0.01 (1%) for the lower limit and 0.007 (0.7%) for the upper limit. 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 (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: fasting plasma glucose (fpg) in mmol/lloge(fpg)loge(fpg) deviation from mean μlogsquared deviationfrom mean subject 15.51.700.0290.000841 subject 25.21.650.0210.000441 subject 35.21.650.0210.000441 subject 45.81.760.0890.007921 subject 55.61.720.0490.002401 subject 64.61.530.1410.019881 subject 75.61.720.0490.002401 subject 85.91.770.0990.009801 subject 94.71.550.1210.014641 subject 105.01.610.0610.003721 subject 115.71.740.0690.004761 subject 125.21.650.0210.000441 mean: 5.33 (m)mean: 1.67 (μlog)sum/(n-1) : 0.068/11 = 0.0062 0.0062 = 0.079 =0.079} = standard deviation of loge(fpg) (σlog) subsequently, the still logarithmized lower limit of the reference range is calculated as: ln ⁡ ( lower limit ) = μ log − t 0.975 , n − 1 × n + 1 n × σ log = 1.67 − 2.20 × 13 12 × 0.079 = 1.49 , \ln()&=\mu _-t_\times \times \sigma _\\&=1.67-2.20\times \times 0.079=1.49,\end and the upper limit of the reference range as: ln ⁡ ( upper limit ) = μ log + t 0.975 , n − 1 × n + 1 n × σ log = 1.67 + 2.20 × 13 12 × 0.079 = 1.85 \ln()&=\mu _+t_\times \times \sigma _\\&=1.67+2.20\times \times 0.079=1.85\end conversion back to non-logarithmized values are subsequently performed as: lower limit = e ln ⁡ ( lower limit ) = e 1.49 = 4.4 =e^)}=e^=4.4} upper limit = e ln ⁡ ( upper limit ) = e 1.85 = 6.4 =e^)}=e^=6.4} thus, the standard reference range for this example is estimated to be 4.4 to 6.4. from arithmetic mean and variance an alternative method of establishing a reference range with the assumption of log-normal distribution is to use the arithmetic mean and arithmetic value of standard deviation. this is somewhat more tedious to perform, but may be useful in cases where a study presents only the arithmetic mean and standard deviation, while leaving out the source data. if the original assumption of arithmetically normal distribution is less appropriate than the log-normal one, then, using the arithmetic mean and standard deviation may be the only available parameters to determine the reference range. by assuming that the expected value can represent the arithmetic mean in this case, the parameters μlog and σlog can be estimated from the arithmetic mean (m) and standard deviation (s.d.) as: μ log = ln ⁡ ( m ) − 1 2 ln ( 1 + ( s.d. m ) 2 ) =\ln(m)-\ln \!\left(1+\!\left(\right)^\right)} σ log = ln ( 1 + ( s.d. m ) 2 ) =\right)^\right)}}} following the exampled reference group from the previous section: μ log = ln ⁡ ( 5.33 ) − 1 2 ln ( 1 + ( 0.42 5.33 ) 2 ) = 1.67 =\ln(5.33)-\ln \!\left(1+\!\left(\right)^\right)=1.67} σ log = ln ( 1 + ( 0.42 5.33 ) 2 ) = 0.079 =\right)^\right)}}=0.079} subsequently, the logarithmized, and later non-logarithmized, lower and upper limit are calculated just as by logarithmized sample values. directly from percentages of interest reference ranges can also be established directly from the 2.5th and 97.5th percentile of the measurements in the reference group. for example, if the reference group consists of 200 people, and counting from the measurement with lowest value to highest, the lower limit of the reference range would correspond to the 5th measurement and the upper limit would correspond to the 195th measurement. this method can be used even when measurement values do not appear to conform conveniently to any form of normal distribution or other function. however, the reference range limits as estimated in this way have higher variance, and therefore less reliability, than those estimated by an arithmetic or log-normal distribution (when such is applicable), because the latter ones acquire statistical power from the measurements of the whole reference group rather than just the measurements at the 2.5th and 97.5th percentiles. still, this variance decreases with increasing size of the reference group, and therefore, this method may be optimal where a large reference group easily can be gathered, and the distribution mode of the measurements is uncertain. bimodal distribution bimodal distribution in case of a bimodal distribution (seen at right), it is useful to find out why this is the case. two reference ranges can be established for the two different groups of people, making it possible to assume a normal distribution for each group. this bimodal pattern is commonly seen in tests that differ between men and women, such as prostate specific antigen. interpretation of standard ranges in medical tests in case of medical tests whose results are of continuous values, reference ranges can be used in the interpretation of an individual test result. this is primarily used for diagnostic tests and screening tests, while monitoring tests may optimally be interpreted from previous tests of the same individual instead. probability of random variability reference ranges aid in the evaluation of whether a test result's deviation from the mean is a result of random variability or a result of an underlying disease or condition. if the reference group used to establish the reference range can be assumed to be representative of the individual person in a healthy state, then a test result from that individual that turns out to be lower or higher than the reference range can be interpreted as that there is less than 2.5% probability that this would have occurred by random variability in the absence of disease or other condition, which, in turn, is strongly indicative for considering an underlying disease or condition as a cause. such further consideration can be performed, for example, by an epidemiology-based differential diagnostic procedure, where potential candidate conditions are listed that may explain the finding, followed by calculations of how probable they are to have occurred in the first place, in turn followed by a comparison with the probability that the result would have occurred by random variability. if the establishment of the reference range could have been made assuming a normal distribution, then the probability that the result would be an effect of random variability can be further specified as follows: the standard deviation, if not given already, can be inversely calculated by the fact that the absolute value of the difference between the mean and either the upper or lower limit of the reference range is approximately 2 standard deviations (more accurately 1.96), and thus: standard deviation (s.d.) ≈ | (mean) - (upper limit) |/2. the standard score for the individual's test can subsequently be calculated as: standard score (z) = | (mean) - (individual measurement) |/s.d.. the probability that a value is of a certain distance from the mean can subsequently be calculated from the relation between standard score and prediction intervals. for example, a standard score of 2.58 corresponds to a prediction interval of 99%, corresponding to a probability of 0.5% that a result is at least such far from the mean in the absence of disease. example the method is described in further detail at differential diagnosis. let's say, for example, that an individual takes a test that measures the ionized calcium in the blood, resulting in a value of 1.30 mmol/l, and a reference group that appropriately represents the individual has established a reference range of 1.05 to 1.25 mmol/l. the individual's value is higher than the upper limit of the reference range, and therefore has less than 2.5% probability of being a result of random variability, constituting a strong indication to make a differential diagnosis of possible causative conditions. in this case, an epidemiology-based differential diagnostic procedure is used, and its first step is to find candidate conditions that can explain the finding. hypercalcemia (usually defined as a calcium level above the reference range) is mostly caused by either primary hyperparathyroidism or malignancy, and therefore, it is reasonable to include these in the differential diagnosis. using for example epidemiology and the individual's risk factors, let's say that the probability that the hypercalcemia would have been caused by primary hyperparathyroidism in the first place is estimated to be 0.00125 (or 0.125%), the equivalent probability for cancer is 0.0002, and 0.0005 for other conditions. with a probability given as less than 0.025 of no disease, this corresponds to a probability that the hypercalcemia would have occurred in the first place of up to 0.02695. however, the hypercalcemia has occurred with a probability of 100%, resulting adjusted probabilities of at least 4.6% that primary hyperparathyroidism has caused the hypercalcemia, at least 0.7% for cancer, at least 1.9% for other conditions and up to 92.8% for that there is no disease and the hypercalcemia is caused by random variability. in this case, further processing benefits from specification of the probability of random variability: the value is assumed to conform acceptably to a normal distribution, so the mean can be assumed to be 1.15 in the reference group. the standard deviation, if not given already, can be inversely calculated by knowing that the absolute value of the difference between the mean and, for example, the upper limit of the reference range, is approximately 2 standard deviations (more accurately 1.96), and thus: standard deviation (s.d.) ≈ | (mean) - (upper limit) |/2 = | 1.15 - 1.25 |/2 = 0.1/2 = 0.05. the standard score for the individual's test is subsequently calculated as: standard score (z) = | (mean) - (individual measurement) |/s.d. = | 1.15 - 1.30 |/0.05 = 0.15/0.05 = 3. the probability that a value is of so much larger value than the mean as having a standard score of 3 corresponds to a probability of approximately 0.14% (given by (100% − 99.7%)/2, with 99.7% here being given from the 68–95–99.7 rule). using the same probabilities that the hypercalcemia would have occurred in the first place by the other candidate conditions, the probability that hypercalcemia would have occurred in the first place is 0.00335, and given the fact that hypercalcemia has occurred gives adjusted probabilities of 37.3%, 6.0%, 14.9% and 41.8%, respectively, for primary hyperparathyroidism, cancer, other conditions and no disease. optimal health range optimal (health) range or therapeutic target (not to be confused with biological target) is a reference range or limit that is based on concentrations or levels that are associated with optimal health or minimal risk of related complications and diseases, rather than the standard range based on normal distribution in the population. it may be more appropriate to use for e.g. folate, since approximately 90 percent of north americans may actually suffer more or less from folate deficiency, but only the 2.5 percent that have the lowest levels will fall below the standard reference range. in this case, the actual folate ranges for optimal health are substantially higher than the standard reference ranges. vitamin d has a similar tendency. in contrast, for e.g. uric acid, having a level not exceeding the standard reference range still does not exclude the risk of getting gout or kidney stones. furthermore, for most toxins, the standard reference range is generally lower than the level of toxic effect. a problem with optimal health range is a lack of a standard method of estimating the ranges. the limits may be defined as those where the health risks exceed a certain threshold, but with various risk profiles between different measurements (such as folate and vitamin d), and even different risk aspects for one and the same measurement (such as both deficiency and toxicity of vitamin a) it is difficult to standardize. subsequently, optimal health ranges, when given by various sources, have an additional variability caused by various definitions of the parameter. also, as with standard reference ranges, there should be specific ranges for different determinants that affects the values, such as sex, age etc. ideally, there should rather be an estimation of what is the optimal value for every