn’t either 5% or 7.5% but somewhere in-between. As they continued to berate me, I finally blurted out, “Look, the answer is somewhere in between 6% and 7%!”.
After some more back and forth, I convinced them put it all in the calculator again. 6.5% jumped out and that launched my reputation as a “numbers guy.” To this day they still probably still think I calculated their crazy formula in my head!
Compounding Interest with the Rule of 72
Okay, that one was simple (although a surprising number of people don’t do it, because numbers make people freeze). However, compounding interest rates are harder. They seem like they should be straightforward, but the numbers tend to runaway from you. The formula isn’t that tough in Excel, but way too difficult to do in your head.
Luckily, there’s a simple way to cheat here as well. Divide the number 72 by any interest rate and your will get roughly the amount of years it takes to double your money. So at 10% your money will double in about 7.2 years (72 divided by 10).
In the same training program, the instructor remarked that the values for radio stations were going crazy (this was in 1996, right after the new telecom act allowed larger groups to form). He gave the example of some guy who bought a station for $3 million and sold it for $75 million thirty years later.
“That’s no big deal” I said, “It’s about 11% annual growth.” This time no long explanations, just amazed looks. They thought I was some kind of Rain Man.
In reality, I just used the “rule of 72” backwards. It was pretty easy to see that the money doubled 4-5 times:
(6, 12, 24, 48, 96) and 75 is somewhere in between 48 and 96
And given that it happened over 30 years:
30 years / 4 = 7.5 years to double= about 10% of 72
30 years / 5 = 6 years to double= exactly 12% of 72
So again, it’s wasn’t that hard to guess 11%. (The actual answer 11.3% – but who cares?). That’s a good return, but not at all unusual. However going from $3 million to $75 million makes it seem like it is because 30 years is a long time. Who says numbers can’t lie?
Sample size is like organic food. Nobody really seems to understand it, but there’s always some little snot around to tell us how important it is. Anytime research is cited, you can be sure someone will ask, “what’s the sample size?” Whatever it is, people who disagree with the study will say it’s too small.
For my part, I’m convinced that the sample size issue is a vast conspiracy cooked up by research companies. You see, it’s really the only source of error that they can’t be blamed for and that they can charge clients big money to correct. That’s a double play in any man’s league!
In actuality, it’s very easy to ballpark sample error: 1/√sample size . So if the sample size is 100:
1/√sample size = 1/√100 = 10% or +/- 5%
If the sample size is 900:
1/√sample size = 1/√900 = 1/30 = 3% or +/- 1.5%
That’s a very small difference given the variance in cost of interviewing an additional 800 people. They usually tell us that anything under 100 respondents yields disaster, what would happen if the sample size was a paltry 50?
1/√sample size = 1/ √49 = 1/7 = 14% or +/- 7%
Again, in most situations that kind of error wouldn’t cause much of a problem. So unless your sample size is very, very low, it shouldn’t affect the outcome of an analysis. Mention this next time someone wants to crap all over your research. The blank stare you get in return will be a thing of beauty!
Lean back and Tell a Story
We human beings (except for a few freaks) are very bad at calculating. That’s why computers were invented in the first place. We are, however, extremely good at interpreting narratives.
Luckily, most of the numbers we come across in business life tell a story (if they don’t, they’re usually either wrong or irrelevant). They go up or down, left or right or they don’t really go anywhere at all. Keep this in mind and you’re unlikely to be confused by numbers.
Unfortunately, most people do just the opposite. They lean in and actually try to calculate (or worse, remember) numbers, which is how things often get confused. Instead, lean back and watch the story unfold. If there isn’t one, your bullshit alarm should be going off. Chances are, something is very wrong.
You Don’t Need to Know The Right Answer, Just The Wrong One
Life isn’t like math class. There is rarely a “right” answer. Most of the numbers we come across are aggregations of estimations. No matter how many decimal places they include, they’re not very precise and shouldn’t be taken that seriously.
It is the failure to realize this simple fact that causes people to screw up numbers so royally. They plug numbers into Excel and then take them as Gospel. That’s always a mistake.
You can’t compete with a computer to get the right answer, but you should be able to notice a number that’s wildly off. If it seems wrong, it usually is. If not, you’ve told yourself the wrong story and need to find the right one. Either way, some simple ballparking will save you an enormous amount of time and embarrassment.
You can live by numbers and you can die by numbers. I, however, would much rather cheat them.