Do Extra Sig Figs Affect Uncertainty in Calculation Results?

[billabuwl50]

When your doing sig figs and you make an action such as average densities, and you end up with an extra sig fig, is there an actual change in the uncertainty if the quantity or is it just a rule of Math.

If I was forced to guess I would say the it was just a mathematical rule.

 

[Office_Shredder]

Sig figs aren't actually a math rule, since math doesn't deal with sig figs. They're a concept derived in inductive sciences, where measurements aren't 100% precise. Using significant figures, you give others an idea as to how precise your measurements are.

You shouldn't be ending up with extra sig figs that are more precise than your original... i.e., if your measurements looked like:

2.78
3.14

then calculating and coming up with a number such as

14.63

can be allowed, depending on the operation (there is an extra sig fig there), but something like:

1.463

shouldn't show up usually

 

[billabuwl50]

for example

I have 2.73, 2.73, 2.78, 2.74, and 2.54. I want to average these usign sig figs. Currently they each have three. If I add them up you get 13.52. So now if you divide it by five you get 2.704, and you have still used sig fig rules.

Would this indicate an actual change in the uncertainty of the quanity, or is this a mathematical fact?

 

[DaveC426913]

Don't you drop the extra once you're done?

So your final would be 2.70?

 

[billabuwl50]

Nope, it stays. I just don't know what it means.

 

[CRGreathouse]

In general, significant figures (which aren't really math at all) are supposed to give a rough bound on the error. In particular, a number is presumed accurate to within 1/2 of a unit in the last place in general. With that assumption in place for the data, it's easy to see that the real average must then lie in the interval [2.699, 2.709]. Thus 2.70 is justified because the error is within 0.9 ULP (not as good as the 0.5 ULP of the data, but good enough). 2.704 isn't justified at all, since then the error would be within 5 ULPs, which is pretty bad.

Of course these are worst-case -- the errors usually cancel out, giving a better precision than interval arithmatic would suggest.