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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.
If the chronometer error rate is 1%, averaging two will give you a 2% error rate.
Either they agree, or they disagree.
If they agree, they're either both telling the truth or both lying. All you can do is go with what they agreed on. In this case, picking what they agreed on is the same as picking what one of them said (say, Alice).
If they disagree, then one is telling the truth and one is lying and you have no way to tell which. So just pick one, and it makes no difference if you pick the same one every time (say, Alice).
So you end up just listening to Alice all the time anyway.
Earlier quoted context omitted.
> In a perfect world they drift less than a minute per day... 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…
I’ve never heard the bring one or three, I’ve always just heard three. I think that exact saying implies that if you have two and one isn’t working out you’ll go crazy but if you have one you’ll be oblivious until it’s too late. 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…
Now replace fail with lying and you have the exact same problem.
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.
To spell out the point: 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.
Earlier quoted context omitted.
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 failur…
> In a perfect world they drift less than a minute per day... 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…
Earlier quoted context omitted.
> In a perfect world they drift less than a minute per day... 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…
I’ve never heard the bring one or three, I’ve always just heard three. I think that exact saying implies that if you have two and one isn’t working out you’ll go crazy but if you have one you’ll be oblivious until it’s too late. 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…
Earlier quoted context omitted.
I’ve never heard the bring one or three, I’ve always just heard three. I think that exact saying implies that if you have two and one isn’t working out you’ll go crazy but if you have one you’ll be oblivious until it’s too late. 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…
15s per day sounds EXTREME. Are you even serious? Even the cheapest Chinese hand watch has 50x better precision.
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.
To spell out the point: If the chronometer error rate is 1%, averaging two will give you a 2% error rate.
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.