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Standard Deviation, Standard Error. Which should never be summarized with the stan¬ Standard Deviation, dard error of the mean.3*"25™ A closer look at the source and mean¬ ing of SD and SE may clarify why Standard Error medical investigators, journal review¬ ers, and editors should scrutinize their usage with considerable care. Which 'Standard' Should We Use? DISPERSION An essential function of "descriptive George W. Brown, MD statistics" is the presentation of con¬ densed, shorthand symbols that epito¬ mize the important features of a collec¬ \s=b\Standard deviation (SD) and standard shorthand expression" in 1968; Fein- tion of data. The idea of a central value error (SE) are quietly but extensively used stein2 later again warned about the is intuitively satisfactory to anyone who biomedical These terms in publications. fatuity and confusion contained in any needs to summarize a group of measure¬ and notations are used as sta- descriptive a ± b statements where b is not defined. ments, or counts. The traditional indica¬ tistics (summarizing numerical data), and Warnings notwithstanding, a glance tors of a central are the mode they are used as inferential statistics (esti- tendency almost medical will most the median mating population parameters from sam- through any journal (the frequent value), of this value between the lowest ples). I review the use and misuse of SD show examples usage. (the midway and SE in several authoritative medical Medical journals seldom state why and the highest value), and the mean journals and make suggestions to help SD or SE is selected to summarize data (the average). Each has its special uses, clarify the usage and meaning of SD and in a given report. A search of the three but the mean has great convenience and SE in biomedical reports. major pediatrie journals for 1981 (Amer¬ flexibility for many purposes. (Am J Dis Child 1982;136:937-941) ican Journal of Diseases of Children, The dispersion of a collection of values Journal of Pediatrics, and Pediatrics) can be shown in several ways; some are deviation (SD) and stan¬ failed to turn up a single article in which simple and concise, and others are com¬ Standarddard error (SE) have surface simi¬ the selection of SD or SE was explained. plex and esoteric. The range is a simple, larities; yet, they are conceptually so There seems to be no uniformity in the direct way to indicate the spread of a different that we must wonder why they use of SD or SE in these journals or in collection of values, but it does not tell are used almost interchangeably in the The Journal of the American Medical how the values are distributed. Knowl¬ medical literature. Both are usually Association (JAMA), the New England edge of the mean adds considerably to preceded by a plus-minus symbol (±), Journal of Medicine, or Science. The the information carried by the range. suggesting that they define a sym¬ use of SD and SE in the journals will be Another index of dispersion is pro¬ metric interval or range of some sort. discussed further. vided by the differences (deviations) of They both appear almost always with a If these respected, well-edited jour¬ each value from the mean of the values. mean (average) of a set of measure¬ nals do not demand consistent use of The trouble with this approach is that ments or counts of something. The med¬ either SD or SE, are there really any some deviations will be positive, and ical literature is replete with statements important differences between them? some will be negative, and their sum like, "The serum cholesterol measure¬ Yes, they are remarkably different, will be zero. We could ignore the sign of ments were distributed with a mean of despite their superficial similarities. each deviation, ie, use the "absolute 180±30 mg/dL (SD)." They are so different in fact that some mean deviation," but mathematicians In the same journal, perhaps in the authorities have recommended that SE tell us that working with absolute num¬ same article, a different statement may should rarely or never be used to sum¬ bers is extremely difficult and fraught appear: "The weight gains of the sub¬ marize medical research data. Fein- with technical disadvantages. jects averaged 720 (mean) ±32 g/mo stein2 noted the following: A neglected method for summarizing the of data is the calculation (SE)." Sometimes, as discussed further, A standard error has nothing to do with dispersion the summary data are presented as the standards, with errors, or with the commu¬ of percentiles (or deciles, or quartiles). "mean of 120 mg/dL ±12" without the nication of scientific data. The concept is an Percentiles are used more frequently in "12" being defined as SD or SE, or as abstract idea, spawned by the imaginary pediatrics than in other branches of some other index of dispersion. Eisen¬ world of statistical inference and pertinent medicine, usually in growth charts or in hart1 warned against this "peril of only when certain operations of that imagi¬ other data arrays that are clearly not world are met in scientific nary reality.2(p336) symmetric or bell shaped. In the gen¬ Glantz3 also has made the following rec¬ eral medical literature, percentiles are From the Los Lunas Hospital and Training ommendation: because of a School, New Mexico, and the Department of Pedi- sparsely used, apparently atrics, University of New Mexico School of Medi- Most medical investigators summarize their common, but erroneous, assumption ± cine, Albuquerque. data with the standard error because it is that the mean SD or SE is satisfactory Reprint requests to Los Lunas Hospital and central and Training School, Box 1269, Los Lunas, NM 87031 always smaller than the standard deviation. for summarizing tendency (Dr Brown). It makes their data look better data dispersion of all sorts of data. STANDARD DEVIATION The generally accepted answer to the a for ( -µ)' - )7 need for concise expression the SD = dispersion of data is to square the differ¬ "V< ence of each value from the group mean, giving all positive values. When these SD of Population Estimate of Population SD From Sample squared deviations are added up and then divided by the number of values in µ = Mean of Population X = Mean of Sample the group, the result is the variance. = Number in Population = Number in Sample The variance is always a positive num¬ ber, but it is in different units than the mean. The way around this inconve¬ Fig 1.—Standard deviation (SD) of population is shown at left. Estimate of population SD derived from is shown at nience is to use the square root of the sample right. variance, which is the population stan¬ dard deviation ( ), which for conve¬ nience will be called SD. Thus, the SD is the square root of the averaged squared QT) - ) pq mean. The is = / ( deviations from the SD SD =_ = SEM SE sometimes called by the shorthand * s/a term, "root-mean-square." The SD, calculated in this way, is in the same as the values and units original SEM SE of Proportion the mean. The SD has additional prop¬ erties that make it attractive for sum¬ SD = Estimate of Population SD = Proportion Estimated From Sample if the = (1 marizing dispersion, especially = Sample Size q -P) data are distributed symmetrically = Sample Size in the revered bell-shaped, gaussian curve. Although there are an infinite Fig 2.—Standard error of mean (SEM) is shown at left. Note that SD is estimate of population SD number of gaussian curves, the one for (not , actual SD of population). Sample size used to calculate SEM is n. Standard error of the data at hand is described completely proportion is shown at right. by the mean and SD. For example, the mean+ 1.96 SD will enclose 95% of the values; the mean ±2.58 SD will enclose that no matter how many times the die determine the deviations is concep¬ 99% of the values. It is this symmetry is thrown, it will never show its aver¬ tualized as an estimate of the mean, x, and elegance that contribute to our age score of 3.5.) rather than as a true and exact popula¬ admiration of the gaussian curve. The SD wears two hats. So far, we tion mean (µ). Both means are calcu¬ The bad news, especially for biologic have looked at its role as a descriptive lated in the same way, but a population data, is that many collections of mea¬ statistic for measurements or counts mean, µ, stands for itself and is a pa¬ surements or counts are not sym¬ that are representative only of them¬ rameter; a sample mean, x, is an esti¬ metric or bell shaped. Biologic data selves, ie, the data being summarized mate of the mean of a larger population tend to be skewed or double humped, J are not a sample representing a larger and is a statistic. shaped, U shaped, or flat on top. Re¬ (and itself unmeasurable) universe or The second change in calculation is in gardless of the shape of the distribu¬ population. the arithmetic: the sum of the squared tion, it is still possible by rote arithme¬ The second hat involves the use of SD deviations from the (estimated) mean is tic to calculate an SD although it may from a random sample as an estimate of divided by -1, rather than by N. (This be inappropriate and misleading. the population standard deviation ( ). makes sense intuitively when we recall For example, one can imagine The formal statistical language says that a sample would not show as great a throwing a six-sided die several hun¬ that the sample statistic, SD, is an spread of values as the source popula¬ dred times and recording the score at unbiased estimate of a population pa¬ tion.
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