In medicine and health-related fields, a reference range or reference interval is the range or the interval of values that is deemed normal for a physiological measurement in healthy persons (for example, the amount of creatinine in the blood, or the partial pressure of oxygen). It is a basis for comparison for a physician or other health professional to interpret a set of test results for a particular patient. Some important reference ranges in medicine are reference ranges for blood tests and reference ranges for urine tests.

The standard definition of a reference range (usually referred to if not otherwise specified) originates in what is most prevalent in a reference group taken from the general (i.e. total) population. This is the general reference range. However, there are also optimal health ranges (ranges that appear to have the optimal health impact) and ranges for particular conditions or statuses (such as pregnancy reference ranges for hormone levels).

Values within the reference range (WRR) are those within normal limits (WNL). The limits are called the upper reference limit (URL) or upper limit of normal (ULN) and the lower reference limit (LRL) or lower limit of normal (LLN). In health care–related publishing, style sheets sometimes prefer the word reference over the word normal to prevent the nontechnical senses of normal from being conflated with the statistical sense. Values outside a reference range are not necessarily pathologic, and they are not necessarily abnormal in any sense other than statistically. Nonetheless, they are indicators of probable pathosis. Sometimes the underlying cause is obvious; in other cases, challenging differential diagnosis is required to determine what is wrong and thus how to treat it.

A cutoff or threshold is a limit used for binary classification, mainly between normal versus pathological (or probably pathological). Establishment methods for cutoffs include using an upper or a lower limit of a reference range.

The standard definition of a reference range for a particular measurement is defined as the interval between which 95% of values of a reference population fall into, in such a way that 2.5% of the time a value will be less than the lower limit of this interval, and 2.5% of the time it will be larger than the upper limit of this interval, whatever the distribution of these values.^[1]

Reference ranges that are given by this definition are sometimes referred as standard ranges.

Since a range is a defined statistical value (Range (statistics)) that describes the interval between the smallest and largest values, many, including the International Federation of Clinical Chemistry prefer to use the expression reference interval rather than reference range.^[2]

Regarding the target population, if not otherwise specified, a standard reference range generally denotes the one in healthy individuals, or without any known condition that directly affects the ranges being established. These are likewise established using reference groups from the healthy population, and are sometimes termed normal ranges or normal values (and sometimes "usual" ranges/values). However, using the term normal may not be appropriate as not everyone outside the interval is abnormal, and people who have a particular condition may still fall within this interval.

However, reference ranges may also be established by taking samples from the whole population, with or without diseases and conditions. In some cases, diseased individuals are taken as the population, establishing reference ranges among those having a disease or condition. Preferably, there should be specific reference ranges for each subgroup of the population that has any factor that affects the measurement, such as, for example, specific ranges for each sex, age group, race or any other general determinant.

Methods for establishing reference ranges can be based on assuming a normal distribution or a log-normal distribution, or directly from percentages of interest, as detailed respectively in following sections. When establishing reference ranges from bilateral organs (e.g., vision or hearing), both results from the same individual can be used, although intra-subject correlation must be taken into account.^[3]

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^[4] (often rounded up to 2) population standard deviations 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:

95% PI = mean ± t_0.975,n−1·√(n+1)/n·sd,

where ${\displaystyle t_{0.975,n-1}}$ is the 97.5% quantile of a Student's t-distribution with n−1 degrees of freedom.

When the sample size is large (n≥30) ${\displaystyle t_{0.975,n-1}\simeq 2.}$

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:^[5]

	Fasting plasma glucose (FPG) in mmol/L	Deviation from mean m	Squared deviation 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) n=12	Mean = 0.00	Sum/(n−1) = 1.95/11 =0.18 ${\displaystyle {\sqrt {0.18}}=0.42}$ = standard deviation (s.d.)

As can be given from, for example, a table of selected values of Student's t-distribution, the 97.5% percentile with (12-1) degrees of freedom corresponds to ${\displaystyle t_{0.975,11}=2.20}$

Subsequently, the lower and upper limits of the standard reference range are calculated as:

${\displaystyle Lower~limit=m-t_{0.975,11}\times {\sqrt {\frac {n+1}{n}}}\times s.d.=5.33-2.20\times {\sqrt {\frac {13}{12}}}\times 0.42=4.4}$

${\displaystyle Upper~limit=m+t_{0.975,11}\times {\sqrt {\frac {n+1}{n}}}\times s.d.=5.33+2.20\times {\sqrt {\frac {13}{12}}}\times 0.42=6.3.}$

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

The 90% confidence interval of a standard reference range limit as estimated assuming a normal distribution can be calculated by:^[6]

Lower limit of the confidence interval = percentile limit - 2.81 × SD⁄√n

Upper limit of the confidence interval = percentile limit + 2.81 × SD⁄√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 mmol/L, resulting in:

Lower limit of the confidence interval of the lower limit of the standard reference range = 4.4 - 2.81 × 0.42⁄√12 ≈ 4.1

Upper limit of the confidence interval of the lower limit of the standard reference range = 4.4 + 2.81 × 0.42⁄√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^[7] to 4.0,^[8] and an upper limit of approximately 6.0^[8] to 6.1.^[9]

In reality, biological parameters tend to have a log-normal distribution,^[10] rather than the normal distribution or Gaussian distribution.

An explanation for this log-normal distribution for biological parameters is: The event where a sample has half the value of the mean or median tends to have almost equal probability to occur as the event where a sample has twice the value of the mean or median. Also, only a log-normal distribution can compensate for the inability of almost all biological parameters to be of negative numbers (at least when measured on absolute scales), with the consequence that there is no definite limit to the size of outliers (extreme values) on the high side, but, on the other hand, they can never be less than zero, resulting in a positive skewness.

As shown in diagram at right, this phenomenon has relatively small effect if the standard deviation (as compared to the mean) is relatively small, as it makes the log-normal distribution appear similar to a normal distribution. Thus, the normal distribution may be more appropriate to use with small standard deviations for convenience, and the log-normal distribution with large standard deviations.

In a log-normal distribution, the geometric standard deviations and geometric mean more accurately estimate the 95% prediction interval than their arithmetic counterparts.

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^[11] 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:

Difference ratio = ⁠| Limit_log-normal - Limit_normal |/ Limit_log-normal ⁠

where:

• Limit_log-normal is the (lower or upper) limit as estimated by assuming log-normal distribution
• Limit_normal is the (lower or upper) limit as estimated by assuming normal distribution.

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

Coefficient of variation = ⁠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.

A method to estimate the reference range for a parameter with log-normal distribution is to logarithmize all the measurements with an arbitrary base (for example 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 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 as a base:^[5]

	Fasting plasma glucose (FPG) in mmol/L	loge(FPG)	loge(FPG) deviation from mean μlog	Squared deviation 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 (m)	Mean: 1.67 (μlog)		Sum/(n-1) : 0.068/11 = 0.0062 ${\displaystyle {\sqrt {0.0062}}=0.079}$ = standard deviation of loge(FPG) (σlog)

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

${\displaystyle {\begin{aligned}\ln({\text{lower limit}})&=\mu _{\log }-t_{0.975,n-1}\times {\sqrt {\frac {n+1}{n}}}\times \sigma _{\log }\\&=1.67-2.20\times {\sqrt {\frac {13}{12}}}\times 0.079=1.49,\end{aligned}}}$

and the upper limit of the reference range as:

Zdroj:https://en.wikipedia.org?pojem=Reference_range
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