5 ms·

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° (±0.00

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))

JBorrow 7y ago
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)
logfromblammo 7y ago
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.
busyant 7y ago
You can see it done here: https://en.wikipedia.org/wiki/Standard_atomic_weight#List_of_atomic_weights https://en.wikipedia.org/wiki/Standard_atomic_weight#List_of...
- logfromblammo 7y ago
  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 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.
gus_massa 7y ago
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 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.

JBorrow 7y ago

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)

logfromblammo 7y ago

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.

busyant 7y ago

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

logfromblammo 7y ago
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 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.

gus_massa 7y ago

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 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.