Fifteen-unit rule for rounding numerical results

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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,

• 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).