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

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.

- Greg

32 Responses leave one →
1. July 6, 2011

Ha, this is great – thanks!

Thx:-)

July 6, 2011

Hi Greg!

Very good post, like usual. It’s probably not my best day today, but I don’t quite follow your sample size calculation:

1. If the formula is 1/sample size, how is it equal with it’s square root (1/10=1/100, 1/900=1/30, etc)?
2. Sample size 900, what do you mean by “10% or +/-3%”?

Tnx for coping with slow readers!

Dragos,

1/sample size = 1/900 = 1/30 = 3% or +/- 1.5%

Thanks for pointing it out.

- Greg

Greg,

you have a conflict there: elsewhere in another blog item you say it should be 1 / “the square root of the sample size”

i.e. 1 / sqrt (900) = 1/30

which is the true expression of sample error approximations.

Robert,

I’m not sure that I get you. If the sample size is 900, the total sample error is 1/30 or about 3%. The true expression for sample error at 95% confidence is .98/ sqrt sample size, but and at 99% confidence it’s something like 1.29, but using the approximate formula is good enough for just about any practical purpose.

- Greg

Greg

On this site you have used two different formulae to express the deviation.

In another blog item you [correctly] express it as

1/√ 900 = 1/30

but here in THIS blog item you have expressed the formula as

1/900 = 1/30

they cannot both be correct. The first form is the correct one.

Robert,

You’re right. Sorry, I’m in the middle of a trans-continental move and am screwing some things up.

Thanks, I’ll correct.

- Greg

Greg

Norra problem. Good luck with the move

Thanks. I’m returning to the US after nearly 15 years abroad. Lots to do on little sleep:-)

- Greg

3. July 6, 2011

Great post, makes me miss my slide rule (oops I think I just dated myself.)

Calculators and computers are great tools, but without some basic math skills you have no clue if the answer they are giving you is right or wrong.

True:-)

- Greg

4. July 7, 2011

As usual a great article and a damn good lesson for anyone who thinks they need to rely on their CA to tell them 2+2=4!

Thx Megha.

- Greg

5. July 8, 2011

Hi Greg,

As always, a great post!

Having previously studied (nuclear) physics, am a great fun of Raymond Smullyan and his books such as “Lady or the Tiger.” I love math puzzles.

A great read for business savvy peeps who care about numbers could be also, with special emphasis on number sin our lives, economy, etc., book by Leonard Mlodinow – “Drunkard’s Walk.” And of course not to forget the amazing John Allen Paulos with his “Illiteracy.”

Why you moving back to the US, if you don’t mind me asking?

Cheers,
H.

Hayk,

Thanks for the suggestions. As for my move back to the US, after nearly 15 years and with a 2 year old daughter, it really felt like time to come home.

- Greg

July 10, 2011

Greg, greatly enjoyed the read (I know, that sounds geeky ).

When I was in school I loved Trachtenberg Speed Math precisely for some of the reasons you mention. One does not need to be always accurate for the task at hand. Also, as you said folks need to see what the impact of the “error” is going to be and if it is worth laboring to precise decimal places.

Great review and highly recommended for kids & adults alike

Best,
Ned

Thanks, Ned. Have a great week!

- Greg

7. July 11, 2011

I really wish this was more standard in schools – as it is, in my courses (college level) I bring things like this up ALL-THE-TIME to give students that “feel” for what makes sense. Unfortunately, many of them will blindly type into a calculator or computer and write down whatever comes up without thinking about it. This of course is not a reflection on them, but a bad habit picked up over time.

The kind of thinking you are showing here is really about developing your intuition for numbers and not only helps you in day to day “dont need the exact number” situations but also as a check on work when you do.

Thanks Jerimi!

For everyone else, check out Jerimi’s site: http://www.mathbootcamps.com Lots of great stuff there!

- Greg

Hey thanks! It is a work in progress.

Good luck with it!

July 12, 2011

A dumb post and only can impress a guy from US…

9. July 14, 2011

Along with Trachtenberg Speed Math you can take a look at
Vedic Maths which is one of the World’s Fastest Mental Maths System.

Thanks. I’ll check it out.

- Greg

10. July 18, 2011

A funny post, Greg, lmao. Wish my university professors (B.Sc. in math) would have approved your rough linear approximation approach in their exams
Someone up here made a nasty comment saying such a post can only impress US guys. While I disagree with his general conclusion (and tone), that guy has some points about Americans and math.
1. Americans are stats-freaks. You cannot even watch a ball game on TV without hearing a detailed comparison between the teams of the average RBI of left-handed blond batters, who weigh more than 150lb, since the great depression in 1929 (did I mention I hate baseball?
2. When it comes to business, Americans try to translate everything into \$ signs. Even things which cannot be monetized seriously. It’s of course not RBI, but rather ROI. I have lately heard the idiotic question “what is the ROI on social media?”. Can one monetize reputation, market penetration and the like in vast crowds which one doesn’t even know. Yes, you can guesstimate till your Excel sheet will itself dial 911 and file a complaint against abuse, you can fart numbers out of your … as you like, but that’s what these numbers will be – brain-farts that you must present to top management to get what you want, while there’s nothing really serious behind these numbers.
That’s something very deep in the American business culture. Everything must be “measured”, for the false sense of security that “if we measure, we will most likely take the right decisions”. The fact that a lot of these “measurements” are useless are a trivial issue…
Just my 2.71828… cents (yes, you may round it to whatever amount you like .

Thanks Roy, although I’m going to have to disagree with your 2nd observation (the first is spot on). I’ve lived overseas for 15 years and only just returned to the US. The notion that Americans want to quantify everything strikes me as very similar to the idea that “all Americans eat at McDonalds.”

I think people often confuse what is Americans with large organizations. For instance, I certainly don’t think that either Unilever or Nestle is any less metrics driven than P&G, but because they are large, multinational organizations, they are often assumed to be influenced by American trends therefore everything they do is somewhat “American.”

- Greg

I knew that stepping into the twilight zone of generalizations, let alone nationality-based ones, would probably lead to controversy. You cannot be right about everyone, and I know not all Americans eat at that famous establishment I personally dislike quite a bit.
Myself, I have been living – and working in IT – in three different countries (and continents): Israel (wannabe-American business culture), Holland (a very European one) and currently Canada (I should be careful here, my colleagues would hate me for throwing them into the same pot with their Southern neighbours, but the business culture is pretty similar).
I agree with you, Greg, the large organizations – wherever they are – are all metrics-driven. But in North America also the medium-size and the small business are very much so, which is quite different from Europe (that’s at least my observation).
I could see the difference when I had to adapt my European CV into an American-style resume. Except for the fact that an American resume is all “me-me-me” (yes, you have to write you are a team player, but all of the achievements you have achieved alone), things need to be quantified. For example, I have never seen anything like “Fortune 500″ (or the local version thereof) mentioned in a European CV, and I have read quite a lot of these.
In my second job interview in this continent, with a small business, I started to describe the technical details of a recent project (in Holland) I was busy with (and quite proud of). The interviewer cut into me, and bluntly asked what the project budget was. Which was totally irrelevant, in my IT career (nearly two decades) I have had projects of “merely” few hundreds of thousands, in which my technical role was far more significant than in other projects with budgets of 7 or 8 digits. Not to mention that a project budget in a different country, with different labour and other costs, means very little unless you are very familiar with the IT sector in that country (which the interviewer wasn’t).
So I guess we are in disagreement about this one. Which is OK, I promise not to try to convince you with a dazzling presentation strewn with numbers

October 16, 2011

Very interesting point of view. I’ve discovered your blog today and I’ve found it very interesting. Thank you. Only one thing: when you talk about sample error, you should fix the “confidence level” of your estimation. In market research usually we take 95%. A sample size of 100 have a maximum of +/- 10 percentage points of error at 95% CL. That 10 points are +/- 10%, you can’t divide. ;-D

Thanks Manuel. Glad to see you’re enjoying the site.

However, I do believe you are mistaken. If you calculate it out, the two-tailed sample error would be 10%, which means +/- 5%. It doesn’t matter what “market research” or anybody else does. The math is the math.

- Greg

Hi Greg!

You are so assertive that you made me doubt (only a bit). ;-D

You are correct: the math is the math.

If you want, you can check several websites with “margin error calculators” that explain the key concepts better than me. I’ve found this one (http://faculty.vassar.edu/lowry/polls/calcs.html) it’s from an emeritus professor of Vassar College.

Margin of error, confidence limits… related to sample size, are concepts usually misunderstood. I don’t want to polemicize, just pointing out that you can’t oversimplify. A margin of error of 10 percentage points gives you a confidence interval of +/10 percentage points around the estimation.

Another key concept that it’s difficult and usually misunderstood is that the “error margin” of a sample is different regarding to the data you are measuring. If you are measuring the “Proportion of people with Product X” and “Proportion of people with Product Y” and data shows that X is 50%, the margin of error of a sample of a 100 is around 10 percentage points (with 95% confidence), so the “real” value must be between 40% and 60%. In the same study, product Y has only 7% (for the same population and the same sample) the margin of error of this second estimation it’s aprox. 5%, so the real value of product Y in the population have to be between 2% and 12%. The same sample size, different estimation of margin of error.

The first time you are going to study a phenomenon usually you don’t know the proportion of the population that have the product or feature of interest. In this situation, when you have to calculate a margin of error for a given sample size, you should consider the worst case scenario, the maximum error always is that the proportion were 50%. This is “a priory “calculation, and you should check and calculate with the final data.

English is not my mother tongue, forgive my odd writing style.

Manuel