Standard deviations and standard errors
BMJ 2005; 331 doi: https://doi.org/10.1136/bmj.331.7521.903 (Published 13 October 2005) Cite this as: BMJ 2005;331:903
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Altman and Bland state that "about 95% of observations of any
distribution usually fall within the 2 standard deviation limits".
However, using the Chebyshev Inequality, we know that only 75% (or more)
of observations fall between these limits. To be sure of 95% or more of
the observations we need 4.5 standard deviations.
In line with the attempt to explain difficult statistics in a simple
way, as the authors always do in an excellent manner, I see no problem
with the following sentence. "If you don't know what your distribution
really looks like, whether it is unimodal or not, symmetric or not, ...
you can use the following rule: there is always at least 1-(1/k square)
percent of observations between k standard deviation limits. Thus for
values of k = 2, 3 and 4 we know that at least 1 - 1/4 = 75%, 1 - 1/9 =
89% and 1 - 1/16 = 93% of observations respectively, can be found in the
area between 2, 3 and 4 standard deviation limits."
Competing interests:
None declared
Competing interests: No competing interests
Doug Altman and Martin Bland’s clear and useful description of the
difference between a standard deviation and a standard error nevertheless
omits one important point. The formula for deriving the standard error of
a mean as the standard deviation divided by the square root of the sample
size is only valid if the observations are independently sampled from a
population of interest. Now it is commonly the case in medical
applications that to the extent that the observations are independently
sampled from a population this is not the population of interest and to
the extent that they are sampled from a population of interest they are
not independent1-3.
Consider for simplicity a small clinical trial in asthma in a single
centre. If we regard the centre as being representative of the population
of centres treating asthmatics, then the patients are not independently
sampled from this population. On the other hand, the patients might be
regarded as independent observations of some hypothetical population of
patients attending this centre but that is of little interest.
It turns out, however, that the standard error of a treatment effect,
the difference between two means, can be reasonably estimated for a
randomised clinical trial and given a useful causal interpretation, even
if the standard errors of the individual group means could not1 3. Thus,
for clinical trials I would go even further than Altman and Bland. Only
standard deviations should be produced for individual groups and in my
view to quote a standard error of a mean for a clinical trial is, in fact,
an error, albeit a standard one.
References
1. Senn SJ, Auclair P. The graphical representation of clinical
trials with particular reference to measurements over time [published
erratum appears in Statistics in Medicine 1991 Mar;10(3):487]. Statistics
in Medicine 1990;9(11):1287-302.
2. Ludbrook J, Dudley H. Issues in Biomedical Statistics - Statistical-
Inference. Australian and New Zealand Journal of Surgery 1994;64(9):630-
636.
3. Senn SJ. Statistical Issues in Drug Development. Chichester: John
Wiley, 1997.
Competing interests:
My academic career is furthered by publication. I consult regularly for the pharmaceutical industry.
Competing interests: No competing interests
Standard error refers to standard error of mean
The term standard error(SE) refers to standard error of mean(SEM)
unless specified otherwise. Those who understand the meaning of the term
standard deviation(SD) may see SE as the SD of sample mean. In even more
simple words it may be statd that SD measures the variation in the
observations of a given sample wheareas SE measures the variation in the
means of possible samples. SD comes under descriptive statistics while SE
clearly belongs to inferential statistics.
Competing interests:
None declared
Competing interests: No competing interests