Open Journal of Statistics, 2012, 2, 213-218
http://dx.doi.org/10.4236/ojs.2012.22026 Published Online April 2012 (http://www.SciRP.org/journal/ojs) 213
Design of Cusum Scheme for M onitori ng Road
Accident Fatalities
Kayode Samuel Adekeye1*, Omololu Stephen Aluko2
1Department of Mathematical Sciences, Redeemer’s University, Redemption City, Nigeria
2Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria
Received December 9, 2011; revised January 12, 2012; accepted January 25, 2012
ABSTRACT
In recent years, road accident fatalities in Nigeria have continued to be on the increase. Thus, urgent attention is needed
to reduce or eliminate road acciden ts fatalities. To achieve this goal, the cumulative sum (Cusum) control chart scheme
was designed for monitoring the road accident fatalities using the recorded occurrence of road accident fatalities in a
state in the western part of Nigeria. The designed Cusum detects the period of the years when the highest occurrence of
road accident fatalities occurred. These periods were observed to be festive periods such as Christmas, Easter, Eid-el-
Kabir, and Eid-el-Moluod. Therefore, the festive periods of the year should be used as benchmark by road managers as
periods where more attention or precaution measure should be put in place on the roads to drastically reduce or elimi-
nate high occurrence of road accident fatalities. The designed Cusum control chart can be adapted for other states in the
country and also for the larger society for detecting the periods when the rate of death as a result of road accidents was
prevalent.
Keywords: Cusum; Fatalities; Monitoring; Prevalent; Ro ad Accident
1. Introduction
The problem of deaths and injury as a result of road ac-
cidents is now acknowledge to be a global phenomenon
with authorities in virtually all countries of the world
concerned about the growth in the number of people
killed and seriously injured on their roads. Fatality means,
a death that is caused in an accident or war or the fact
that a particular disease or accident will result in death”.
In recent years there have been two major studies of
causes of death worldwide which have been published in
the “Global Burden of Disease” (2008). This publication
show that road accidents as a cause of death or disability
were by no means insignificant as it is in the ninth place
out of a total of over 100 separately identified causes of
death [1].
In recent years, road accident fatalities in Nigeria have
continued to be on the increase. This study is motivated
by recent interest of Nigerian government to minimize
the road fatal accidents by 50% in year 2015. Thus, ur-
gent attention is needed to be given to our road transpor-
tation system in order to reduce or eliminate road acci-
dents fatalities and save lives.
In this paper, our focus is to develop a Cusum control
chart that will be able to assist the government to achiev e
the above stated goal. The cumulative sum (Cusum) con-
trol chart scheme was designed for monitoring the road
accident fatalities using the recorded occurrence of road
accident fatalities in a state in the western part of Nigeria.
The designed Cusum control chart can be adapted for
other states in the country and also for the larger society
for detecting the time point (month) of the year when the
rate of death as a result of road accidents was prevalent
(high occurrence) and the cumulative trend for road ac-
cident fatalities so that appropriate actions would be
taken to forestall re-occurrence. The designed scheme is
expected to serve as a useful tool for road managers in
their effort to reduce road accident fatalities.
2. The Cumulative Sum (Cusum) Control
Chart
A Cumulative Sum (Cusum) control chart is a graphical
representation of the trend in the outcome of a series of
consecutive procedures performed over time. The Cusum
control charts attempt to device a simple procedure that
will make use of all available information. The unique
and important feature of Cusum is its ability to indicate
the trend of deteriorating performance early, prompting
preventive and corrective measures. A major advantage
of this chart over the ordinary Shewart control chart is
*Corresponding a uthor.
C
opyright © 2012 SciRes. OJS

K. S. ADEKEYE ET AL.
214
that it is very effective in detecting relative small shifts.
They are more meaningful graphically, as process shifts
are often easy to detect and points of change can be eas-
ily located [2]. The Cusum control chart is a technique
for identifying whether a real problem has arisen and
provides a means of estimating the time at which the
problem arose. Such information assists in the identifica-
tion of management changes which might have caused a
problem or, alternatively, an improvement, in the attrib-
ute of interest. In effect, the Cusum chart functions as a
significance test in attempting to distinguish real effects
from sampling variance. It was introduced by Page [3],
which provided integral equations for approximately the
Average Run Length (ARL). Many authors like Lucas
[4], Ott et al. [5] and Montgomery [6] have worked on
the Cusum control chart. Lucas [4] gave a detailed pro-
cedure for designing a counted data Cusum chart based
on a poisson counts and implemented it on the occur-
rence of industrial accidents. Osanaiye and Talabi [7]
considered a versatile dimension on the application of
Cusum chart in a non-manufacturing sector and imple-
mented it on the cases of diabetes patients. Adekeye [8]
applied the concept of cusu m control chart to monitor the
crime rate in Nigeria. Nix, Rowland and Kemp [9] used
Cusum control chart to monitor rare congenital malfor-
mations, while Lin and Adams [10] applied Cusum to
monitor surgical performance.
Let 12 m
,,,
X
XX be a sequence of observed inde-
pendent valu es from a process. Without loss of generality,
the in-control process mean is assumed to be zero. The
principal feature of the Cusum control chart is that the
values of the random variable i
X
, are
compared with a pre-determined target or reference value
T. The cumulative sum of the deviation of the variable X
from T is
1,2, 3,,im

1ii i
SXTS

 (2.1)
It should be noted that when T is unknown, then an es-
timate of T given by
1
m
i
i
X
Tm


1im
(2.2)
is often used. In the literatures, S0 = 0 is the ideal value
used. To determine the trend or process shift, the values
of Si, are plotted on a chart or presented in a
table to detect an upward shift or a downward shift in
process quality (one-sided Cusum) or in both directions
(Two-sided Cusum).
3. Designing of Cusum Scheme for Accident
Fatality
The Cusum chart is often used to detect an upward or
downward shift in the process quality (one-sided Cusum
chart) or shift in both directions (Two-sided Cusum
Chart). To monitor a positive shift from the target value,
the Cusum statistic is given as:


1
max 0,
iii
CXkC



 (2.3)


For monitoring upward movement and for monitoring
downward movement, the statistic is


**
1
min 0,
iii
CXkC






Ch
*
Ch
(2.4)
The process is taken to be out-of-control if i
for an upward shift or i for a downward shift.
The procedures for determination of the parameters of
the Cusum chart (K and h) as discussed in the literatures
(see [3,4]) are presented below.
For a counted data Cusum, the parameters to be de-
termined are the reference value (k) and the decision in-
terval (h) The value k can be described as the reference
value for the process; which is usually chosen between
the acceptable process mean value (0

) and the mean
level that the Cusum scheme is inherited to detect
quickly (1

), otherwise known as the rejectable mean
value. The values of 0

and 1

are mean numbers of
counts per sampling interval. The reference value k for
the counted Cusum should be chosen close to
10
10
mm
knm nm



(2.5)
The decision interval, h, is determined by specifying
the in-Control (IC) and out-of-Control (OC) average run
length (ARL). The IC-ARL is the average number of
consecutive procedures required for a Cusum chart to
cross a decision interval or signal during the period when
the process is performing at an acceptable level. This is
analogous to Type I error or false-positive error in hy-
pothesis testing. On the other hand, the OC-ARL is the
average number of procedures performed before the
Cusum chart signals, during the period when the process
is performing at an unacceptable level. It is a measure of
sensitivity and is analogous to power, Type II error or
false-negative error in hypothesis testing.
A design with short IC-ARL (large type I error) is
prone to false alarm while a design with short OC-ARL
(high power) will quickly detect poor performance. Ide-
ally, Cusum monitoring requires long IC-ARL (small
type I error) and short OC-ARL (high power) before the
chart signals an actual deterioration in performance. Un-
fortunately, thus ideal could not be reached, as a desira-
bly long IC-ARL (small type I error) will lead to unac-
ceptably long OC-ARL (low power). On the other hand,
the desired short OC-ARL (high power) will lead to more
frequent false alarm of short IC-ARL (large type I error).
Hence, a trade-off is made between them. For normally

K. S. ADEKEYE ET AL.
215
distributed random variables, Ewan & Kemp [11] deter-
mined the ARL at 0

(call it L0) and the ARL at 1

(call it L1). These ARLs are functions of ,
h
01
kk

 
and n

. Thus, when the reference
value, k is obtained using Equation (2.5) the correspond-
ing decision interval, h is obtained using the nomograph
table (see [5]) or the tables of C1 and C2 schemes for
Poisson variable as presented in British Standard 5703,
part 4 [12].
The data used in this work was obtained from the De-
partment of Policy, Research and Statistics of the Federal
Road Safety Corps (FRSC), Ondo State Command, Akure,
Nigeria for the period January 2001 to December 2010.
Thus it covers the period of 120 months. The data is on
road accident fatalities that occurred within the State for
the period under consideration as recorded on the spot of
the road accidents.
The global expectation on road accident fatalities is to
reduce or minimize the occurrence of death from road
accidents. Therefore, a downward trend in the CUSUM
path will reflect the expectation of the globe. However, a
plot of the Cusum value of the recorded road accident
fatalities reflected an upward movement as shown in
Figure 1. Hence, we design a Cusum chart for an upward
shift. This will enable us to monitor the upward trend and
locate the point at which an unbearable number of oc-
currences are likely to happen.
From the data for the 120 months, the overall mean per
month is 9X8.2 and the standard deviation


.
Thus, in other to detect changes in the mean level of road
fatalities during this period, the acceptable mean level is
chosen nearer to the current mean level, i.e. 09


and
standard deviation 8.2

. Suppose the authority de-
mands a shift of 1.5σ from 0

to be the rejectable level,
then a shift of 1.5σ in the positive direction yields
121

. Using Equation (2.4), the reference value k is
computed. Thus,
 
21 912
21 9
knn




hTo obtain the decision interval, , the corresponding
value of h fro m table C2 of BS 5703, Part 4, when k = 12,
is equal to 11. Thus, the parameters of the designed
Cusum are k = 12 and h = 11. Therefore, an out-of-con-
trol signal will be indicated when


1iii
max 0,1211CXC

 . The plot of the Cusum
values with the obtained decision interval, h is plotted in
Figure 1.
From Figure 1, it is observed from the Cusum chart
that the Cusum values for the first 71 months are less
than the decision interval (h). From the 72nd to the 89th
months, an upward movement was observed. Thereafter
from the 90th to the 108th months a slight downward shift
in the Cusum values were observed. The last segment of
the Cusum control chart reflect an upward movement. It
should be noted that from the 72nd month to the 120th
month, the Cusum values were above the decision inter-
val. Thus, the process of the monthly road accident fa-
talities can be adjudged to be out-of-control. An impor-
tant feature of the Cusum Control chart is reflected in
Figure 1, the point of change is clearly visible.
In order to know the exact time of the year when the
road accident fatality is more prominent, the data were
broken down into yearly basis and the corresponding
Cusum statistic for each year are implemented to detect
increase in road accident occurrence as shown in Tables
1-10.
4. Discussion of Results
The design Cusum with reference value k = 12 and deci-
sion interval, h = 11 for the period of one hundred and
twenty (120) months as plotted in Figure 1 reflect that
the period under consideration can be categorized into
three segments. The first segment is the period 2001 to
November, 2006 where the road accident fatalities were
less than the decision interval. This implies that, though
Figure 1. Cusum control chart for road accident fatalities.

K. S. ADEKEYE ET AL.
216
Table 1. Cusum tabulation for road accident fatality for
year 2001.
i
X  k
i
C i Xk i X
Jan 5 –7 –7 0
Feb 0 –12 –19 0
Mar 6 –6 –25 0
Apr 8 –4 –29 0
May 4 –8 –37 0
Jun 0 –12 –49 0
Jul 3 –9 –58 0
Aug 7 –5 –63 0
Sept 2 –10 –73 0
Oct 3 –9 –82 0
Nov 4 –8 –90 0
Dec 9 –3 –93 0
Table 2. Cusum tabulation for road accident fatality for
year 2002.
i
X

k
i
C i Xk i X
Jan 3 –9 –9 0
Feb 2 –10 –19 0
Mar 4 –8 –27 0
Apr 6 –6 –33 0
May 0 –12 –45 0
Jun 7 –5 –50 0
Jul 3 –9 –59 0
Aug 3 –9 –68 0
Sept 5 –7 –75 0
Oct 2 –10 –85 0
Nov 8 –4 –89 0
Dec 6 –6 –95 0
Table 3. Cusum tabulation for road accident fatality for
year 2003.
Table 4. Cusum tabulation for road accident fatality for
year 2004.
i
Xi Xk


i
Xk
i
C
Jan 11 –1 –1 0
Feb 16 4 3 4
Mar 0 –12 –9 0
Apr 13 1 –10 1
May 2 –10 –20 0
Jun 9 –3 –23 0
Jul 8 –4 –27 0
Aug 5 –7 –34 0
Sept 4 –8 –42 0
Oct 9 –3 –45 0
Nov 7 –5 –50 0
Dec 4 –8 –58 0
Table 5. Cusum tabulation for road accident fatality for
year 2005.
i
Xi Xk


i
Xk
i
C
Jan 6 –6 –6 0
Feb 2 –10 –16 0
Mar 8 –4 –20 0
Apr 4 –8 –28 0
May 7 –5 –33 0
Jun 3 –9 –42 0
Jul 5 –7 –49 0
Aug 0 –12 –61 0
Sept 6 –6 –67 0
Oct 3 –9 –76 0
Nov 9 –3 –79 0
Dec 11 –1 –80 0
Table 6. Cusum tabulation for road accident fatality for
year 2006.
i
X  k
i
C i Xk i X
Jan 7 –5 –5 0
Feb 0 –12 –17 0
Mar 0 –12 –29 0
Apr 3 –9 –38 0
May 6 –6 –44 0
Jun 4 –8 –52 0
Jul 5 –7 –59 0
Aug 6 –6 –65 0
Sept 0 –12 –77 0
Oct 7 –5 –82 0
Nov 3 –9 –91 0
Dec 10 –2 –93 0
i
Xi Xk


i
Xk
i
C
Jan 4 –8 –8 0
Feb 0 –12 –20 0
Mar 2 –10 –30 0
Apr 2 –10 –40 0
May 11 –1 –41 0
Jun 6 –6 –47 0
Jul 2 –10 –57 0
Aug 0 –12 –69 0
Sept 8 –4 –73 0
Oct 10 –2 –75 0
Nov 0 –12 –87 0
Dec 26 14 –73 14*

K. S. ADEKEYE ET AL. 217
Table 7. Cusum Tabulation for road accident fatality for
year 2007.
i
XXk

Xk
i
C i i
Jan 4 –8 –8 0
Feb 18 6 –2 0
Mar 14 2 0 0
Apr 24 12 12 12*
May 16 4 16 16*
Jun 9 –3 13 13*
Jul 26 14 27 27*
Aug 6 –6 21 21*
Sept 17 5 26 26*
Oct 10 –2 24 24*
Nov 14 2 26 26*
Dec 18 6 32 32*
Table 8. Cusum tabulation for road accident fatality for
year 2008.
i
XXk  Xk
i
C i i
Jan 13 1 1 1
Feb 15 3 4 4
Mar 15 3 7 7
Apr 20 8 15 15*
May 12 0 15 15*
Jun 4 –8 7 7
Jul 13 1 8 8
Aug 5 –7 1 1
Sept 1 –11 –10 0
Oct 14 2 –8 0
Nov 9 –3 –11 0
Dec 12 0 –11 0
fatalities exist within the said period but the rate is not
alarming. In the second and third segment, the Cusum
values were outside the decision interv al. This implies an
upward shift in the process. The cumulative sum of the
fatalities increases from December, 2006 to May, 2008,
thereafter a downward shift was observed from June
2008 to December, 2009. It should be noted that the
downward shift is not an improvement or reduction per
se, since the values at this period were still far above the
decision interval. The Cusum tabulation in Table 1 through
Table 10 shows that the road accident fatalities during
the period January 2001 to November 2007 (see Tables
1-6) were close and no alarming rate of occurrence was
observed. For the periods where alarming rate was dis-
played, it is very easy to detect the period of the year
when the highest occurrence of road accident fatalities
Table 9. Cusum tabulation for road accident fatality for
year 2009.
i
Xi Xk


i
Xk
i
C
Jan 24 12 12 12*
Feb 26 14 26 26*
Mar 24 12 38 38*
Apr 28 16 54 54*
May 20 8 62 62*
Jun 19 7 69 69*
Jul 43 31 100 100*
Aug 10 –2 98 98*
Sept 5 –7 91 91*
Oct 33 21 112 112*
Nov 17 5 117 117*
Dec 0 –12 105 105*
Table 10. Cusum Tabulation for road accident fatality for
year 2010.
i
Xi Xk


i
Xk
i
C
Jan 24 12 12 12*
Feb 26 14 26 26*
Mar 24 12 38 38*
Apr 28 16 54 54*
May 20 8 62 62*
Jun 19 7 69 69*
Jul 43 31 100 100*
Aug 10 –2 98 98*
Sept 5 –7 91 91*
Oct 33 21 112 112*
Nov 17 5 117 117*
Dec 0 –12 105 105*
occurred. From Table 6, December 2006 was the only
month that has a Cusum value that is more than the deci-
sion interval. For year 2007 (Table 7), December also
has the highest Cusum value. In year 2008 (see Table 8),
the highest Cusum values was in April and May. From
Tables 9 and 10 the highest occurrence is in the month of
November. It should be noted that the months where the
highest occurrence of road accident fatalities were ob-
served for the years are all festive periods such as
Christmas, Easter, Eid-el-Kabir, and Eid-el-Moluod.
5. Conclusion
The Cusum chart plotted for road accident fatality data is
capable of detecting small shift from the mean level.
More so, the points of change in the process are clearly
identified and points at which these occurred are easily

K. S. ADEKEYE ET AL.
218
located on the CUSUM in Figure 1. Thus, CUSUM chart
provides visual aids diagnosis and identifies the situa-
tions that require quick attentio n. The months where high
occurrence of road accident fatalities are identified should
be used as benchmark by road manager especially Fed-
eral Road safety Corps (FRSC) as periods where more
attention or precaution measure should be put in place on
the roads to drastically reduce or eliminate high occur-
rence of road accident fatalities. The design Cusum con-
trol chart can be adapted for other states in the country,
the entire country and for other countries of the world .
REFERENCES
[1] World Health Organization, “The Health Report—Global
Burden of Diseases,” 2008.
[2] Y. Zhao, F. Tsung and Z. J. Wang, “Dual CUSUM Con-
trol Schemes for a Mean Range Detection CUSUM,” IIE
Transactions, Vol. 37, No. 11, 2005, pp. 1047-1057.
doi:10.1080/07408170500232321
[3] E. S. Page, “Continuous Inspection Schemes,” Biometrika,
Vol. 41, No. 1-2, 1954, pp.100-114.
doi:10.1093/biomet/41.1-2.100
[4] J. M. Lucas, “Counted Data CUSUM’s,” Technometrics,
Vol. 27, No. 2, 1985, pp. 199-244.
[5] E. R. Ott, E. G. Schilling and D. V. Neubauer, “Process
Control: Trouble Shooting and Interpretation of Data,”
4th Edition, Quality Press, Milwaukee, 2005.
[6] D. C. Montgomery, “Introduction to Statistical Quality
Control,” 4th Edition, John Wiley and Sons, New York,
2005.
[7] P. A. Osanaiye and C. O. Talabi, “On Some Non-Manu-
facturing Application of Counted Data Cumulative Sum
(Cusum) Control Chart Scheme,” The Statistician, Vol.
38, 1989, pp. 251-257. doi:10.2307/2349058
[8] K. S. Adekeye, “Use of Cumulative Sum Control Chart
for Detection of Increase in Crime Rate,” Nigerian Jour-
nal of Mathematics and Application, Vol. 15, 2002, pp.
188-196.
[9] A. B. Nix, R. J. Rowlands and K. W. Kemp, “Internal
Quality Control,” Clinical Medicine, Vol. 6, 1986, pp.
425-440.
[10] W. S. W. Lin and B. M. Adams, “Combined Control
Chart for Forecast-Based Monitoring Schemes,” Journal
of Quality Technology, Vol. 28, No. 3, 1996, pp. 289-301.
[11] K. W. Kemp, “The Use of Cumula tive Sum for Sampling
Inspection Schemes,” Applied Statistics, Vol. 11, No. 1,
1962, pp. 16-31. doi:10.2307/2985287
[12] British Standards Institution, “British Standards Guide to
Data Analysis and Quality Control Using CUSUM Tech-
niques, Part 4,” 1982.