Sailors in the past had a similar adage: “Never go to sea with two chronometers; take one or three.” They relied on precise clocks to calculate longitude.
But the chronometers are will sync with each other if you don't store them apart, which would result correlated noise that an average won't fix.
In a perfect world they drift less than a minute per day and you’re relatively close to the time with an average or just by picking one and knowing that you don’t have massive time skew.
I believe this saying was first made about compasses which also had mechanical failures. Having three lets you know which one failed. The same goes for mechanical watches, which can fail in inconsistent ways, slow one day and fast the next is problematic the same goes for a compass that is wildly off, how do you know which one of the two is off?
A minute per day would be far too much drift for navigation, wouldn't it?
From Wikipedia [1]:
> For every four seconds that the time source is in error, the east–west position may be off by up to just over one nautical mile as the angular speed of Earth is latitude dependent.
That makes me think a minute might be your budget for an entire voyage? But I don't know much about navigation. And it is beside the point, your argument isn't changed if we put in a different constant, so I only mention out of interest.
> Having three lets you know which one failed.
I guess I hadn't considered when it stops for a minute and then continues ticking steadily, and you would want to discard the measurement from the faulty watch.
But if I just bring one watch as the expression councils, isn't that even worse? I don't even know it malfunctioned and if it failed entirely I don't have any reference for the time at the port.
My interpretation had been that you look back and forth between the watches unable to make a decision, which doesn't matter if you always split the difference, but I see your point.
A well serviced rolex in 2026 with laser cut gears drifts +/- 15sec per day.
One with hand filed gears is going to be +/- a minute on a good day, and that’s what early navigation was using. I have watches with hand filed gears and they can be a bit rough.
Prior to that, it was dead reckoning, dragging a string every now and again to calculate speed and heading and the current and then guesstimating your location on a twice daily basis.
Those two wildly inaccurate systems mapped most of the world for us.
Though not without significant errors, the most amusing to me being that islands had a tendency to multiply because different maps would be combined and the cartographer would mistake the same island on two maps as being separate islands due to errors. A weird case of aliasing I suppose.
Modern Rolex (and Omega et al) are more like +/-2s.
"The precision of a COSC-certified chronometer must be between -4 and +6 seconds per day."
15s per day is not unusual for a good mechanical watch.
Even that was much better than the dead-reckoning they had to do in bluewater before working chronometers were invented. Your ship's "position" would be a triangle that might have sides ten miles long at lower latitudes.
It's possible to navigate without being able to measure your longitude. Like if you're looking for an island, you should first navigate to the correct latitude and then sail along that latitude until you hit the island. The route is longer, obviously. But that's what you should do if your chronometers disagree.
If the chronometer error rate is 1%, averaging two will give you a 2% error rate.
You wouldn't be well served by averaging a measurement with a 1% error and a measurement with a 90% error, but you will have still have less than or equal to 90% error in the result.
If the errors are correlated, you could end up with a 1% error still. The degenerate case of this is averaging a measurement with itself. This is something clocks are especially prone to; if you do not inertially isolate them, they will sync up [1]. But that still doesn't result in a greater error.
You could introduce more error if you encountered precision issues. Eg, you used `(A+B)/2` instead of `A/2 + B/2`; because floating point has less precision for higher numbers, the former will introduce more rounding error. But that's not a function of the clocks, that's a numerics bug. (And this is normally encountered when averaging many measurements rather than two.)
There are different ways to define error but this is true whether you consider it to be MSE or variance.
The average of a right and a wrong clock is wrong. Half as wrong as the wrong one, but still wrong.
If this is a good mental model for dealing with clock malfunctions depends on the failure modes of the clocks.
The result in the original article only applies when there are discrete choices. For stuff you can actually average, more is always better.
Oh, and even with discrete choices (like heads vs tails), if you had to give a distribution and not just a single highest likelihood outcome, and we'd judge you by the cross-entry, then going from one to two is an improvement. And going from odd n to the next even n is an improvement in general in this setting.
Or bring only two, but step on one immediately, to get rid of the cursed pair situation, and also to get the clumsiness out of the way early. Old sailor's trick.
This saying must originate with a landlubber...