`
`6 The Cardiac Arrhythmia Suppression Trial I. Preliminary Report. Effect of
`encainide and flecainide on mortality in a randomized trial of arrhythmia
`suppression after myocardial infarction. N Engl J Med 1989;321:406-12.
`7 Dickert N, Grady C. What’s the price of a research subject? Approaches to
`payment for research participation. N Engl J Med 1999;341:198-203.
`8 Grady C. Money for research participation: does it jeopardize informed
`consent? Am J Bioethics 2001;1:40-4.
`9 Macklin R. “Due” and “undue” inducements: On paying money to
`research subjects. IRB: a review of human subjects research 1981;3:1-6.
`10 McGee G. Subject to payment? JAMA 1997;278:199-200.
`in research. Bioethics
`11 McNeil P. Paying people
`to participate
`1997;11:390-6.
`12 Wilkenson M, Moore A. Inducement in research. Bioethics 1997;11:
`373-89.
`13 Halpern SD, Karlawish JHT, Casarett D, Berlin JA, Asch DA. Empirical
`assessment of whether moderate payments are undue or unjust induce-
`ments for participation in clinical trials. Arch Intern Med 2004;164:801-3.
`14 Bentley JP, Thacker PG. The influence of risk and monetary payment on
`J Med Ethics
`the research participation decision making process.
`2004;30:293-8.
`15 Viens AM. Socio-economic status and inducement to participate. Am J
`Bioethics 2001;1.
`
`16 Beauchamp TL, Childress JF. Respect for autonomy. Principles of biomedical
`ethics. 4th ed. New York: Oxford University Press, 1994:120-88.
`17 Roberts LW. Evidence-based ethics and informed consent in mental
`illness research. Arch Gen Psychiatry 2000;57:540-2.
`18 Bayer R, Oppenheimer GM. Toward a more democratic medicine: sharing the
`burden of ignorance. AIDS Doctors: voices from the epidemic. New York: Oxford
`University Press, 2000:156-69.
`19 Coulter A, Rozansky D. Full engagement in health. BMJ 2004;329:1197-8.
`20 Joffe S, Manocchia M, Weeks JC, Cleary PD. What do patients value in
`their hospital care? An empirical perspective on autonomy centred
`bioethics. J Med Ethics 2003;29:103-8.
`21 Heesen C, Kasper J, Segal J, Kopke S, Muhlhauser I. Decisional role pref-
`erences, risk knowledge and information interests in patients with multi-
`ple sclerosis. Mult Scler 2004;10:643-50.
`22 Azoulay E, Pochard F, Chevret S, et al. Half the family members of inten-
`sive care unit patients do not want to share in the decision-making proc-
`ess: a study in 78 French intensive care units. Crit Care Med
`2004;32:1832-8.
`23 Dunn LB, Gordon NE. Improving informed consent and enhancing
`recruitment for research by understanding economic behavior. JAMA
`2005;293:609-12.
`
`Statistics Notes
`Standard deviations and standard errors
`Douglas G Altman, J Martin Bland
`
`The terms “standard error” and “standard deviation”
`are often confused.1 The contrast between these two
`terms reflects the important distinction between data
`description and inference, one that all researchers
`should appreciate.
`The standard deviation (often SD) is a measure of
`variability. When we calculate the standard deviation of a
`sample, we are using it as an estimate of the variability of
`the population from which the sample was drawn. For
`data with a normal distribution,2 about 95% of individu-
`als will have values within 2 standard deviations of the
`mean, the other 5% being equally scattered above and
`below these limits. Contrary to popular misconception,
`the standard deviation is a valid measure of variability
`regardless of the distribution. About 95% of observa-
`tions of any distribution usually fall within the 2 standard
`deviation limits, though those outside may all be at one
`end. We may choose a different summary statistic, how-
`ever, when data have a skewed distribution.3
`When we calculate the sample mean we are usually
`interested not in the mean of this particular sample, but
`in the mean for individuals of this type—in statistical
`terms, of the population from which the sample comes.
`We usually collect data in order to generalise from them
`and so use the sample mean as an estimate of the mean
`for the whole population. Now the sample mean will
`vary from sample to sample; the way this variation
`occurs is described by the “sampling distribution” of the
`mean. We can estimate how much sample means will
`vary from the standard deviation of this sampling distri-
`bution, which we call the standard error (SE) of the esti-
`mate of the mean. As the standard error is a type of
`standard deviation,
`confusion is understandable.
`Another way of considering the standard error is as a
`measure of the precision of the sample mean.
`The standard error of the sample mean depends
`on both the standard deviation and the sample size, by
`the simple relation SE = SD/√(sample size). The stand-
`ard error falls as the sample size increases, as the extent
`of chance variation is reduced—this idea underlies the
`sample size calculation for a controlled trial,
`for
`
`example. By contrast the standard deviation will not
`tend to change as we increase the size of our sample.
`So, if we want to say how widely scattered some
`measurements are, we use the standard deviation. If we
`want to indicate the uncertainty around the estimate of
`the mean measurement, we quote the standard error of
`the mean. The standard error is most useful as a means
`of calculating a confidence interval. For a large sample,
`a 95% confidence interval is obtained as the values
`1.96×SE either side of the mean. We will discuss confi-
`dence intervals in more detail in a subsequent Statistics
`Note. The standard error is also used to calculate P val-
`ues in many circumstances.
`The principle of a sampling distribution applies to
`other quantities that we may estimate from a sample,
`such as a proportion or regression coefficient, and to
`contrasts between two samples, such as a risk ratio or
`the difference between two means or proportions. All
`such quantities have uncertainty due to sampling vari-
`ation, and for all such estimates a standard error can be
`calculated to indicate the degree of uncertainty.
`In many publications a ± sign is used to join the
`standard deviation (SD) or standard error (SE) to an
`observed mean—for example, 69.4±9.3 kg. That
`notation gives no indication whether the second figure
`is the standard deviation or the standard error (or
`indeed something else). A review of 88 articles
`published in 2002 found that 12 (14%)
`failed to
`identify which measure of dispersion was reported
`(and three failed to report any measure of variability).4
`The policy of the BMJ and many other journals is to
`remove ± signs and request authors to indicate clearly
`whether the standard deviation or standard error is
`being quoted. All journals should follow this practice.
`
`Competing interests: None declared.
`
`1 Nagele P. Misuse of standard error of the mean (SEM) when reporting
`variability of a sample. A critical evaluation of four anaesthesia journals.
`Br J Anaesthesiol 2003;90:514-6.
`2 Altman DG, Bland JM. The normal distribution. BMJ 1995;310:298.
`3 Altman DG, Bland JM. Quartiles, quintiles, centiles, and other quantiles.
`BMJ 1994;309:996.
`4 Olsen CH. Review of the use of statistics in Infection and Immunity. Infect
`Immun 2003;71:6689-92.
`
`Cancer Research
`UK/NHS Centre
`for Statistics in
`Medicine, Wolfson
`College, Oxford
`OX2 6UD
`Douglas G Altman
`professor of statistics
`in medicine
`
`Department of
`Health Sciences,
`University of York,
`York YO10 5DD
`J Martin Bland
`professor of health
`statistics
`
`Correspondence to:
`Prof Altman
`doug.altman@
`cancer.org.uk
`
`BMJ 2005;331:903
`
`BMJ VOLUME 331 15 OCTOBER 2005 bmj.com
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`903
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`Opiant Exhibit 2222
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