Endgames Statistical Question

# Reference and normal ranges

BMJ 2010; 341 (Published 24 November 2010) Cite this as: BMJ 2010;341:c6666
1. Philip Sedgwick, senior lecturer in medical statistics
1. 1Section of Medical and Healthcare Education, St George’s, University of London, Tooting, London, UK
1. p.sedgwick{at}sgul.ac.uk

Researchers recorded the forehead temperature of 1000 apparently healthy subjects aged between 18 and 65 years using a handheld infrared thermometer.1 The sample mean forehead temperature was 33.3°C, with a standard deviation of 1.18°C. The normal range for forehead temperature, calculated as 1.96 standard deviations either side of the sample mean, was reported as 31.0-35.6°C.

A separate study measured a series of immunohaematological variables in 150 healthy adults to establish haematological reference ranges for HIV negative adults from the Central African Republic.2 Ranges for each variable were derived as the interval from the 2.5th to 97.5th centile of the sample measurements. The reference range for erythrocyte counts was reported to be 4.50-6.10×1012/l for men and 3.42-5.44×1012/l for women.

Which of the following statements, if any, are true?

• a) People with a forehead temperature outside the normal range have abnormal measurements

• b) Only healthy people have a forehead temperature in the normal range

• c) The calculation of the normal range for forehead temperature assumed the sample measurements were normally distributed

• d) The reference range for erythrocyte counts contains the central 95% of the sample measurements

Answers c and d are true, whereas a and b are false.

A normal range, also known as a reference range or interval, is calculated for a single physical, biological, or social continuous variable. The interval is derived from a large sample of healthy people and typically intends to include 95% of such individuals. Sometimes separate ranges are calculated for different age groups or either sex (or both). Each interval provides a range of values that are expected for healthy people in the population and is used to interpret patients’ test results or measurements. Reference ranges can be derived in several ways as described below.

A patient’s forehead temperature can be compared with the normal range. Anyone outside the interval would be considered to have a low or high temperature compared with healthy people. But this does not automatically mean that the person’s forehead temperature is abnormal (a is false). Furthermore, a patient with a health problem may have a measurement within the normal range (b is false). All measurements should be interpreted in the context of the patient and his or her medical history. Therefore, the term “normal range” is considered potentially misleading and “reference range” or “reference interval” is preferred.

Calculation of the reference range for forehead temperature assumed that the sample measurements were normally distributed (c is true). The theoretical Normal distribution, described in a previous question,3 has the familiar bell shaped curve described by its mean and standard deviation. Exactly 95% of the area under the curve, that is 95% of the distribution, is within the range of 1.96 standard deviations either side of the mean. The normal range for forehead temperature was therefore calculated as (33.3°C−1.96×1.18) to (33.3oC+1.96×1.18), that is 31.0-35.6°C. Therefore, 2.5% of people would be expected to have a measurement below this range and 2.5% above.

The sample measurements of forehead temperature are an empirical distribution and would only approximate a Normal distribution. Therefore, the proportion of the sample included in the reference range would approximate 95%. Obviously, the proportion of the sample that the reference range will contain, or the proportion that will be above or below it, is not guaranteed. Therefore, references ranges are often calculated as the range that is no further than two standard deviations either side of the sample mean, which contains about 95% or more of sample measurements.4

Calculation of the reference range for erythrocyte counts does not assume that the sample measurements approximate a Normal distribution. The range was derived from the 2.5th to the 97.5th centile of the sample measurements. It contains the central 95% of the sample measurements of erythrocyte counts, with 2.5% of people having a measurement below and 2.5% above this range (d is true). Separate ranges were calculated for men and women because the sample distribution of erythrocyte counts differed between the sexes. A reference range based on centiles of the sample measurements allows for variables with skewed distributions that cannot be adequately described by the sample mean and standard deviation.4 5

## Notes

Cite this as: BMJ 2010;341:c6666

## Footnotes

• Competing interests: None declared.

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