Editing Reference range (section)

=====Necessity=====
Reference ranges for substances that are usually within relatively narrow limits (coefficient of variation less than 0.213, as detailed below) such as [[electrolytes]] can be estimated by assuming normal distribution, whereas reference ranges for those that vary significantly (coefficient of variation generally over 0.213) such as most [[hormones]]<ref name="pmid19758299">{{cite journal| author=Levitt H, Smith KG, Rosner MH| title=Variability in calcium, phosphorus, and parathyroid hormone in patients on hemodialysis. | journal=Hemodial Int | year= 2009 | volume= 13 | issue= 4 | pages= 518–25 | pmid=19758299 | doi=10.1111/j.1542-4758.2009.00393.x | pmc= | s2cid=24963421 | url=https://pubmed.ncbi.nlm.nih.gov/19758299  }}</ref> are more accurately established by log-normal distribution.

The necessity to establish a reference range by log-normal distribution rather than normal distribution can be regarded as depending on how much difference it would make to ''not'' do so, which can be described as the  ratio:

:{{math|1=Difference ratio = {{sfrac| {{mabs| Limit{{sub|log-normal}} - Limit{{sub|normal}} }} | Limit{{sub|log-normal}} }} }}

where:
* ''Limit<sub>log-normal</sub>'' is the (lower or upper) limit as estimated by assuming log-normal distribution
* ''Limit<sub>normal</sub>'' is the (lower or upper) limit as estimated by assuming normal distribution.

[[File:Diagram of coefficient of variation versus deviation in reference ranges erroneously not established by log-normal distribution.png|thumb|350px|Coefficient of variation versus deviation in reference ranges established by assuming normal distribution when there is actually a log-normal distribution.]]

This difference can be put solely in relation to the [[coefficient of variation]], as in the diagram at right, where:

:{{math|1=Coefficient of variation = {{sfrac|s.d.|m}}}}

where:
* ''s.d.'' is the standard deviation
* ''m'' is the arithmetic mean

In practice, it can be regarded as necessary to use the establishment methods of a log-normal distribution if the difference ratio becomes more than 0.1, meaning that a (lower or upper) limit estimated from an assumed normal distribution would be more than 10% different from the corresponding limit as estimated from a (more accurate) log-normal distribution. As seen in the diagram, a difference ratio of 0.1 is reached for the lower limit at a coefficient of variation of 0.213 (or 21.3%), and for the upper limit at a coefficient of variation at 0.413 (41.3%). The lower limit is more affected by increasing coefficient of variation, and its "critical" coefficient of variation of 0.213 corresponds to a ratio of (upper limit)/(lower limit) of 2.43, so as a rule of thumb, if the upper limit is more than 2.4 times the lower limit when estimated by assuming normal distribution, then it should be considered to do the calculations again by log-normal distribution.

Taking the example from previous section, the standard deviation (s.d.) is estimated at 0.42 and the arithmetic mean (m) is estimated at 5.33. Thus the coefficient of variation is 0.079. This is less than both 0.213 and 0.413, and thus both the lower and upper limit of fasting blood glucose can most likely be estimated by assuming normal distribution. More specifically, the coefficient of variation of 0.079 corresponds to a difference ratio of 0.01 (1%) for the lower limit and 0.007 (0.7%) for the upper limit.