Fifteen-unit rule for rounding numerical results: Difference between revisions

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==Definition==
==Definition==


Usually, from scientific studies (whether based on physical measurements or mathematical modelling), numerical results are obtained that consist of a an average value or a best estimate (e.g. 4.5678 meters) and a measure of the probable uncertainty (e.g. ± 0.2345 meters). However, either or both of these figures may be too precise (i.e. contain more digits than is justifiable or meaningful).  
Usually, from scientific studies (whether based on physical measurements or mathematical modelling), numerical results are obtained that consist of an average value or a best estimate (e.g. 4.5678 meters) and a measure of the probable uncertainty (e.g. ±0.2345 meters). However, as raw figures, either or both of these may be too precise (i.e. contain more digits than is justifiable or meaningful).  


A systematic method for rounding those results to a justifiable and meaningful precision is needed, that will not convey a misconception of excessively (in)accurate results.
Thus, before publishing, a systematic method for rounding those results to a justifiable and meaningful precision is needed, in order to avoid a misconception of excessively (in)accurate results.


==Result==
==Result==
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Please see the following examples:
Please see the following examples:


* '''(1062 ± 41) meters''' is '''incorrect''', because the LSD of the average value (2) is associated with an uncertainty of 41 units (and the uncertainty is expressed using a precision of 41 units)
* <font color=red>(1062 ± 41) meters</font> is '''incorrect''', because the LSD of the average value (2) is associated with an uncertainty of 41 units (and the uncertainty is expressed using a precision of 41 units)
* '''(1060 ± 50) meters''' is '''correct''', because the LSD of the average value (6) is now associated with an uncertainty of 5 units only (and the uncertainty is expressed using a precision of 5 units)
* <font color=green>(1060 ± 50) meters</font> is '''correct''', because the LSD of the average value (6) is now associated with an uncertainty of 5 units only (and the uncertainty is expressed using a precision of 5 units)
:: - Note that the original uncertainty (41) has been rounded upwards (to 50).
:: - Note that the original uncertainty (41) has been rounded upwards (to 50).


* '''(0.8765 ± 0.0132) kg''' is '''incorrect''', because the LSD of the average value (5) is associated with an uncertainty of 132 units (and the uncertainty is expressed using a precision of 132 units)
* <font color=red>(0.8765 ± 0.0132) kg</font> is '''incorrect''', because the LSD of the average value (5) is associated with an uncertainty of 132 units (and the uncertainty is expressed using a precision of 132 units)
* '''(0.876 ± 0.014) kg''' is '''correct''', because the LSD of the average value (6) is now associated with an uncertainty of 14 units only (and the uncertainty is expressed using a precision of 14 units)
* <font color=green>(0.876 ± 0.014) kg</font> is '''correct''', because the LSD of the average value (6) is now associated with an uncertainty of 14 units only (and the uncertainty is expressed using a precision of 14 units)
:: - Note that the original uncertainty (0.0132) has been rounded upwards (to 0.014).
:: - Note that the original uncertainty (0.0132) has been rounded upwards (to 0.014).

Revision as of 15:22, 8 August 2011

Scope

When presenting the final results of a study, what is the proper way of rounding numerical results of the form "average ± probable uncertainty"?

Definition

Usually, from scientific studies (whether based on physical measurements or mathematical modelling), numerical results are obtained that consist of an average value or a best estimate (e.g. 4.5678 meters) and a measure of the probable uncertainty (e.g. ±0.2345 meters). However, as raw figures, either or both of these may be too precise (i.e. contain more digits than is justifiable or meaningful).

Thus, before publishing, a systematic method for rounding those results to a justifiable and meaningful precision is needed, in order to avoid a misconception of excessively (in)accurate results.

Result

The 15-unit rule says that:

  • in the average value, the uncertainty of the least significant digit (LSD) must not exceed 15 units,
  • the probable uncertainty must not exceed 15 units (where one unit pertains to the LSD of the average value)
  • the probable uncertainty is always rounded upwards.

Please see the following examples:

  • (1062 ± 41) meters is incorrect, because the LSD of the average value (2) is associated with an uncertainty of 41 units (and the uncertainty is expressed using a precision of 41 units)
  • (1060 ± 50) meters is correct, because the LSD of the average value (6) is now associated with an uncertainty of 5 units only (and the uncertainty is expressed using a precision of 5 units)
- Note that the original uncertainty (41) has been rounded upwards (to 50).
  • (0.8765 ± 0.0132) kg is incorrect, because the LSD of the average value (5) is associated with an uncertainty of 132 units (and the uncertainty is expressed using a precision of 132 units)
  • (0.876 ± 0.014) kg is correct, because the LSD of the average value (6) is now associated with an uncertainty of 14 units only (and the uncertainty is expressed using a precision of 14 units)
- Note that the original uncertainty (0.0132) has been rounded upwards (to 0.014).
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