Measures of Central Tendency in
Rehabilitation Research
What Do They Mean?
Gonzales VA, Ottenbacher KJ: Measures of central tendency in rehabilitation research: what do they mean? Am J Phys Med Rehabil
2001;80:141–146.
ABSTRACT
Measures of central tendency including the mean, median, and mode
are commonly reported in rehabilitation research. It is believed that the
relationship among the mean, median, and mode changes in a specific
way when the distribution being analyzed is skewed. A number of
widely used textbooks were reviewed to determine how the relationship among the mean, median, and mode is presented in the health
sciences and rehabilitation literature. We report a potential misinterpretation of the relationship between measures of central tendency that
was identified in several research and statistical textbooks on the subject of rehabilitation. The misinterpretation involves measures of central
tendency derived from skewed unimodal sample distributions. The reviewed textbooks state or imply that in asymmetrical distributions, the
median is always located between the mode and mean. An example is
presented illustrating the fallacy of this assumption. The mean and
median will always be to the right of the mode in a positively skewed
unimodal distribution and to the left of the mode in a negatively skewed
distribution; the order of the mean and median is impossible to predict
or generalize. The assumption that the median always falls between the
mode and mean in the calculation of coefficients of skewness has
implications for the interpretation of exploratory and confirmatory data
analysis in rehabilitation research.
Key Words: Skewness, Measurement, Statistics, Averages
February 2001 Measures of Central Tendency 141
Research Series Article
Measurement
A description of the mean, median, and mode is included in every
biomedical or health sciences research textbook presenting elementary data analysis procedures. In a
review documenting the use of various statistical methods reported in
the medical rehabilitation research
literature, Schwartz and colleagues1
stated, “Descriptive statistics were so
frequently encountered that, although they were considered a statistical method, their incidence was not
separately tabulated.” Other investigators2 have reported that the mean,
standard deviation, and related measures of central tendency are the
commonest statistics reported in the
biomedical literature. Unfortunately,
some descriptions of central tendency associated with data that are
not normally distributed are misleading and may contribute to statistical
misinterpretations. These misinterpretations exist at two different levels
of analysis, namely, exploratory or
confirmatory.
In describing data analysis procedures, Tukey3 distinguished between exploratory and confirmatory
data analysis. Exploratory data analysis (EDA) uses descriptive procedures
and pattern recognition to examine
the characteristics of data. In contrast, confirmatory data analysis relies on statistical hypothesis testing
to evaluate quantitative data.4–6 Discussion of data analytic issues in rehabilitation and health sciences research often focuses on statistical
inference and confirmatory analysis
rather than description and exploratory analysis.5 Hoaglin, Mosteller,
Tukey,4 and others7 have argued that
one result of this emphasis on confirmatory data analysis is that clinical
investigators know P values and F
ratios, but often they do not understand or ignore patterns and basic
relationships in data.
Tukey3 contended that EDA
should be a component of all research studies involving quantitative
data. Exploratory analysis and descriptive examination should be the
first step in any rehabilitation research or evaluation study.7 Royeen
and Geiger7 suggested that EDA be
used in clinical research and evaluation as a “preliminary step to determine what the data reveal” (p. 72).
They suggested that “subsequent to
exploratory data analysis, confirmatory or inferential statistical procedures can be selected based on the
findings of the exploratory data analysis.”7 Sinacore, Chang, and Falconer8 provided an excellent example of
EDA in healthcare program evaluation. Their illustration demonstrates
how the use of EDA can reveal important information relevant to program effectiveness that might be
overlooked if only conventional confirmatory statistical methods are
used.
EDA frequently involves examination of various measures of central
tendency and the distributional characteristics of symmetry and skewness.9 Measures of central tendency
also have important implications regarding the development of accurate
inferences in confirmatory data
analysis.8
Measures of Central Tendency
The computational characteristics of the mean, median, and mode
are familiar to researchers in medical
rehabilitation. They will not be described here. The potential error involving measures of central tendency
occurs when the mean, median, and
mode are derived from skewed unimodal sample distributions. The error involves a misinterpretation regarding relative placement or order
of the three measures of central tendency in a skewed distribution. This
error is contained in numerous introductory statistics texts. For example,
in a text titled “Basic Statistics for
the Health Sciences,” Kuzma10 stated
that: In a symmetrical distribution
the three measures of central tendency are identical. In an asymmetrical (Fig. 4.2) distribution the mode
remains located (by definition) at the
peak; the mean is off to the right; and
the median is in-between. Leftskewed distributions are the same,
but in a mirror-image (p. 38).
Graphs illustrating this relationship are presented in Figure 1.
This presentation is commonly
found in rehabilitation and health
sciences research textbooks either in
a narrative or graphic format. The
implication is that in a skewed, unimodal sample distribution, the mean
is always pulled furthest toward the
tail of the distribution and the median is located between the mode and
mean. In such cases, the authors argued that the median is the most
appropriate (accurate) statistic to use
for descriptive purposes. For examFigure 1. Measures of central tendency in symmetrical and asymmetrical distribution.
142 Gonzales and Ottenbacher Am. J. Phys. Med. Rehabil. ● Vol. 80, No. 2
ple, Vercruyssen and Hendrick11
stated, “when a distribution is badly
skewed the mean is pulled toward the
tail and is always higher (or lower)
than the median and the mode” (p. 58).
They further stated that in cases of
skewed data, the median is the preferred measure of central tendency.
In their text, “Essentials of Biostatistics,” Elston and Johnson12 presented graphs similar to those in Figure 1 (see p. 53) and stated that in
“symmetric unimodal distributions
the mean, the median and the mode
are all equal. In a unimodal asymmetric distribution the median always
lies between the mean and the mode”
(p. 52). Portney and Watkins,13 in
their widely used text “Foundations
of Clinical Research: Applications to
Practice,” also provided graphs similar to those in Figure 1 and stated,
“The median will always fall between
the mode and the mean in a skewed
curve, and the mean is pulled toward
the tail” (p. 322).
Although it is true that the median may lie between the mode and
mean in many cases when the data
are skewed, it is not true for all
skewed unimodal distributions. The
relative order of the mean and median for unimodal, skewed sample
distributions is not predictable in the
absolute manner presented in many
health and behavioral sciences textbooks. As an illustration of the variation in distributional placement of
the median and mean, consider the
following numbers 18, 20, 22, 24, 26,
29, 35, 35, and 39. The descriptive
data and graphic distribution of the
nine numbers are presented in Figure 2. Examination of Figure 2 reveals that the distribution is negatively skewed. The relative order of
the three measures of central tendency is not in agreement with the
narrative descriptions and graphic
distributions commonly included in
textbooks presenting basic statistical
procedures.7, 10–16
The error might be considered
trivial if it were not so widespread
and contrary to the spirit of EDA.
Correction of the error involves a
simple qualification of the interpretation presented above. For skewed,
unimodal sample distributions, the
mean and median will always be to
the right of the mode in a positively
skewed distribution, and to the left of
the mode in a negatively skewed distribution. The relative position of the
mean and median, however, is impossible to predict or generalize. Although it is usually assumed that the
median will fall between the mode
and mean in skewed distributions,
there is no empiric evidence to support the assumption that this is always true, or even true in a majority
of cases in the medical rehabilitation
research literature.8
Implications
The assumption that the mean
will always be the extreme value in a
skewed, unimodal sample distribution has practical implications in the
mathematical calculation of frequently used coefficients of skewness.
Coefficients of skewness and other
descriptive characteristics of a distribution are often referred to as “moments” of the distribution. Moments
are simply the expectations of different powers of the random variable.17
For example, the first moment about
the origin of a random variable X is
the mean. The second moment about
the mean is the variance and the
third moment about the mean is the
skewness. The third moment will be
zero for a symmetric distribution,
negative for skewness to the left, and
positive for skewness to the right.
The statistical characteristics of
skewness (third moments) were extensively described by Pearson18–20 in
a series of articles published .70 yrs
ago. Several formulas exist to compute third moment (skewness) coefficients. These formulas are based on
the mathematical description presented by Pearson. The formula most
commonly used in current statistical
software packages to compute skewness is shown below:
Skewness (Sk)
5 1/n O
n
i51(xi 2 m)
3
s3 (1)
This formula involves computing
the average of the cubed deviations
from the mean and then dividing the
average by the cube of the standard
deviation. By cubing the deviations
the signs are preserved providing an
indication of whether the distribution is negatively or positively
skewed.
Two other formulas used to compute coefficients of skewness (Sk),
originally proposed by Pearson,18–20
are presented below:
Sk 5 3~m 2 median)
s (2)
Sk 5 m 2 mode
s (3)
In these formulas, Sk equals the
coefficient of skewness, m is population mean (or its estimate), and s is
the population standard deviation (or
its estimate).
When the data from the example
presented in Figure 2 are analyzed
using Equation 1, the result is: 90.04/
340.06 5 10.26. This value suggests
a positive coefficient of skewness, although the distribution is negatively
skewed. When the second equation is
used, the result is: [3(27.55 2 26]/
6.98 5 10.67. When the data in FigFigure 2. Example of data and relative placement of mode, median, and
mean.
February 2001 Measures of Central Tendency 143
ure 2 are analyzed using the third
equation, the coefficient of skewness
is (27.55 2 35)/6.98 5 21.07. This
value accurately reflects the negatively skewed characteristic of the
sample distribution.
The third equation provides a
more accurate coefficient of skewness
for the data presented in Figure 2
because it does not depend on the
relative order of the mean and median. Rather, the third equation is
sensitive to which side of the mode
the mean is located on. Unfortunately, the third equation is infrequently used or recommended in basic statistics textbooks or statistical
software packages because of the
problem encountered when the data
are from a bimodal distribution.
It is often recommended that the
median be reported as the preferred
measure of central tendency if the
data are from a non-normal (skewed)
distribution. The recommendation is
based on the assumption that the median will always be a value between
the mode and mean. As Figure 2 illustrates, this is not always the case.
Determining which measure of
central tendency is most appropriate
for describing a distribution depends
on several factors. The scale of measurement of the variable is an important consideration. All three measures of central tendency can be
applied to variables on the interval or
ratio scales. For data on the nominal
scale, only the mode is meaningful. If
data are ordinal, both the median and
mode can be applied. It is also necessary to consider how the summary
measure will be used statistically. The
mean is considered the most stable of
the three measures of central tendency. If we were to repeatedly draw
random samples from a population,
the means of those samples would
fluctuate less than the mode or median. Each of the measures of central
tendency is, in its way, a best guess
about any score, but the sense of
“best” differs with the way error is
regarded. If both the size of the errors
and their signs are considered important, and the investigator wants zero
error in the long run, then the mean
serves as the best “guess.” If the investigator wants to be exactly right as
often as possible, then the mode is
indicated. If, on the other hand, the
researcher wants to come as close as
possible on the average, irrespective
of sign or error, then the median is
the best guess. From the point of
view of purely descriptive statistics,
as apart from inferential work, the
median is a most serviceable measure. Its property of representing the
typical (most nearly like) score makes
it fit the requirements of simple and
effective communication better than
the mean in many contexts.
Rehabilitation researchers dealing with skewed data should compute
all three measures of central tendency to determine which will be the
most representative for a given set of
numbers. Portney and Watkins13 noted: “The choice of which index to
report with skewed distributions depends on what facet of information is
appropriate to the analysis. It is often
reasonable to report all three values,
to present a complete picture of a
distribution’s characteristics.” The
techniques of EDA should also be
used to “present a complete picture
of a distribution’s characteristics”
when the data are skewed. Tukey3
and others4 pioneered the use of EDA
methods that provide detailed information regarding data patterns and
characteristics.
Box plots. In the box plot, the upper
and lower quartiles of the data are
portrayed by the top and bottom of a
rectangle and the median portrayed
by a horizontal line segment within
the rectangle (Fig. 3). The mean can
be designated as a plus (1) in the
rectangle. The lines from the rectangle, referred to as “whiskers,” can extend to represent the minimum and
maximum values or the 10th and
90th percentiles (Fig. 3). Outliers
may be identified using a defined outlier detection rule and displayed as
asterisks or other symbol. Advantages
of the box plot include the following:
(1) the central rectangle includes
50% of the data, (2) the whiskers
show the range of data, and (3) symmetry is indicated by the box and
whisker relationship and location of
the mean and median. It is easy to
compare distributions of more than
one sample by constructing side-byside box plots. Two disadvantages are
that box plots do not depict the mode
and do not include raw values.
Stem-and-leaf plots. The stem-andleaf plot is a hybrid between a table
and a graph because it shows numerical values, but also presents a profile
of the data distribution. Figure 4 depicts a stem-and-leaf plot of effect
size values (d-indexes) computed
from a meta-analysis examining the
effectiveness of medical rehabilitation in persons with stroke.21 The
numbers in the vertical column (on
the left) represent the “stem” in Figure 4 and the numbers in the row to
the right represent the “leaf” values.
Each stem-and-leaf combination indicates a number in the data set. For
example, for the stem 0.9 (left column), the following “leaf” values are
found in the row to the right, 0119.
These stem-and-leaf values represent
the following four numbers, 90, 91,
91, and 99. Stem-and-leaf plots can
include the raw data values and reFigure 3. Sample box plot showing
distribution and descriptive statistics.
144 Gonzales and Ottenbacher Am. J. Phys. Med. Rehabil. ● Vol. 80, No. 2
lated descriptive statistics, that is,
mean, median mode (Fig. 4). In addition, the shape of the distribution is
“graphically” displayed in a stemand-leaf plot. Depending on the size
of the data set, it may be necessary to
change the measurement units by
multiplying by some power of 10 or
to truncate the data (that is, to ignore
some digits on the right) to get values suitable for the stem-and-leaf display. There are also methods to transform the data, for example, by taking
logarithms. The rules for making the
diagram can be modified if one finds
that some other variation does a better job for a particular data set. For
example, each leaf can be two digits
rather than one, with a comma separating the leaves on a single row so
that 4 24,38,45 would represent 424,
438, and 445. Tukey3 and others4 provided detailed information and instructions regarding how to create
stem-and-leaf plots.
CONCLUSIONS
Accurate EDA relies on the ability to compute and interpret basic
descriptive statistics. Any error or
misinterpretation at this fundamental level of data analysis may lead to
contradictory results and contribute
to a form of statistical confusion that
Cook and Campbell22 referred to as
statistical conclusion invalidity. In a
discussion of data analysis procedures in medical rehabilitation research, Findley23 advised clinical researchers that “you cannot bypass the
data description phase of analysis” (p.
92). Findley23 noted that “you need to
be sure that your choice of central
value and dispersion truly represent
your data. If you decide to move to
more analyses, such as testing differences between groups, then it is even
more important that the values for
each group that you subject to statistical tests truly represent that group”
(p. 92). If basic statistical values such
as measures of central tendency do
not reflect the intended characteristics, the potential for statistical misinterpretation is increased.
Frequently, errors associated
with statistical conclusion validity in
medical rehabilitation research are
impossible to eliminate. For example,
problems associated with low statistical power and type 2 errors are often difficult to resolve in clinical or
field based studies with relatively
small sample sizes. Obstacles associated with the interpretation of basic
descriptive statistics, such as measures of central tendency, can be resolved more directly if authors and
rehabilitation researchers follow the
logic of sound quantitative reasoning
and treat descriptive and exploratory
statistics with the attention and respect they deserve.
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146 Gonzales and Ottenbacher Am. J. Phys. Med. Rehabi