Education and debate
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`of biomedical
`Respectfor autonomy.
`16 Beauchamp TL, Childress
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`encainide and flecainide on mortality in a randomized trial of arrhythmia
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`17 Roberts LW Evidence-based ethics and informed consent in mental
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`illness research. Arch Gen Psychiatry 2000;57:540-2.
`payment for research participation. NEngl] Med 1999;341:198-203.
`18 Bayer R, Oppenheimer GM. Flu/ard a more democratic medicine: sharing the
`8 Grady C. Money for research participation: does it jeopardize informed
`burden ofignorance AIDS Doctorswoicesfrom the epidemic. New York: Oxford
`consent? Am]Bioethics 2001 ;1 :40-4.
`University Press, 2000:156-69.
`9 Macklin R. “Due” and “undue” inducements: On paying money to
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`research subjects. IRB:a review ofhuman subjects research 1981 ;3: 1-6.
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`P. Paying people
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`in research. Bioethics
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`12 Wilkenson M, Moore A Inducement in research. Bioethics 1997;11:
`373-89.
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`
`
`Statistics Notes
`
`Standard deviations and standard errors
`
`Douglas G Altman,] 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 typegin 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 reducedithis idea underlies the
`
`sample size calculation for a controlled trial, for
`
`BMI VOLUME 331
`
`15 OCTOBER 2005
`
`bmj.c0m
`
`Cancer Research
`UK/NHS Centre
`for Statistics in
`Medicine, Wolfson
`College, Oxford
`OX2 6UD
`Douglas G Altman
`professor ofstatistics
`in medicine
`
`Department of
`Health Sciences,
`University ofYork,
`York YOlO 5DD
`J Martin Bland
`professor ofhealth
`statistics
`
`Correspondence to:
`Prof Altman
`doug.altman@
`cancerorguk
`
`BM] 2005;331:903
`
`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. Ifwe
`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 i sign is used to join the
`standard deviation (SD) or standard error (SE) to an
`observed meanifor example, 69.4i9.3 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 BM] and many other journals is to
`remove i 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 (SElVI) when reporting
`variability of a sample. A critical evaluation of four anaesthesia journals.
`Br]Anaesthesiol 2003;90:514-6.
`Altman DG, BlandJM. The normal distribution BM] 1995;310:298.
`Altman DG, Bland JM. Quartiles, quintiles, centiles, and other quantiles.
`05M
`BM] 1994;309:996.
`4 Olsen CH. Review of the use of smtistics in Infection and Immunity. Infect
`Immun 2003;71:6689-92.
`
`903
`
`InnoPharma Exhibit 1093.0001
`
`