Editing Reference range (section)

=====From logarithmized sample values=====
A method to estimate the reference range for a parameter with log-normal distribution is to logarithmize all the measurements with an arbitrary [[base of a logarithm|base]] (for example [[e (mathematical constant)|''e'']]), derive the mean and standard deviation of these logarithms, determine the logarithms located (for a 95% prediction interval) 1.96 standard deviations below and above that mean, and subsequently [[exponentiation|exponentiate]] using those two logarithms as exponents and using the same base as was used in logarithmizing, with the two resultant values being the lower and upper limit of the 95% prediction interval.

The following example of this method is based on the same values of [[fasting plasma glucose]] as used in the previous section, using [[e (mathematical constant)|''e'']] as a [[base of a logarithm|base]]:<ref name=Keevil1998/>

{|class="wikitable"
|-
! !! [[Fasting plasma glucose]]<br> (FPG) <br>in mmol/L !! log<sub>[[e (mathematical constant)|''e'']]</sub>(FPG) !! log<sub>e</sub>(FPG) deviation from<br> mean ''μ''<sub>log</sub> !! Squared deviation<br>from mean
|-
| Subject 1 || 5.5 || 1.70 || 0.029 || 0.000841
|-
| Subject 2 || 5.2 || 1.65 || 0.021 || 0.000441
|-
| Subject 3 || 5.2 || 1.65 || 0.021 || 0.000441
|-
| Subject 4 || 5.8 || 1.76 || 0.089 || 0.007921
|-
| Subject 5 || 5.6 || 1.72 || 0.049 || 0.002401
|-
| Subject 6 || 4.6 || 1.53 || 0.141 || 0.019881
|-
| Subject 7 || 5.6 || 1.72 || 0.049 || 0.002401
|-
| Subject 8 || 5.9 || 1.77 || 0.099 || 0.009801
|-
| Subject 9 || 4.7 || 1.55 || 0.121 || 0.014641
|-
| Subject 10 || 5.0 || 1.61 || 0.061 || 0.003721
|-
| Subject 11 || 5.7 || 1.74 || 0.069 || 0.004761
|-
| Subject 12 || 5.2 || 1.65 || 0.021 || 0.000441
|-
| || '''Mean: 5.33''' <br> (''m'') || '''Mean: 1.67'''<br> (''μ''<sub>log</sub>) ||  || Sum/(n-1) : 0.068/11 = 0.0062 <br> <math>
    \sqrt{0.0062} = 0.079</math><br>= '''standard deviation of log<sub>e</sub>(FPG)'''<br> (''σ''<sub>log</sub>)
|}

Subsequently, the still logarithmized lower limit of the reference range is calculated as:

: <math>\begin{align} \ln (\text{lower limit}) &= \mu_{\log} - t_{0.975,n-1} \times\sqrt{\frac{n+1}{n}} \times \sigma_{\log}\\ 
&= 1.67 - 2.20\times\sqrt{\frac{13}{12}} \times 0.079 = 1.49, \end{align}</math>

and the upper limit of the reference range as:

: <math>\begin{align} \ln (\text{upper limit}) &= \mu_{\log} + t_{0.975,n-1} \times\sqrt{\frac{n+1}{n}} \times \sigma_{\log}\\ 
&= 1.67 + 2.20\times\sqrt{\frac{13}{12}} \times 0.079 = 1.85 \end{align}</math>

Conversion back to non-logarithmized values are subsequently performed as:

: <math> \text{Lower limit} = e^{\ln (\text{lower limit})} = e^{1.49} = 4.4</math>

: <math> \text{Upper limit} = e^{\ln (\text{upper limit})} = e^{1.85} = 6.4</math>

Thus, the standard reference range for this example is estimated to be 4.4 to 6.4.