Hacker News Reader

The problem is that these lat/lon figures are not displayed with precision information at all, so we can't even know the std dev. Sig figs are a simple and clear way to communicate the precision of a measurement, but if you want to be more statistically accurate, you can use parentheses to indicate the standard deviation. In your example (if I mess this up please correct me, but I think) it would be 45.73490534578(1000000)° Which again is silly because there is no way to know the standard deviation to that level of precision. A more reasonable number would be 45.734905(10)°

Is it difficult to include precision when reporting measurements? No

Is it sometimes valuable? Yes

Is it really too much to ask for? No

I've never seen the convention you're using and I don't think I understand it. Conventions I've seen include:

* Give an error bound like 45.73490534578° (±0.00002°) and indicate in prose that this is a 2σ bound.

* Put non-significant figures in parenthesis, like 45.73490(534578)° (EDIT: possibly I've misinterpreted this one when I've seen it, see logfromblammo's reply)

* Put a bar over the last significant figure, like 45.73490̄534578 (hopefully this one renders properly when I post this... (EDIT: nope))

The value in the parentheses is the symmetric one-sigma bound.

If I say the atomic weight of F is 18.998403163(6) g/mol...

  mean     18.998403163 g/mol
  std.dev.  0.000000006 g/mol

If I say Planck time is 5.391245(60)e−44 s...

  mean     5.391245e−44 s
  std.dev. 0.000060e-44 s

The standard rules for rounding imply that whenever a measurement is given to a certain number of significant figures, you're leaving out "0(5)" from the end. So 1.2345 is 1.23450(5) in parenthetical notation.

Significant figures rules give you a close-enough propagation of error, but in order to be more exact, you need to combine absolute uncertainties when adding or subtracting, and combine relative uncertainties when multiplying or dividing.

The idea of using something like 45.734905° ± 0.000010° is that the information of the other digits is so small that it's better to ignore it. (Assuming that it is unbiases, and the other digits have some information at all.)

An intuitive way to understand this is considering how much work is necessary to extract information from the next digit. You can use statistic to extract more information. As a rule of thumb you need 100 times more measurements for each additional digit. So you need something like 100 (or 400) for the first digit, 10000 (or 40000) for the second digit and so on. (There is a constant here, I never remember the constant, perhaps it's 4, perhaps it's 1.)

To extract some information from the last 8 in 45.73490534578° you need 1000000000000 measurements! So it's better to just ignore the tiny amount of information in the 8.

In the lab in very controlled scenarios you can repeat a measurement very carefully a lot of times automatically and then use statistics. But if you have a handheld GPS, you can't repeat the measurement more than a few hundred of times.

A device that use a somewhat similar process is the https://en.wikipedia.org/wiki/Lock-in_amplifier it is not exactly this statistic trick, but note that it needs an stable environment to repeat the measurements. Wikipedia says that it can detect a signal 1 million times smaller than noise, but IIRC (the cheap ones?) can only detect a signal that is 1/100 or 1/1000 of the noise..

PS: Please never use 45.73490534578° (±0.00002°). If you use in a lab in the university, the TA will get mad at you. You can try using 45.73490534578° (±0.00002000000°) that is bad but not so horrible bad, the TA will still get mad at you but you may survive.

You can see it done here: https://en.wikipedia.org/wiki/Standard_atomic_weight#List_of...

The brackets in the table mean that different sources of the element have different proportions of isotopes, so the mean atomic weights for specific deposits may cover a range that differs from the overall mean for every deposit of the element ever measured.

I.e. if you measure carbon from the upper atmosphere, it's going to have more C-14 in it, from cosmic rays flipping protons in N-14 to neutrons. And if you measure carbon buried for thousands of years, it's going to have less C-14, from natural decay.

If you look at https://en.wikipedia.org/wiki/List_of_physical_constants you can see that the parentheses are omitted from defined constants, and included for measured constants.

The parentheses show something similar to your first, where this is the +- read from the right hand side of the number. So you could write yours:

45.73490534578(2000000)

The question isn't about mathematical rigor. It's about utility and distraction.

In much of the world, phone numbers work even if you add extra numbers that aren't needed. So I could give you my phone number plus 10 extra digits that change nothing. The end result would be the same utility to you (you can contact me) but with an increased cost in recording, memorizing, chance of error.

Using lat/lon to an unnecessary level of detail is the same thing. More digits are more chances to make mistakes, more cognitive load.

Exactly. Extraneous, useless, information is not no-value, it's negative-value, because of this.

> slightly more than half the time, 45.73490534578° is still going to be closer to the true value than 45.7349053457° is. By performing significant figure rounding,

That's not rounding, so I'll assume you meant "slightly more than half the time, 45.73490534578° is still going to be closer to the true value than 45.7349053458° is".

And that may be true, but if 50.0001% of the time one is closer to the true value than the other that's essentially meaningless.

> if your measurements are unbiased

This is a big IF which you can't actually know because its beyond your level of precision, by definition.

This is completely knowable, by taking repeated measurements of a reference object, one which was either checked by a more precise instrument or is definitionally correct (e.g. the old reference kilogram or the Greenwich meridian)

In the time it takes you to note down those extra digits, you could have improved your actual measurement by the same minuscule fraction of a bit ten times over.