A Rule Of Thumb That Actually Works If Your Thumb Is Good At Math#
A Rule Of Thumb#
What started all of this was my rule of thumb about whether my team’s lead was “safe”:
2 times the number of minutes remaining = pretty safe
3 times the number of minutes remaining = very safe
And I wanted to see how well it held up against real data. So now, after some effort, we can plot the percent point plots like this one:
And then we can plot those 2x and 3x guides on top of it and we get:
So, in short, my rule of thumb was very poor.
So – in googling about to see if someone had a better one – I was delighted to have stumbled on this explanation:
“An NBA team leading by twice the square root of minutes left in the game has an 80% chance of winning.”
This turns out to be very accurate, regardless of which era you examine. In fact, leveraging the observation that win probabilities versus point deficit are normally distributed, we can derive the multiplier for any given win probability:
Where \(\Phi^{-1}\) is the inverse of the standard normal cumulative distribution
function and the 2.49 constant applies when looking at all data from 1996 to now
(if you change the conditions, that number changes, usually slightly, as
explained below). This allows us to expand this rule for the (more
interesting) 5% chance and the snowball-in-hell 1% chance:
Percent Chance |
Rule of Thumb |
|---|---|
20% of coming back (or 80% chance of holding the lead) |
\(\approx 2.0 \cdot \sqrt{t}\) |
5% of coming back (or 95% chance of holding the lead) |
\(\approx 4.0 \cdot \sqrt{t}\) |
1% of coming back (or 99% chance of holding the lead) |
\(\approx 6.0 \cdot \sqrt{t}\) |
Looking at all the years from 1996 to 2024 we get:
Which you can see holds up very nicely. (In fact, even later on, I stumbled upon another example of the square root rule but for the 90% probability case).
For many numbers, it’s difficult to calculate square roots mentally, but for 16, 9, 4 and 1 it’s easy:
Time Left |
20% Chance of Coming Back |
5% Chance of Coming Back |
1% Chance of Coming Back |
|---|---|---|---|
16 Minutes |
2 * √16 = 8 Points |
4 * √16 = 16 Points |
6 * √16 = 24 Points |
9 Minutes |
2 * √9 = 6 Points |
4 * √9 = 12 Points |
6 * √9 = 18 Points |
4 Minutes |
2 * √4 = 4 Points |
4 * √4 = 8 Points |
6 * √4 = 12 Points |
1 Minute |
2 * √1 = 2 Points |
4 * √1 = 4 Points |
6 * √1 = 6 Points |
Best Fit Guides#
Now, you can – for any given situation – calculate the best fit guides that fit a little better than the 2, 4, 6 times the square root of minutes remaining. For example, for all eras you get:
Which is very close to the 2, 4, 6 times the square root of minutes remaining number, but fits a little bit better.
As you change conditions, the constant changes, but usually only slightly. For example, if we look just at the “old school” era (1996-2016), we get:
Which is this plot:
And if we look at just the “modern era” (2017-2024), we get:
Which is this plot:
Showing there is a slight difference in the constants. But the rule of thumb is still very close to accurate.
You can use the dashboard page to see how it works for any given situation and add the ‘Calculated Guides’ to your conditions. Normally, the 2, 4, 6 times the square root of minutes remaining guides are very close to optimal. But for some conditions – like a top 10 team playing a bottom 10 team – this rule of thumb does not hold up at all.