The buffer hour: why a day planned from honest estimates still overruns¶

Level: 101 · for anyone whose to-do list for today is still half full at midnight

One line: A task can hardly finish before its estimate and can run far past it, so estimation errors all point one way and do not cancel. A day planned to capacity from honest estimates overruns almost every time; a reserved hour, and estimates multiplied by your own recorded overrun, fix it.

The advice¶

Chapter 3 of Todd Zakrajsek's The New Science of Learning (third edition, 2022), "Developing Your Learning Strategy", gives two tips about schedules and to-do lists that look like caution:

Reserve an hour every day in case something takes longer than expected. If everything goes as planned, enjoy the unplanned "you time".
Put an estimated completion time on each item. As you estimate, your estimates will improve. (The author adds: "I still struggle with this.")

Both are arithmetic. This page shows why, with a model of task times and exact probabilities.

The shape of a task's time¶

The program's model: a task estimated at one hour takes one hour with chance 6/10, an hour and a half with chance 3/10, and three hours with chance 1/10. The numbers are made up for the example. The shape is the claim: a task can barely finish early, since the work still has to be done, but it can run far over, when the printer jams, the reading is harder than it looked, or office hours has a queue. Psychologists call the habit of underestimating the planning fallacy (Kahneman and Tversky, 1979; Buehler and colleagues, 1994, who asked students when they would finish their theses).

Verified output of the_buffer_hour.py — regenerated by tools/run_examples.py, never hand-typed.

1. One task, estimated at one hour

   takes 1.00 h   chance  60.0%
   takes 1.50 h   chance  30.0%
   takes 3.00 h   chance  10.0%

   Most likely time: 1 hour, the estimate. Average time: 1.35 hours.
   The estimate is the most common outcome and still 35% short of the
   average, because the misses are all on one side.

2. A day planned to capacity: n one-hour tasks in n hours

   tasks   planned   average needed   chance the day fits
       1       1 h         1.35 h       60.0%
       2       2 h         2.70 h       36.0%
       3       3 h         4.05 h       21.6%
       4       4 h         5.40 h       13.0%
       5       5 h         6.75 h        7.8%
       6       6 h         8.10 h        4.7%

   Every task is estimated honestly, and a five-task day fits one time
   in thirteen. The day fits only if every task hits its estimate.

3. The buffer hour: five tasks, with b spare hours in the plan

   buffer   chance the day fits
      0 h     7.8%
      1 h    46.7%
      2 h    65.3%
      3 h    88.2%

   One reserved hour turns a day that fits one time in thirteen into one
   that fits nearly half the time. A plan that holds most days needs more
   slack than feels reasonable, or fewer tasks. When nothing goes wrong,
   the book says, enjoy the unplanned 'you time'.

4. Why the errors do not cancel: skewed against symmetric

   The same five tasks, if a task were as likely to finish half an hour
   early as half an hour late (x 1/2, 1, 3/2 with chances 2/10, 6/10, 2/10):

                      average day   fits in 5 h   fits in 6 h
   symmetric errors       5.00 h     64.0%         96.4%
   skewed errors          6.75 h      7.8%         46.7%

   With symmetric errors an early task pays for a late one, and a plan
   of five hours is right on average. Real tasks cannot run much under
   their estimate, so nothing pays for the late ones.

5. Estimates that improve: keep the ratio actual / estimated

   Suppose you write down, for twenty past tasks, estimated and actual time.
   Twelve took the estimate, six took 1.5 times it, two took 3 times it.
   Average ratio: 1.35. Multiply every new estimate by it:

   five one-hour tasks are planned as 6.75 hours
   chance that fits:  56.4%  (unscaled:   7.8%)

   The ratio is metacognition applied to time: compare what you
   predicted with what happened, and correct the prediction, not the
   feeling. The plan now matches the average day; the buffer hour is
   for the days worse than average.

Honest estimates, and a plan that fails¶

Section 1 has the surprise in miniature. The estimate of one hour is the most likely outcome, so it is an honest estimate. But the average time is 1.35 hours, because all the misses are on one side. An estimate can be the right answer to "how long does this usually take?" and the wrong answer to "how long should I plan for?".

Section 2 plans a day to capacity: n one-hour tasks in n hours. The day fits only if every task hits its estimate, so its chance is (6/10)ⁿ: 60% for one task, 7.8% for five, about one day in thirteen. No single estimate was wrong; the plan was.

The buffer hour¶

Section 3 adds spare hours to a five-task day. One reserved hour raises the chance that the day fits from 7.8% to 46.7%; three raise it to 88%. The book's hour is not pessimism. It is the minimum that makes a full day workable at all, and the table says a plan that holds on most days needs either more slack or fewer tasks.

Section 4 shows where the problem comes from. If a task were as likely to finish half an hour early as half an hour late, early tasks would pay for late ones, the average day would be exactly five hours, and a five-hour plan would fit 64% of the time. Skew, not error, breaks the plan. It is the same lopsidedness that separates a mean from a typical value in a share above a cutoff ↗.

Estimates that improve¶

"Your estimates will improve" only if you keep score. Section 5 records twenty past tasks, estimated and actual, and takes the average ratio, 1.35. Multiplying new estimates by it plans five tasks as 6.75 hours, and the chance the day fits goes from 7.8% to 56%. What remains is variation around the average, and that is what the buffer hour is for.

This is metacognition applied to time. In metacognition, a student compares the confidence they wrote down with the answers they got, and corrects the confidence. Here, you compare the time you wrote down with the time it took, and correct the estimate. In both cases a feeling ("this will take an hour", "I know this") is a prediction, and a prediction can be checked.

Try it¶

Keep the ratio for a week. Write an estimate next to every to-do item, and the actual time when you cross it off. Divide.
Plan tomorrow with it. Multiply every estimate by your ratio, then add the book's hour.
Change the model. Make the bad case likelier (3/10 instead of 1/10, taking the extra from the first line) and rerun: how many buffer hours does a five-task day need to fit half the time?
The cards.

Flashcards¶

The page as a deck of Anki cards: the_buffer_hour.txt. Import with File → Import. Tags: definition, skew, buffer, calibration, trap.

Po polsku, w skrócie¶

Książka Todda Zakrajska The New Science of Learning radzi zostawiać codziennie godzinę rezerwy i zapisywać przy każdym zadaniu szacowany czas. Obie rady to arytmetyka. Zadanie prawie nigdy nie kończy się dużo szybciej, niż zakładaliśmy, ale bywa, że trwa dwa albo trzy razy dłużej. Błędy szacunku idą więc w jedną stronę i się nie znoszą.

W modelu programu zadanie „na godzinę" trwa godzinę z szansą 60%, półtorej z szansą 30% i trzy godziny z szansą 10%. Szacunek jest uczciwy, bo godzina to najczęstszy wynik, ale średnio zadanie trwa 1,35 godziny. Dzień z pięcioma takimi zadaniami, zaplanowany na pięć godzin, mieści się w planie tylko w 7,8% przypadków. Godzina rezerwy podnosi to do 46,7%. Gdyby błędy były symetryczne, pięciogodzinny plan wystarczałby w 64% dni. To asymetria, a nie sam błąd, psuje plan.

Szacunki poprawiają się tylko wtedy, gdy się je sprawdza: zapisuj czas przewidywany i rzeczywisty, policz średni stosunek i mnóż przez niego nowe szacunki. To metakognicja zastosowana do czasu: porównaj przewidywanie z wynikiem i popraw przewidywanie, a nie uczucie.