The Rule of 3 Significant Digits

Two people have shaped how I guess an answer.

Their comments and guidance have tailored how to form a quick estimate, my ability to articulate a hunch and the effectiveness of those guesses.

You probably guess or make a rough estimate regularly. How good is your gut feel? Do you keep track and score yourself?

Making an estimate should be second nature for you. It’s not something to do in public, too often. The practice can aid you in numerous ways.

Physics, Calculations, and Estimates

In my first physics class in college, we regularly ‘enjoyed’ pop quizzes. One, in particular, provided a lesson that has stuck with me for my entire life.

Here is the quiz question (as far as I recall it’s wording)

“How many piano tuners are there in New York City? 3 minutes, show your work.”

The homework for the class from the previous class included calculating acceleration given force on mass or something like that. Not a hint that we would be tested on census values.

There were a few groans across the room. A few pencils started to scratch out some answer.

How would you answer? What work would you show? Keep in mind I went to college a long time ago, and we did not enjoy the benefits of a Google search or anything even remotely similar. I had not lived in NYC nor played piano.

Give yourself three minutes, do not use the internet (if you haven’t already), and add you answer to the comments section below. Show your work.

The lecture after the quiz discussed the value of making a reasonable estimate or educated guess before performing the experiment or calculation. Physics involves math and just knowing the formulas does not guarantee you will get the right answer.

He likely mentioned a quote, or I ran across it later, by John Archibald Wheeler attributed to his book, Spacetime Physics.

Never make a calculation until you know the answer. Make an estimate before every calculation, try a simple physical argument (symmetry! invariance! conservation!) before every derivation, guess the answer to every paradox and puzzle. Courage: No one else needs to know what the guess is. Therefore make it quickly, by instinct. A right guess reinforces this instinct. A wrong guess brings the refreshment of surprise. In either case life as a spacetime expert, however long, is more fun! – John Archibald Wheeler, Spacetime Physics

This process helped me throughout my career. More than once helping me catch a missing sign that altered a calculated result. When I would guess the result of a calculation should show an increasing speed, and my calculation shows the falling rock has a decreasing speed over time, I would find the dropped negative sign and correct my calculation.

Another exercise we did in class was to sum a set of numbers, quickly. The first step was to jot down a guess, an order of magnitude or rough estimate. If the list is ten positive three-digit numbers, the result is going to be greater than 1,000. How much more, roughly? If the results of you addition work is -237, could that possibly be right?

Having the estimate allows you to compare your answer to your hunch. It provides a check step for your calculation.

Just because Excel churns out a number doesn’t mean it’s right. How do you know or at least how do you check?

Improve the Effectiveness of Your “Back of the Envelope” Estimates

I’d been making these educated guesses regularly for years. Then I shared an estimated value of a proposed project with Helen.

Helen had the office next to mine at HP at the time. She is an inventive and wicked smart engineer. I often shared ideas and proposals with her as her insights and advice always improved my work.

The proposal was exciting for me as I expected $10 million in value for a rather modest investment of time.

She stopped me right there. She said, “$10 million, really? I don’t believe that.” Or something to the effect.

Continuing the discussion, I quickly went over the assumptions and back of the envelope calculations that supported the claim.

She didn’t question the logic nor the actual results. Rather she wondered if the nice round numbers I used could be slightly altered. Sure, they only rough estimates, such as 100,000 customers or $1,000 in the cost of each failure, etc.

She said when she hears a nice round number she instantly knows it is a guess or estimate. Her guard goes up as she becomes distracted by the round number rather than focusing on the logic and assumptions.

Helen recommended I alter the final result to include three significant digits. Instead of $10 million, how about $9.87 million. It’s about the same as the result I got using a sequence of round numbers, yet Helen suggested it was “a bit more believable.”

Hum. Never thought of that. I always focused on the assumptions and logic, not the result. I thought the order of magnitude was sufficient to convey the result.

As you know, everyone filters what they hear and accept. Recognizing some recoil when they hear a nice round number meant I could lessen the effects of that kind of filter by simply using a three digit estimate.

So, I tried it. Over the next year each time I presented a guest or estimate I alternated between nice round numbers and estimates with three significant digits.

The round number estimates always generated questions about the values and numbers that went into the result. The three significant digit estimate often were not questioned or enjoy questions about the assumptions and logic only. I kept track that year and twice as many proposals using the three significant digit rule moved forward.

You will have to make quick estimates, work out a rough return on investment, or forecast return on investment values. In these cases you need to make a guess, then using the set of assumptions and a bit logic refine your estimate in just a short time (never having all the data you need). When presenting the results try using three significant digits.

Record these estimates and remind yourself to check how well it turns out if possible. Also, note how your audience responds to the ‘believable’ estimate you present.

Your Next Estimate

How many will view, share, and respond to this article?

Please, right now, without too much thought, add your guesses to the comment section below.

A month after this article posts I’ll tally up the shares and post the actual answer. I think I can find counts for views and the comment count will suffice for the last element.

About Fred Schenkelberg

I am an experienced reliability engineering and management consultant with my firm FMS Reliability. My passion is working with teams to create cost-effective reliability programs that solve problems, create durable and reliable products, increase customer satisfaction, and reduce warranty costs.

21 thoughts on “The Rule of 3 Significant Digits

      1. It depends on what assumptions do we make. My assumptions were based on that in our days less people are interested to play piano, there are not that many professional players that would need piano tuning, many piano playing students are using electronic piano, which does not need regular tuning as the string piano, so I assumed that the demand of piano tuners will be really low. I could be wrong, but my assumptions are based on my perception of this topic.

  1. It would be interesting to find out how wide would be the spread. There are at least five assumptions to make.

    1. There certainly are different ways to form an estimate in this case. I do not recall how my classmates answered so don’t have the spread from that experience. May have to do this in a class setting as an exercise at some point.

      1. In my opinion, the “Fermi Questions” approach is good to find out if someone’s way of thinking will be suitable for or specific task or position. the capability of someone to solve non-traditional problem. It will show rather what information one would need to solve a problem, than get a good estimated number without proper information.

  2. My estimate of piano tuners: 135
    There are many suburbs around NYC that would need access to at least 5 or more to their location …. the amount of high schools, drama courses and individuals would continually utilize a piano tuner on a regular basis.

    1. Thanks Tom – thanks – my estimate is in this range. I think the idea is layout the assumptions so you know what specific data you need to actually get a good estimate of the number of interest. The way I approached it I started with the population of NYC and an assumption of how many people or what percentage of people have a piano that needs tuning. A few more assumptions and I could create a back of the envelope estimate.

      The percentage of people with pianos that get tuned would be difficult to find – maybe a search for a survey or census data. Maybe run my own survey, etc.



        1. You know we’re not really interested in the answer, rather the approach and assumption you take to form an answer. Today such a question is a good google query away. Yet, framing the information you need to make an estimate is very good practice. Get an estimate before the search. I would say the list in the link is just a subset as many tuners do not advertise (the Carnegie Hall team probably doesn’t need to advertise and are fully employed, for example.)

  3. Let’s do some assumptions….
    There are ~8M people in NYC
    Let’s assume 1 out of 1000 is playing actively at home a piano
    So at least 8000 pianos per year needs to be tuned.
    Tuning a piano needs ~4 hours including transport of the tuner to the piano and back home.

    So a tuner can tune 2 pianos per day if he is occupied for 100% in tuning…..

    Let’s assume a duty cycle of 0.5….

    So there are 8000 working days per year, at least 40 tuners are necessary to get the pianos tuned.

    With a lot of people who are able to tune a piano without being active as tuner, let’s go for at least 97 tuners in NYC

  4. Without even living in the USA – so guesses at everything including population.
    1) NYC 5,000,000 people
    2) 1/100 has a piano ie 50,000 pianos
    3) Pianos should be tuned 1/yr; say actual average achieved is 1 every 2 yr.
    4) Therefore work to be done is 50,000/2 = 25,000 pianos tuned per year.
    5) A piano tuner could feasibly tune 2 pianos/day.
    6) Say he achieves 75% utilisation, ie 1.5 pianos tuned/day
    7) He is able to work for say 220 days per year. ie 220×1.5 = 330 pianos per year
    8) Therefore the requirement is for 25,000/330 = 75 piano tuners working within NYC.
    9) Much piano tuning is done on a repeat basis over decades so I expect that the efficiency of piano tuners in finding and scheduling work would be quite high. I have no reason to increase my estimate to account for these type of inefficiencies.

  5. My estimate is 250.

    At the core of the estimate, is the question how many piano tuners would be needed to service at population the size of NYC?

    I know NYC has about 8.5 million people. That’s a pretty inconceivable number, and it’s difficult to conceive as a population. So instead I will substitute a city with a population that is more intuitive. My neighboring city of Santa Rosa has about 170,000 people. That will work. I have some understanding of how many specialists of various types service the population of Santa Rosa.

    Next comes the guess. Without some rather complicated sampling, we don’t know how many pianos there are in a population nor how often they need tuning nor how fast a piano tuner can tune a piano. I would need estimates for all these values to accurately answer the question, but we are only looking for an estimate of how many piano tuners serve the community. And if I compound too many estimates my error will multiply. So instead I am going to guess. My intuitive guess is that 5 piano tuners can service a city the size of Santa Rosa.

    Since NYC is 50 times the size of Santa Rosa, NYC would, therefore, have 250 piano tuners.

Leave a Reply

Your email address will not be published.