Significant Figures

[dttom]

I am just a bit confused about whether we should use a truncated value in calculation, personally I think the answer should be 'no' (as multiple rounding-off or truncation adds uncertainty), but I just want a confirm. Say for example, I worked on raw data and got an eqation of:

y = 0.784x(plus or minus 0.002)+0.059(plus or minus 0.008), so here 0.784 and 0.059 are truncated values based on the standard deviation (0.002 and 0.008). Now if y is 0.224, shall I use the raw values (that is, the original values not truncated) or truncated values of 0.784 and 0.059? If if use the truncated ones, then,

0.224 = 0.784x+0.059, x=0.210(3 sig fig). I just feel a little bit weird not to use numbers provided in an equation but to trace back to their (the numbers') origins. Because two methods give different result, I just want a check, thanks for any help.

Edited September 22, 2010 by dttom

[cypress]

If I understand your questions, I don't see the answer as so much should it be done or not but rather when is it most appropriate. I'll try to explain my thoughts as follows:

 

I am just a bit confused about whether we should use a truncated value in calculation, personally I think the answer should be 'no' (as multiple rounding-off or truncation adds uncertainty), but I just want a confirm.

 

I think in these situations, it is not that rounding adds uncertainty as it introduces error, since the source of the difference is known and is due to use of imprecise values. with this in mind multiple rounding-off should be avoided until final results are obtained in nearly every case.

 

Say for example, I worked on raw data and got an eqation of:

y = 0.784x(plus or minus 0.002)+0.059(plus or minus 0.008), so here 0.784 and 0.059 are truncated values based on the standard deviation (0.002 and 0.008).Now if y is 0.224, shall I use the raw values (that is, the original values not truncated) or truncated values of 0.784 and 0.059? If if use the truncated ones, then, 0.224 = 0.784x+0.059, x=0.210(3 sig fig). I just feel a little bit weird not to use numbers provided in an equation but to trace back to their (the numbers') origins. Because two methods give different result, I just want a check, thanks for any help.

 

Indeed they do. The final result after rounding could range from 0.210 to 0.211 and as you suspected is a result of using rounded numbers in a mixed equation where the imprecision is propagated to the solution through multiple roundings. However what is a little less clear is which of these final results is "more correct". Use of intermediate rounding has introduced additional imprecision or more correctly a bias that in this example happens to span the midpoint. This is one of the limitations of using rounding and truncation to express the precision of an equation.

 

If you are uncomfortable with this "effect" then use the unrounded numbers until the final result is obtained, that method should never lead you astray.

[Mr Skeptic]

Use the most precise numbers you have for your calculations, then factor in the errors for the final answer.