21
Chapter 2
Truth and Central
Tendency
We have seen three statistical expressions for central tendency: mean, median, and
mode. Mean is the arithmetic average of all observations. Each data point con-
tributes to the mean. Median is the middle value of the data array when data are
arranged in an order—either increasing order or decreasing order. It is the value
of a middle position of the ordered array and does not enjoy contribution from all
observations as the mean does. Mode is the most often repeated value. e three
are equal for symmetrical distributions such as the normal distribution. In fact,
equality of the three values can be used to test if the data are skewed or not. Skew
is proportional to the difference between mean and mode.
Mean
Use of mean as the central tendency of data is most common. e mean is the true
value while making repeated measurements of an entity. e way to obtain truth
is to repeat the observation several times and take the mean value. e influence
of random errors in the observations cancel out, and the true value appears as the
mean. e central tendency mean is used in normal distribution to represent data,
even if it was an approximation. Mean is the basis for normal distribution; it is
one of the two parameters of normal distribution (the other parameter is standard
deviation). One would expect the mean value of project variance data such as effort
variance, schedule variance, and size variance to reveal the true error in estimation.
22 ◾ Simple Statistical Methods for Software Engineering
Once the true error is found out, the estimation can be calibrated as a measurement
process.
It is customary to take a sample data and consider the mean of the sample as
the true observation. It makes no statistical sense to judge based on a single obser-
vation. We need to think with “sample mean” and not with stray single points.
“Sample mean” is more reliable than any individual observation. “Sample mean”
dominates statistical analysis.
Uncertainty in Mean: Standard Error
e term “sample mean” must be seen with more care; it simply refers to the mean
of observed data. Say we collect data about effort variance from several releases
in a development project. ese data form a sample from which we can compute
the mean effort variance in the project. Individual effort variance data are used to
measure and control events; sample mean is used to measure and control central
capability. Central tendency is used to judge process capability.
Now the Software Engineering Process Group (SEPG) would be interested in
estimating process capability from an organizational perspective. ey can collect
sample means from several projects and construct a grand mean. We can call the
grand mean by another term, the population mean. Here population refers to the
collective experience of all projects in the organization. e population mean rep-
resents the true capability of organization.
If we go back to the usage of the term truth, we find there are several discoveries
of truth; each project discovers effort variance using sample mean. e organiza-
tion discovers truth from population mean.
Now we can estimate the population mean (the central tendency of the organi-
zational process) from the sample mean from one project (the central tendency of
the local process). We cannot pinpoint the population mean, but we can fix a band
of values where population mean may reside. ere is an uncertainty associated
with this estimation. It is customary to define this uncertainty by a statistic called
standard error. Let us look further into this concept.
It is known that the mean values gathered from different projects—the sample
means—vary according to the normal distribution. e theorem that propounds
this is known as the central limit theorem. e standard deviation of this normal
distribution is known as the standard error.
If we have just collected sample data from one project with n data points, and
with a standard deviation s, then we can estimate standard error with reasonable
accuracy using the relation
Truth and Central Tendency ◾ 23
SE =
s
n
Defining an uncertainty interval for mean is further explained in Chapter 25.
Median
e physical median divides a highway into two, and the statistical median divides
data into two halves. One half of the data have values greater than the median. e
other half of the data have values smaller than the median. It is a rule of thumb that
if data are nonnormal, use median as the central tendency. If data are normally dis-
tributed, median is equal to mean in any case. Hence, median is a robust expression
of the central tendency, true for all kinds of data. For example, customer satisfac-
tion data—known as CSAT data—are usually obtained in an ordinal scale known
as the Likert scale. One should not take the mean value of CSAT data; median is
the right choice. (It is a commonly made mistake to take the mean of CSAT data.)
In fact, only median is a relevant expression of central tendency for all subjective
data. Median is a truer expression of central tendency than mean in engineering
data, such as data obtained from measurements of software complexity, productiv-
ity, and defect density.
While the mean is used in the design of normal distribution, the median is
used in the design of skewed distributions such as the Weibull distribution.
Median value is used to develop the scale parameter that controls width.
Box 2.1 Hanging a Beam
ink of mean as a center of gravity. In Figure 2.1, the center of gravity
coincides with the geometric center, which is analogous to the median of the
beam, and as a result, the beam achieves equilibrium. In Figure 2.2, the cen-
ter of gravity shifts because of asymmetrical load distribution; the beam tilts
in the direction of center of gravity. e median, however, is still the same
old point. e distance between median and center of gravity is like the dif-
ference between median and mean. Such a difference makes the beam tilt; in
the case of a data array, the difference between median and mean is a signal
of data “skew” or asymmetry.
24 ◾ Simple Statistical Methods for Software Engineering
Geometric middle point
(analogous to median)
Center of gravity
(analogous to mean)
e uniform beam balances at the middle
point. e center of gravity (analogous to
mean) and the middle point (analogous to
median) coincide.
Figure 2.1 Geometric middle point and center of gravity coincides and the
beam is balanced.
Geometric middle point
(analogous to median)
Center of gravity
(analogous to mean)
e asymmetrically loaded beam tilts. is is
analogous to data skew.
Rider upsets
balance of the
beam
Figure 2.2 Asymmetry is introduced by additional weight on the rightside
of the beam. The mean shifts to the right.
Truth and Central Tendency ◾ 25
Mode
Mode, the most often repeated value in data, appears as the peak in the data dis-
tribution. Certain judgments are best made with mode. e arrival time of an
employee varies, and the arrival data are skewed as indicated in the three expres-
sions of central tendency: mean = 10:00 a.m., confidence interval of the mean =
10:00 a.m. ± 20 minutes, median = 9:30 a.m., and mode = 9:00 a.m. e expected
arrival time is 9:00 a.m. Let us answer the question, is the employee on time?
e question presumes that we have already decided not to bother with individual
arrival data but wish to respond to the central tendency. Extreme values are not
counted in the judgment. We choose the mode for some good reasons. Mean is
biased by extremely late arrivals. Median is insensitive to best performances. Mode
is more appropriate in this case.
Geometric Mean
When the data are positive, as is the case with bug repair time, we have a more
rigorous way of avoiding the influence of extreme values. We can use the concept
of geometric mean.
e geometric mean of n numbers is the nth root of the product of the n num-
bers, that is,
GM = x x x
n
n
1 2
Box 2.2 a RoBust RefeRence
Median is a robust reference that can serve as a baseline much better than
mean serves. If we wish to monitor a process, say test effectiveness, first we
need to establish a baseline value that is fair. Median value is a fair central line
of the process, although many tend to use mean. Mean is already influenced
by extreme values and is “prejudiced.” Median reflects true performance of
the process. Untrimmed mean reflects the exact location of the process with-
out any discrimination. Median effectively filters away prejudices and offers a
fair and robust judgment of process tendency. For example, the median score
of a class in a given subject is the true performance of the class, and the mean
score does not reflect the true performance.
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