Punctuality - Arriving on Time and Math

by Xachariah 4 min read3rd May 201240 comments


In hindsight, this post seems incredibly obvious.  The meat of it already exists in sayings which we all know we ought to listen to: "Always arrive 10 minutes earlier than you think early is," "If you arrive on time, then you're late," or "Better three hours too soon than one minute too late." Yet even with these sayings, I still never trusted them nor arrived on time.  I'd miss deadlines, show up late, and just be generally tardy.  The reason is that I never truly understood what it took to arrive on time until I grokked the math of it.  So, while this may be remedial reading for most of you, I'm posting this because maybe there's someone out there who missed the same obviousness that I missed.



Statistical Distributions

Everyone here understands that our universe is controlled and explained by math.  Math describes how heavenly bodies move.  Math describes how our computers run.  Math describes how other people act in aggregate.  Wait a second, something's not right with that statement... "other people".  The way it comes out it's natural to think that math controls the way that other people act, and not myself.  Intellectually, I am aware that I am not a special snowflake who is exempt from the laws of math.  While I had managed to propagate this thought far enough to crush my belief in libertarian free will, I hadn't propagated it fully through my mind.  Specifically, I hadn't realized I could also use math to describe my actions and reap the benefit of understanding them mathematically.  I was still late to arrive and missing deadlines, and nothing seemed to help.


But wait, I'm a rationalist!  I know all about the planning fallacy; I know to take the outside view!  That's enough to save me right?  Well, not quite.  It seemed I missed one last part of the puzzle... Bell Curves.


When I go to work every day, the time from when I do nothing but getting ready to go to work until the time that I actually arrive there (I'll just call this prep time) usually takes 45 minutes, but sometimes it can take more time or less time.  Weirdly and crazily enough, if you plot all the prep times on a graph, the shape would end up looking roughly like a bell.  Well that's funny.  Math is for other people, but my behavior appears like it can be described statistically.  Some days I will have deviations from the normal routine that help me arrive faster while other days will have things that slow me down.  Some of them happen more often, some of them happen less often.  If I were describable by math, I could almost call these things standard deviations: days where I have almost zero traffic prep time takes 1 standard deviation less, days when I can't find my car keys my prep time takes 1 standard deviation more,  days I realize would be late and skip showering take 2 standard deviations less, and days when there is a terrible accident on the freeway end up requiring +2 or +3 standard deviations more in time.  To put it in other words, my prep time is a bell curve, and I've got 1-sigma and 2-sigma (and occasionally 3-sigma) events speeding me up and slowing me down.


This holds true for more than just going to work.  Everything's time-until-completion can be described this way: project completion times, homework, going to the airport, the duration of foreplay and sex.  Everything.  It's not always bell curves, but it's a probability distribution with respect to completion times, and that can help give useful insights.


Starting 'On Time' Means You Won't be On Time

What do we gain by understanding that our actions are described by a probability distribution?  The first and most important take away is this: If you only allocate the exact amount of time to do something, you'll be late 50% of the time.  I'm going to repeat it and italicize because I think it's that important of a point.  If you only allocate the exact amount of time to do something, you'll be late 50% of the time.  That's the way bell curves work.


I know I've heard jokes about how 90% of the population has above average children, but it wasn't until I really looked at the math of my behavior that I realized I was doing the exact same thing.  I'd say "oh it takes me 45 minutes on average to go to work every day, so I'll leave at 7:15."  Yet I never realized that I was completely ignoring that half the time would take longer than average.  So half the time, I'd end up be pressed for time and have to skip shaving (or something) or I'd end up late. I was terribly unpunctual until I realized I that I had to arrive early to always arrive on time.  "If you arrive on time, then you are late."  Hmm.  You win this one, folk wisdom.


Still, the question remained.  How much early would it take to never be late?  The answer lay in bell curves.



Acceptable Lateness and Standard deviation

Looking at time requirements as a bell curve implies another thing: One can never completely eliminate all lateness; the only option is to make a choice about what probability of lateness is acceptable.  A person must decide what lateness ratio they're willing to take, and then start prepping that many standard deviations beforehand.  And, despite what employers say, 0% is not a probability.

If my prep time averages 45 minutes with a standard deviation of 10 minutes then that means...

  • Starting 45 minutes beforehand will force me to be late or miss services (eg shaving) around 50% of the time or about 10 workdays a month.
  • Starting 55 minutes beforehand will force me to be late or miss services (eg shaving) around 16% of the time or about 3 workdays a month.
  • Starting 65 minutes beforehand will force me to be late or miss services (eg shaving) around 2.3% of the time or about 1 day every other month.

That's really good risk reduction for a small amount of time spent.  (NB, remember that averages are dangerous little things.  Taking this to a meta level, consider that being late to work about 3 times a month isn't helpful if you arrive late only once the first month, then get fired the next month when you arrive late 5 times. Hence, "Always arrive 10 minutes earlier than you think early is."  God I hate folk wisdom, especially when it's right.)


The risk level you're acceptable with dictates how much time you need for padding.  For job interviews, I'm only willing to arrive late to 1 in 1000, so I prepare 3 standard deviations early now.  For first dates, I'm willing to miss about 5%.  For dinners with the family, I'm okay with being late half the time.  It feels similar to the algorithm I used before, which was a sort of ad-hoc thing where I'd prepared earlier for important things.  The main difference is that now I can quantify the risk I'm assuming when I procrastinate.  It causes each procrastination to become more concrete for me, and drastically reduces the chance that I'll be willing to make those tradeoffs.  Instead of being willing to read lesswrong for 10 more minutes in exchange for "oh I might have to rush", I can now see that it would increase my chance of being late from 16% to 50%, which is flatly unacceptable.  Viewing procrastination in terms of the latter tradeoff makes it much easier to get myself moving. 


The last quote is "Better three hours too soon than one minute too late."  I'm glad that at least that one's wrong.  I'm sure Umesh would have some stern words for that saying.  My key to arriving on time is locating your acceptable risk threshold and making an informed decision about how much risk you are willing to take.



The time it takes for you to complete any task is (usually) described by a bell curve.  How much time you think you'll take is a lie, and not just because of the planning fallacy.  Even if you do the sciency-thing and take the outside view, it's still not enough to keep you from getting fired or showing up to your interview late.  To consistently show up on time, you must incorporate padding time.


So I've got a new saying, "If you wish to be late only 2.3% of the time, you must start getting ready at least two standard deviations before the average prep time you have needed historically."  I wish my mom would have told me this one.  It's so much easier to understand than all those other sayings!

(Also my first actual article-thingy, so any comments or suggestions is welcome)