# Significant figures

1. Feb 22, 2017

### Silviu

1. The problem statement, all variables and given/known data
I am given some results of certain experiments with errors and I need to rewrite them correctly in term of significant digits.

2. Relevant equations
The professor explained to us that, for example, 1.12345 ± 0.5231 is not correct (or at least not the right way) because you already have an error on the first digit after . so adding more digits make no sense, so I should rewrite this as 1.1 ± 0.5 (or in some cases 1.12 ± 0.52).

3. The attempt at a solution

4.12734 ± 1.357 --- 4.1 ± 1.4

4.12734 ± 0.0487 --- 4.13 ± 0.05

0.4321273 ± 0.00169 --- 0.4321 ± 0.0017

0.002163 ± 0.00032 --- 0.0022 ± 0.0003

304479 ± 791 --- 3.045e5 ± 800

728 ± 0.422 --- 728.0 ± 0.4

511.24 ± 2.721 --- 511 ± 3

383 ± 61.32 --- 383 ± 61

987.12 ± 62.57 --- 987 ± 63

6974.12734 ± 487 --- 6970 ± 490

123456789 ± 2344 --- 1.23456e8 ± 2000

0.000002723 ± 0.000000317 --- 2.7e-6 ± 3e-7

12.4 ± 7.2 --- 12.4 ± 7.2

These are the ones we have to rewrite and on the right of --- is my solution. Is it correct? I am a bit confused especially about the ones with big numbers and errors of order of 10^2. Thank you

2. Feb 22, 2017

### BvU

Hi,

In general, errors in experimental results are difficult to determine: they are estimates. Even with large numbers of observations and then averaging, statistics show that the relative accuracy of a standard deviation is about $1/\sqrt n$. So with 10 measurements the error is only 30% accurate.
Except in special cases, more than one significant digit is not really achievable.

I learned that an exception is when the first digit of the error is a 1: then you give one more (in order not to have such a big step to the next value)

You follow the guidelines nicely, but sense some discomfort when errors are > 1.

I would present $383\pm 61$ as $380\pm 60$ without hesitation, idem $987\pm 62 \rightarrow 990 \pm 60$.
With bigger numbers, one way around is to report e.g. $6974 \pm 487$ as $(6.9\pm 0.5)\times 10^3$

Here another guideline comes in: powers of ten preferably in steps of 3 . But it's not a strict guideline at all.

(on the next line you show some fatigue: last digit rounds off to 7). A mix of scientific and normal number format is uneasy on the eye; I would prefer $(123.457 \pm 0.02) \times 10^6$.
 oh, 0.002 of course. Thanks mfb - and we see it's a matter of tastes differing (post below) -- not a strict guideline at all. My motivation: there's often a name for such a power, like $M\Omega$ etc.

Then the next line is also uneasy because of the different exponents. I like $(2.7\pm0.3)\times 10^{-7}$ better. Taste ?  oh, sorry $10^{-6}$ - even better. ; thanks mfb)

Last edited: Feb 23, 2017
3. Feb 23, 2017

### Staff: Mentor

Don't use different exponents for central value and uncertainty. I would write it as (2.7±0.3) e-6, writing it as 2.7e-6 ± 0.3e-6 is possible as well.
Should be 0.002, not 0.02. I would not shift the decimal dot around like that just to get a multiple of 3.