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.
I think it is different in the continuous case though, because you can average two (reasonably accurate) chronometers and get a better measurement. But we can't average true and false, at least not in the context of this problem definition.
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.
The saying probably assumes that each chronometer has a certain small probability of malfunctioning, resulting in a significant error (basically a fat-tailed error distribution). With three chronometers, you can use a robust estimator of the true value (consensus value or median). With two, there's no robust estimator and if you use the mean, you have twice the probability of being significantly wrong (though only by half as much).
You can definitely average two relatively accurate chronometers but you if you only have two it’s difficult to tell if one is way fast or way slow.
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?
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.
You will have an error rate of less than or equal to 1%. You can't average two measurements and get a result with a higher error rate than the worst of the original measurements had.
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 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.
You seem to agree that two is not a good number. Better bring four then, so that you're not left with only two after your mishap.
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.
maxbond|1 month ago
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.
yccs27|1 month ago
jethro_tell|1 month ago
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?
patja|1 month ago
jethro_tell|1 month ago
tlb|1 month ago
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.
BurningFrog|1 month ago
If the chronometer error rate is 1%, averaging two will give you a 2% error rate.
maxbond|1 month ago
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.
[1] https://www.youtube.com/watch?v=T58lGKREubo
eru|1 month ago
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.
dsmurrell|1 month ago
dxdm|1 month ago
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.
Luc|1 month ago
This saying must originate with a landlubber...