Statistical Round
`
`pISSN 2005—6419 - eISSN 2005—7563
`
`dummmm
`
`Korean Journal ofAnesthesiology
`
`error of the mean
`
`Standard deviation and standard
`
`Dong Kyu Leel, Iunyong Inz, and Sangseok Lee3
`
`Department ofAnesthesiology and Pain Medicine, IKorea University Guro Hospital, Seoul, ZDongguk University
`Ilsan Medical Center, Goyang, 3Sanggye Paik Hospital, Inje University College ofMedicine, Seoul, Korea
`
`In most clinical and experimental studies, the standard deviation (SD) and the estimated standard error of the mean (SEM)
`are used to present the characteristics of sample data and to explain statistical analysis results. However, some authors oc—
`casionally muddle the distinctive usage between the SD and SEM in medical literature. Because the process of calculating
`the SD and SEM includes different statistical inferences, each of them has its own meaning. SD is the dispersion of data in
`a normal distribution. In other words, SD indicates how accurately the mean represents sample data. However the mean—
`ing of SEM includes statistical inference based on the sampling distribution. SEM is the SD of the theoretical distribution
`of the sample means (the sampling distribution). While either SD or SEM can be applied to describe data and statistical re—
`sults, one should be aware of reasonable methods with which to use SD and SEM. We aim to elucidate the distinctions be—
`
`tween SD and SEM and to provide proper usage guidelines for both, which summarize data and describe statistical results.
`
`Key Words: Standard deviation, Standard error of the mean
`
`propose a standard against which statistical analysis results in
`medial literatures can be evaluated.
`
`Medical studies begin by establishing a hypothesis about a
`population and extracting a sample from the population to test
`the hypothesis. The extracted sample will take a normal distribu—
`tion if the sampling process was conducted Via an appropriate
`randomization method with a sufficient sample size. As with all
`normally distributed data, the characteristics of the sample are
`represented by the mean, variance or SD. The variance or SD
`includes the differences of the observed values from the mean
`
`(Fig. 1); thus, these values represent the variation of the data
`[1—3]. For instance, if the observed values are scattered closely
`around the mean value, the variance — as well as the SD — are
`
`reduced. However, the variance can confuse the interpretation
`of the data because it is computed by squaring the units of the
`observed values. Hence, the SD, which uses the same units used
`
`with the mean, is more appropriate [3] (Equations 1 and 2).
`Sample: x1, x2, x3, ..., xn (sample size = n)
`Tl.
`
`. .. Equation 1
`.
`Mean (9?) = :1 xi
`7.1
`*_ .
`7'1
`’— ' 2
`"—1
`n—l
`Variance = M, Standard Deviation (SD) = M
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`.
`
`A data is said to follow a normal distribution when the values
`
`of the data are dispersed evenly around one representative value.
`A normal distribution is a prerequisite for a parametric statistical
`analysis [1]. The mean in a normally distributed data represents
`the central tendency of the values of the data. However, the mean
`alone is not sufficient when attempting to explain the shape of
`the distribution; therefore, many medical literatures employ the
`standard deviation (SD) and the standard error of the mean (SEM)
`along with the mean to report statistical analysis results [2].
`The objective of this article is to state the differences with
`regard to the use of the SD and SEM, which are used in descrip—
`tive and statistical analysis of normally distributed data, and to
`
`Corresponding author: Dong Kyu Lee, M.D., Ph.D.
`Department of Anesthesiology and Pain Medicine, Korea University
`Guro Hospital, 148, Gurodong-ro, Guro-gu, Seoul 152-703, Korea
`Tel: 82-2-2626-3237, Fax: 82-2-2626-1437
`E-mail: entopic@naver.com
`ORCID: http://orcid.org/0000-0002-4068-2363
`
`Received: April 2, 2015.
`Revised: 1st, April 21, 2015; 2nd, April 24, 2015; 3rd, May 6, 2015.
`Accepted: May 7, 2015.
`
`Korean J Anesthesiol 2015 June 68(3): 220-223
`http://dx.doi.org/10.4097/kjae.2015.68.3220
`
`.
`
`.
`
`. .
`
`.
`
`. .
`
`.
`
`. .
`
`.
`
`. .
`
`.
`
`. .
`
`.
`
`. .
`
`.
`
`. .
`
`.
`
`. .
`
`.
`
`. .
`
`.
`
`. .
`
`.
`
`. .
`
`.
`
`. Equation 2
`
`(-9 This is an open—access article distributed under the terms of the Creative Commons Attribution Non—Commercial License (http://Creativecommons.org/
`licenses/by—nc/4.0/), which permits unrestricted non—commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
`
`Copyright (Q the Korean Society of Anesthesiologists, 201 5
`
`Online access in http://ekja.org
`
`InnoPharma Exhibit 1092.0001
`
`

`

`KOREA JANEST-IESIOL
`
`Leeetal.
`
`
`
`A
`B
`C
`D
`E
`F
`G
`
`
`1
`X
`X
`X
`X
`X
`'--
`X
`
`
`2
`X
`X
`X
`X
`X
`'-'
`X
`
`
`3
`X
`X
`X
`X
`X
`'-'
`X
`
`
`
`
`
`
`
`
`
`
`
`
`4
`
`5
`
`X
`
`X
`
`X
`
`...
`
`...
`
`X
`
`X
`
`X
`
`x
`
`X
`
`
`
`
`
`
`A
`
`B
`
`C
`
`Frequency
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`9(-
`Meanfi):Ln
`
`..
`Standard devnatlon (SD) =
`
`Scale
`
`
`
`X
`x
`
`XXX
`
`
`
`XXXX
`
`
`
`XXXXX
`
`
`
`XXXXXXXXX
`
`
`XXXXXXXXXXX
`
`X XXXXXXXXXXXXXX
`
`
`
`XXXXXXXXXXXXXXXXXXXX
`
`
`
`
`XXXXXXXXXXXXXXXXXXXXXXX
`
`
`
`
`XXXXXXXXXXXXXXXXXXXXXXXXX
`
`
`
`
`
`
`
`V
`
`
`
`Fig. 1. Process of data description. First,
`we gather raw data from the population
`by means of randomization (A). We then
`arrange the each value according to the scale
`(frequency distribution); we can presume
`the shape of the distribution (probability
`distribution) and can calculate the mean and
`standard deviation (B). Using these mean and
`standard deviation, we produce a model of
`the normal distribution (C). This distribution
`represents the characteristics of the data we
`gathered and is the normal distribution, with
`which statistical inferences can be made (3?:
`mean, SD: standard deviation, xi: observation
`value, n: sample size).
`
`
`
` 95.4% of data|
` o0‘_______
`I|
`
`¢
`V
`O
`O
`x /(b% / IQ?
`*4
`
`4&5
`
`$99.7% of déta
`s
`(90
`
`/ x
`*4
`
`v
`(£00
`
`v
`0500
`
`fix
`
`Rx
`
`As mentioned previously, using the SD concurrently with the
`mean can more accurately estimate the variation in a normally
`distributed data. In other words, a normally distributed statisti—
`cal model can be achieved by examining the mean and the SD
`of the data [1] (Fig. 1, Equations 1 and 2). In such models, ap—
`proximately 68.7% of the observed values are placed within one
`SD from the mean, approximately 95.4% of the observed values
`are arranged within two SDs from the mean, and about 99.7%
`of the observed values are positioned within three SDs from the
`mean [1,4]. For this reason, most medical literatures report their
`samples in the form of the mean and SD [5].
`The sample as referred to in medical literature is a set of
`observed values from a population. An experiment must be
`
`conducted on the entire population to acquire a more accurate
`confirmation of a hypothesis, but it is essentially impossible to
`survey an entire population. As a result, an appropriate sampling
`process — a process of extracting a sample that represents the
`characteristics of a population — is essential to acquire reliable
`results. For this purpose, an appropriate sample size is deter—
`mined during the research planning stage and the sampling is
`done Via a randomization method. Nevertheless, the extracted
`
`sample is still a part of the population; thus, the sample mean is
`an estimated value of the population mean. When the samples
`of the same sample size are repeatedly and randomly taken from
`the same population, they are different each other because of
`sampling variation as well as sample means (Fig. 2, Level B). The
`
`Online access in http://ekja.org
`
`221
`
`InnoPharma Exhibit 1092.0002
`
`

`

`SD and SEM
`
`VOL. 68, NO. 3, JUNE 2015
`
`
`
`A
`
`Fig. 2. Process of statistical inference.
`Level A indicates the population. In most
`experiments, we only obtain one set of
`sample data from the population using
`randomization (Level B); the mean and
`standard deviation are calculated from
`
`
`
`The population
`
`
`
` /
`
`
`
`l
`
`
`we
`
`
`
`X1, SD1
`
`
`
`sample data we have. For statistical infe—
`rence purposes, we assume that there are
`several sample data sets from the population
`(Level B); the means of each sample data set
`produce the sampling distribution (Level C).
`Using this sampling distribution, statistical
`analysis can be conducted. In this situation,
`the estimated standard error of the mean
`or the 95% confidence interval has an
`
`c \
`
`
`
`Sam ' 9 distribution (X112,
`
`.. KN)
`
`Estimated Standard Error of the Mean (SEM) =
`
`a;
`95% Confidence Inverval :7 —(1.96 x SEM) ~7 + (1.96 x SEM)
`
`SD
`
`important role during the statistical analysis
`process (3c: mean, SD: standard deviation, n:
`sample size, N: number of sample data sets
`extracted from population).
`
`distribution of different sample means, which is achieved Via
`repeated sampling processes, is referred to as the sampling dis—
`tribution and it takes a normal distribution pattern (Fig. 2, Level
`C) [1,6,7]. Therefore, the SD of the sampling distribution can be
`computed; this value is referred to as the SEM [1,6,7]. The SEM
`is dependent on the variation in the population and the number
`of the extracted samples. A large variation in the population
`causes a large difference in the sample means, ultimately result—
`ing in a larger SEM. However, as more samples are extracted from
`the population, the sample means move closer to the population
`mean, which results in a smaller SEM. In short, the SEM is an in—
`
`dicator of how close the sample mean is to the population mean [7].
`In reality, however, only one sample is extracted from the popula—
`tion. Therefore, the SEM is estimated using the SD and a sample
`size (Estimated SEM). The SEM computed by a statistical program
`is an estimated value calculated Via this process [5] (Equation 3).
`
`Estimated Standard Error ofthe Mean (SEM) = % Equation 3
`
`A confidence interval is set to illustrate the population mean
`intuitively A 95% confidence interval is the most common [3,7].
`The SEM of a sampling distribution is estimated from one sam—
`ple, and a confidence interval is determined from the SEM (Fig. 2,
`Level C). In the strict sense, the 95% confidence interval provides
`the information about a range within which the 95% sample
`means will fall, it is not a range for the population mean with
`95% confidence. For example, a 95% confidence interval signifies
`that when 100 sample means are calculated from 100 samples
`
`from a population, 95 of them are included within the said confi—
`dence interval and 5 are placed outside of the confidence interval.
`In other words, it does not mean that there is a 95% probability
`that the population mean lies within the 95% confidence interval.
`When statistically comparing data sets, researchers estimate
`the population of each sample and examine whether they are
`identical. The SEM — not the SD, which represents the variation
`in the sample — is used to estimate the population mean (Fig. 2)
`[4,8,9]. Via this process, researchers conclude that the sample
`used in their studies appropriately represents the population
`within the error range specified by the pre—set significance level
`[4,6,8].
`The SEM is smaller than the SD, as the SEM is estimated usu—
`
`ally the SD divided by the square root of the sample size (Equa—
`tions 2 and 3). For this reason, researchers are tempted to use the
`SEM when describing their samples. It is acceptable to use either
`the SEM or SD to compare two different groups if the sample
`sizes of the two groups are equal; however, the sample size must
`be stated in order to deliver accurate information. For example,
`when a population has a large amount of variation, the SD of an
`extracted sample from this population must be large. However,
`the SEM will be small if the sample size is deliberately increased.
`In such cases, it would be easy to misinterpret the population
`from using the SEM in descriptive statistics. Such cases are com—
`mon in medical research, because the variables in medical re—
`
`search impose many possible biases originated from inter— and
`intra—individual variations originated from underlying general
`conditions of the patients and so on. When interpreting the SD
`
`222
`
`Online access in http://ekja.org
`
`InnoPharma Exhibit 1092.0003
`
`

`

`KOREAN J AN ESTH ESIOL
`
`Lee et al.
`
`and SEM, however, the exact meaning and purposes of the SD
`and SEM should be considered to deliver correct information.
`
`[3,4,6,7,10].
`
`We examined 36 clinical or experimental studies published
`in Volume 6, Numbers 1 through 6 of the Korean Journal ofAn—
`esthesiology and found that a few of the studies inappropriately
`used the SD and SEM. First, examining the descriptive statistics,
`we found that all of the studies used the mean and SD or the
`
`observed number and percentage. One study suggested a 95%
`confidence interval; this particular study appropriately stated
`the sample size along the confidence interval, offering a clearer
`understanding of the data suggested in the study [11]. Among
`the 36 studies examined, only one study described the results
`of a normality test [12]. Second, all 36 studies used the SD, the
`observed number or the percentage to describe their statistical
`results. One study did not specify what the values in the graphs
`and tables represent (i.e., the mean, SD or interqaartile range).
`There was also a study that used the mean in the text but showed
`an interqaartile range in the graphs. Sixteen studies used either
`the observed number or the percentage, and most of them re—
`ported their results without a confidence interval. Only two stud—
`ies stated confidence intervals, but only one of those two studies
`
`appropriately used the confidence interval [13]. As shown above,
`we found that some of the studies have inappropriately used the
`SD, SEM and confidence intervals in reporting their statistical
`results. Such instances of the inappropriate use of statistics must
`be meticulously screened during manuscript reviews and evalu—
`ations because they may hamper an accurate comprehension of
`a study’s data.
`In conclusion, the SD reflects the variation in a normally
`distributed data, and the SEM represents the variation in the
`sample means of a sampling distribution. With this in mind, it is
`pertinent to use the SD (paired with a normality test) to describe
`the characteristics of a sample; however, the SEM or confidence
`interval can be used for the same purpose if the sample size is
`specified. The SEM, paired with the sample size, is more useful
`when reporting statistical results because it allows an intuitive
`comparison between the estimated populations via graphs or
`tables.
`
`ORCID
`
`Iunyong ln, http://orcid.org/0000—0001—7403—4287
`Sangseok Lee, http://orcid.org/0000—0001—7023—3668
`
`References
`
`1. Curran-Everett D, Taylor S, Kafadar K. Fundamental concepts in statistics: elucidation and illustration. IAppl Physiol (1985) 1998; 85: 775-86.
`2. Curran-Everett D, Benos D]. Guidelines for reporting statistics in journals published by the American Physiological Society: the sequel. Adv
`Physiol Educ 2007; 31: 295-8.
`3. Altman DG, Bland IM. Standard deviations and standard errors. BM] 2005; 331: 903.
`4. Carlin IB, Doyle LW. Statistics for clinicians: 4: Basic concepts of statistical reasoning: hypothesis tests and the t-test. J Paediatr Child Health
`2001; 37: 72-7.
`
`5. Livingston EH. The mean and standard deviation: what does it all mean? J Surg Res 2004; 119: 117-23.
`6. Rosenbaum SH. Statistical methods in anesthesia. In: Miller’s Anesthesia. 8th ed. Edited by Miller RD, Cohen NH, Eriksson LI, Fleisher LA,
`Wiener-Kronish JP, Young WL: Philadelphia, Elsevier Inc. 2015, pp 3247-50.
`7. Curran-Everett D. Explorations in statistics: standard deviations and standard errors. Adv Physiol Educ 2008; 32: 203-8.
`8. Carley S, Lecky F. Statistical consideration for research. Emerg Med] 2003; 20: 258-62.
`9. Mahler DL. Elementary statistics for the anesthesiologist. Anesthesiology 1967; 28: 749-59.
`10. 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 Anaesth 2003; 90: 514-6.
`11. Koh M], Park SY, Park E], Park SH, Ieon HR, Kim MG, et al. The effect of education on decreasing the prevalence and severity of neck and
`shoulder pain: a longitudinal study in Korean male adolescents. Korean J Anesthesiol 2014; 67: 198-204.
`12. Kim HS, Lee DC, Lee MG, Son WR, Kim YB. Effect of pneumoperitoneum on the recovery from intense neuromuscular blockade by
`rocuronium in healthy patients undergoing laparoscopic surgery Korean J Anesthesiol 2014; 67: 20-5.
`13. Lee H, Shon Y], Kim H, Paik H, Park HP. Validation of the APACHE IV model and its comparison with the APACHE II, SAPS 3, and
`Korean SAPS 3 models for the prediction of hospital mortality in a Korean surgical intensive care unit. Korean J Anesthesiol 2014; 67: 115-
`22.
`
`Online access in http://ekja.org
`
`223
`
`InnoPharma Exhibit 1092.0004
`
`