Editing Reference range (section)

====Normal distribution====
{{further|68–95–99.7 rule}}
[[File:Standard deviation diagram.svg|thumb|350px|When assuming a normal distribution, the reference range is obtained by measuring the values in a reference group and taking two standard deviations either side of the mean. This encompasses ~95% of the total population.]]

The 95% interval, is often estimated by assuming a [[normal distribution]] of the measured parameter, in which case it can be defined as the interval limited by 1.96<ref name=MedicalStatistics>Page 48 in: {{cite book |author1=Sterne, Jonathan |author2=Kirkwood, Betty R. |title=Essential medical statistics |publisher=Blackwell Science |location=Oxford |year=2003 |isbn=978-0-86542-871-3 |url-access=registration |url=https://archive.org/details/essentialmedical00kirk }}</ref> (often rounded up to 2) population [[standard deviation]]s from either side of the population mean (also called the [[expected value]]).
However, in the real world, neither the population mean nor the population standard deviation are known. They both need to be estimated from a sample, whose size can be designated ''n''. The population standard deviation is estimated by the sample standard deviation and the population mean is estimated by the sample mean (also called mean or [[arithmetic mean]]). To account for these estimations, the 95% [[prediction interval]] (95% PI) is calculated as:

: {{math|1= 95% PI = mean ± ''t''{{sub|0.975,''n''&minus;1}}·{{sqrt|(''n''+1)/''n''}}·sd}},

where <math>t_{0.975,n-1}</math> is the 97.5% quantile of a [[Student's t-distribution]] with ''n''&minus;1 [[Degrees of freedom (statistics)|degrees of freedom]].

When the sample size is large (''n''≥30) <math>t_{0.975,n-1}\simeq 2.</math>

This method is often acceptably accurate if the standard deviation, as compared to the mean, is not very large. A more accurate method is to perform the calculations on logarithmized values, as described in separate section later.

The following example of this (''not'' logarithmized) method is based on values of [[fasting plasma glucose]] taken from a reference group of 12 subjects:<ref name=Keevil1998>[http://www.clinchem.org/cgi/content-nw/full/44/7/1535/T1 Table 1. Subject characteristics] in: {{Cite journal | last1 = Keevil | first1 = B. G. | last2 = Kilpatrick | first2 = E. S. | last3 = Nichols | first3 = S. P. | last4 = Maylor | first4 = P. W. | title = Biological variation of cystatin C: Implications for the assessment of glomerular filtration rate | journal = Clinical Chemistry | volume = 44 | issue = 7 | pages = 1535–1539 | year = 1998 | doi = 10.1093/clinchem/44.7.1535 | pmid = 9665434| doi-access = free }}</ref>

{|class="wikitable"
|-
! !! [[Fasting plasma glucose]]<br> (FPG) <br>in mmol/L !! Deviation from<br> mean ''m'' !! Squared deviation<br>from mean ''m''
|-
| Subject 1 || 5.5 || 0.17 || 0.029
|-
| Subject 2 || 5.2 || -0.13 || 0.017
|-
| Subject 3 || 5.2 || -0.13 || 0.017
|-
| Subject 4 || 5.8 || 0.47 || 0.221
|-
| Subject 5 || 5.6 || 0.27 || 0.073
|-
| Subject 6 || 4.6 || -0.73 || 0.533
|-
| Subject 7 || 5.6 || 0.27 || 0.073
|-
| Subject 8 || 5.9 || 0.57 || 0.325
|-
| Subject 9 || 4.7 || -0.63 || 0.397
|-
| Subject 10 || 5 || -0.33 || 0.109
|-
| Subject 11 || 5.7 || 0.37 || 0.137
|-
| Subject 12 || 5.2 || -0.13 || 0.017
|-
| || '''Mean = 5.33''' (''m'') <br> ''n''=12 || Mean = 0.00 || Sum/(''n''&minus;1) = 1.95/11 =0.18 <br> <math>
    \sqrt{0.18 } = 0.42 </math><br>= '''standard deviation (s.d.)'''
|}

As can be given from, for example, a [[Student's t-distribution#Table of selected values|table of selected values of Student's t-distribution]], the 97.5% percentile with (12-1) degrees of freedom corresponds to 
<math>t_{0.975,11} = 2.20</math>
 
Subsequently, the lower and upper limits of the standard reference range are calculated as:
:<math> Lower~limit = m - t_{0.975,11} \times\sqrt{\frac{n+1}{n}}\times s.d. = 5.33 - 2.20\times\sqrt{\frac{13}{12}} \times 0.42 = 4.4</math>

:<math> Upper~limit = m + t_{0.975,11} \times\sqrt{\frac{n+1}{n}}\times s.d. = 5.33 + 2.20\times\sqrt{\frac{13}{12}} \times 0.42 = 6.3.</math>

Thus, the standard reference range for this example is estimated to be 4.4 to 6.3&nbsp;mmol/L.

=====Confidence interval of limit=====
The 90% ''confidence interval of a standard reference range limit'' as estimated assuming a normal distribution can be calculated by:<ref>[https://books.google.com/books?id=p7XwAwAAQBAJ&pg=PA65 Page 65] in: {{cite book|title=Tietz Fundamentals of Clinical Chemistry and Molecular Diagnostics|author=Carl A. Burtis, David E. Bruns|edition=7|publisher=Elsevier Health Sciences|year=2014|isbn=9780323292061}}</ref>

: Lower limit of the confidence interval = percentile limit - 2.81 × {{frac|''SD''|{{sqrt|''n''}}}}

: Upper limit of the confidence interval = percentile limit + 2.81 × {{frac|''SD''|{{sqrt|''n''}}}},

where SD is the standard deviation, and n is the number of samples.

Taking the example from the previous section, the number of samples is 12 and the standard deviation is 0.42&nbsp;mmol/L, resulting in:

:''Lower limit of the confidence interval'' of the ''lower limit of the standard reference range'' = 4.4 - 2.81 × {{frac|0.42|{{sqrt|12}}}} ≈ 4.1

:''Upper limit of the confidence interval'' of the ''lower limit of the standard reference range'' = 4.4 + 2.81  × {{frac|0.42|{{sqrt|12}}}} ≈ 4.7

Thus, the lower limit of the reference range can be written as 4.4 (90% CI 4.1–4.7) mmol/L.

Likewise, with similar calculations, the upper limit of the reference range can be written as 6.3 (90% CI 6.0–6.6) mmol/L.

These confidence intervals reflect [[random error]], but do not compensate for [[systematic error]], which in this case can arise from, for example, the reference group not having fasted long enough before blood sampling.

As a comparison, actual reference ranges used clinically for fasting plasma glucose are estimated to have a lower limit of approximately 3.8<ref name=firstaid>Last page of {{cite book |author1=Deepak A. Rao |author2=Le, Tao |author3=Bhushan, Vikas |title=First Aid for the USMLE Step 1 2008 (First Aid for the Usmle Step 1) |publisher=McGraw-Hill Medical |year=2007 |isbn=978-0-07-149868-5 |url-access=registration |url=https://archive.org/details/firstaidforusmle00taol }}</ref> to 4.0,<ref name=uppsala>Reference range list from Uppsala University Hospital ("Laborationslista"). Artnr 40284 Sj74a. Issued on April 22, 2008</ref> and an upper limit of approximately 6.0<ref name=uppsala/> to 6.1.<ref name=Medline-GTT>{{MedlinePlusEncyclopedia|003466|Glucose tolerance test}}</ref>