# Fifteen-unit rule for rounding numerical results

## Scope

When presenting the final results of a study, what is a scientifically sound method for rounding numerical results of the form "average value ± 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.

The 15-unit rule is a practical method (used in e.g. the teaching of university-level physics and engineering) that ensures that neither the average value nor the probable uncertainty will be expressed with a meaninglessly high precision.

## Result

The 15-unit rule says that:

• in the average value, the uncertainty of the least significant digit (LSD) must not exceed 15 units,
• and the probable uncertainty is always rounded upwards, to the same decimal position as the average value.
TODO: {{#todo:Tämähän on hyvä ohje. Voisitko tehdä koodin, jossa R:llä tehdään tuo pyöristys automaattisesti? Varmaan toimisi helpoiten funktiona, jolle annetaan 2 parametria eli estimate ja uncertainty. --Jouni 13:46, 10 August 2011 (EEST)|Erkki Kuusisto|Opasnet}}

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 also 41 units)
• (1060 ± 41) meters is also incorrect, because the uncertainty contains a lower significant decimal position ("ones") than the average value ("tens")
• (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 has been rounded to the same decimal position as the average value)
- 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
• (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 has been rounded to the same decimal position as the average value)
- Note that the original uncertainty (0.0132) has been rounded upwards (to 0.014).

• (0.9765 ± 0.0172) kg is incorrect, because the LSD of the average value (5) is associated with an uncertainty of 172 units
• (0.976 ± 0.018) kg is still incorrect, because the LSD of the average value (6) is associated with an uncertainty of 18 units
• (0.98 ± 0.02) kg is correct, because the LSD of the average value (8) is now associated with an uncertainty of 2 units only (and the uncertainty has been rounded to the same decimal position as the average value).