EXERCISE AND QUALITY OF LIFE
Research article
Volume 2, No. 2, 2010, 6374
UDC 796.325055.2:796.015.68]:519.8
A SIMPLE MATHEMATICAL MODEL FOR ESTIMATING
GENERAL JUMPING PREPAREDNESS OF SENIOR
FEMALE VOLLEYBALL PLAYERS
Milivoj Dopsaj, Nemanja Ãopi„^{}, Goran Neöi„ and Milan Sikimi„
Faculty of Sport and Physical Education
University of Belgrade
Abstract
The aim of this paper is to define a simple mathematical model for estimating general
jumping preparedness of senior female volleyball players (SCOREpoint) by applying a set of 7
variables measured by standardized tests within the field measuring conditions. The sensitivity of
this model is determined on the basis of achieved points and differences in general jumping
preparedness among female volleyball players competing in elite international competitions, elite
national competitions, statelevel and regionallevel competitions. Applied battery of tests has
cumulatively explained
80.64% of the total variant of measurement. The defined model
explained the criterion of general jumping preparedness of senior female volleyball players at the
level of 100% (Adj. R^{2} = 1.000), and with the minor error in prediction (Std. Err. Est. = 0.003
SCORE_{point} points). The obtained model has the following form: SCOREpoint = 86.762 +
(0.4595 · CMJARM
+ 0.5158 · CMJNOARM + 0.4620 · SJCONARM + 0.4812 · SJCONCNOARM
+ 0.5431 ·
CM_{BJ } + 0.5626 · SJCONCBJ + 0.138 · SLJ. The defined model has a satisfactory level of
discrimination and it is proposed for further practical use.
Keywords: volleyball, jumping ability, mathematical model, general jumping preparedness
Introduction
Volleyball is a team sport played at all levels by both genders (e.g., youth, Olympic,
professional) and places an emphasis on fast and explosive movements such as jumping, hitting,
and blocking (Spence, Disch, Fred, & Coleman, 1980; Marques, GonzalezBadillo, & Kluka,
2006; Stojanovi„ & Kosti„, 2002; Sheppard et al., 2007; Raji„, Dopsaj, & PablosAbella, 2008).
Volleyball is commonly described as a complex, high speed, explosive and powerful sport.
Success in a volleyball game depends to a great extent on the movement speed without
the ball, the speed of the rhythm change and direction of movement, agility and jumping ability
(Neöi„, 2008; Suzovi„ and Nedeljkovi„, 2009). Repeated maximal or nearmaximal vertical
^{} Corresponding author. Faculty of Sport and Physical Education, University of Belgrade, Blagoja Parovi„a 156,
11000 Belgrade, Serbia, Email: nemanjacopic@yahoo.com
© 2010 Faculty of Sport and Physical Education, University of Novi Sad, Serbia
63
M. Dopsaj et al.
jumps, frequent changeofdirection sprints, diving to make a save, and repeated overt head
movements when spiking or blocking are among the movements that make up the game (Black,
1995; Gandeken, 1999). A volleyball match can be played up to five sets, meaning that the
duration of the match may be up to around 90 minutes. During that time a volleyball player
performs 250300 actions in which the explosiveness of the leg muscles dominates. Out of the
total number of actions, jumps comprise 5060%, fast movements and direction change in space
amount to around 30% and falls make around 15%. The latest data point out that the average
body height of a contemporary volleyball player is between 195 and 200cm (Ercolessi, 1999)
and for female volleyball players from 185.41±7.88 cm for the Olympic level, up to 180.88±3.03
cm for the First League level and 174.25±3.07 cm for the regionallevel (Second League ñ
North). (Dopsaj, Neöi„ & oki„, 2010). The average height of the vertical jump of the spike
receiver, spiker and middle blocker is from 345 to 355cm and in the block from 320 to 335cm
(Ercolessi,
1999). The explosive power and speed strength is dominant in spike and block
actions and in most cases it is the key factor in winning points or the quality of defense actions in
a block.
An average action in volleyball lasts about 6 seconds, and is followed by an average rest
period of 14 seconds, not including player substitutions or timeouts (Gandeken, 1999). This
action workrest ratio suggests that athletes primarily use the adenosine triphosphate
phosphocreatine system. There are about 50 rallies per a game. As a result, energysystem train
ing for volleyball should consist of 50 or more repeats lasting 510 seconds. These efforts should
consist of jumping, running, and/or diving, involving frequent changes of direction, followed by
1015 seconds of rest (Black, 1995).
Bearing in mind that during a match volleyball players make jumps applying various
jumping techniques (with and without arm swings), that they make jumps with a dominant
vertical or horizontal component of movement direction of the gravity center of the body, and
that the given elements are made in relation with various regimes of muscle contractions of leg
stretchers (Stretch Shortening Cycle i.e. excentricconcentric muscle regime type of contraction,
only concentric muscle type of contraction and different combination of contraction as well as
isometricconcentric, excentricisometricconcentric, prestretch deep jump impactconcentric
etc.), it is supposed that from a methodological aspect, the sum of the results of various types and
kinds of jumps would give a better general (summary) estimation about the level of general
jumping preparedness of the female players in relation with the information obtained by an
individual test. Also, the given kind of testing could be realized within the spacetime conditions
of usual trainings without disturbing the training rhythm, where the information on current level
of development of the examined level of preparedness could be done by applying the method of
field testing by means of the model load (Dopsaj, 2010).
The aim of this paper is to define a simple mathematical model for estimation of general
level of jumping preparedness of senior female volleyball players. The sensitivity of this model
will be determined on the basis of achieved points and differences in general jumping
preparedness among the female volleyball players who have been successful in volleyball on
various levels: in elite international competitions (the Olympics), elite national competitions
(Super League), and statelevel (First League) and regionallevel (Second League ñ North)
competitions. In that way a simple and operational method of testing female volleyball players
will be defined, and the obtained results can be used in the process of staged training tests of
jumping readiness in the function of longterm training process.
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A mathematical model for estimating jumping preparedness
Method
The main method used in this research is the field testing method (Dopsaj, 2010). As far
as measuring is concerned, a method of direct measuring was used where the Abalakís method
was applied (Zaciorski & Kraemer, 2006), that is, Belt Jump Test (Klavora, 2000) (Figure 1).
Prior to each testing, the players were explained the purpose and ways of the measuring, and
each of them gave an oral consent to participate in the study in accordance with the norms of the
Ethical Committee of the Faculty of Sport and Physical Education of the University in Belgrade.
All measuring were performed in the afternoons in training gyms with the same type of flooring
(wooden parquet floor). All the measuring was done by qualified experts ñ three Physical
Education teachers.
Figure 1. A device for measuring height of vertical jump (Abalakís method).
Sample of the examined players
Sample consisted of 59 female players (N): 15 players of the Olympic selection (OS), 12
players of the VC RadniËki (SL) who participated in the competitions of the Super League of
Serbia, 20 players of the VC Vizura (FL) who played in the First League of Serbia and 12
players of the VC Kikinda (SLN) who took part in the competitions of the Second League ñ
North.
An average age and length of training period of the female volleyball players from the
Olympic selection was 22.6±3.2 and 12.3±2.5 years. The players of the VC RadniËki were
20.0±2.1 and 7.4±1.7 years of age. The players of the VC Vizura 19.4±2.6 and 8.4±3.0 and with
the players VC Kikinda 18.4±2.7 i 6.2±2.9 years respectively.
All players had been informed about the subject, goal and objectives of the research and
in cooperation with their trainers gave a voluntary consent to participate in the research.
Table 1
Basic descriptive indicators of the sample of all four teams
Volleyball
Average age
Training period
Teams
N
players
(mean±sd)
(mean±sd)
1
Olympic Selection
15
22.6±3.2
12.3±2.5
2
Super league
12
20.0±2.1
7.4±1.7
3
First league
20
19.4±2.6
8.4±3.0
4
Second league North
12
18.4±2.7
6.2±2.9
65
M. Dopsaj et al.
Variables samples
In order to define a simple mathematical model, tests were selected by the criterion of
simplicity, informativeness and practical usage in function of application from the aspect of field
measuring. From the aspect of motoric structure, the jumps were selected which are the most
representative from the analytical, diagnostical and situational aspect in function of defining the
profile of jumping ability in volleyball (Spence et al., 1980; Fleck et al., 1985; Klavora, 2000;
Young et al., 2001; Sheppard et al., 2007; Marques et al., 2004; Ãopi„, Dopsaj, Neöi„, &
Sikimi„, 2010).
A battery of tests by which jumping ability was assessed, both in function of basic
vertical and in function of basic horizontal component, contained the following 7 different types
of jumps which represented individual variables of the measured space:
Seven variables for assessment of different types of jumps are:

double leg vertical countermovement jump with arm swing (CMJARM)
measured in cm (Figure 2)

double leg vertical countermovement jumpno arm swing allowed
(CMJNOARM) measured in cm
(Figure 3)

double leg concentric vertical squat jump (SJCONARM )
measured in cm (Figure 4)

double leg concentric vertical squat jump (SJCONCNOARM)
measured in cm (Figure 5)

double leg vertical countermovement standing block jump (CM_{BJ})
measured in cm (Figure 6)

double leg vertical squat standing block jump (SJCONCBJ)
measured in cm (Figure 7)

Standing long jump (SLJ) measured in cm.
A
B
C
D
Figure 2. Double leg vertical countermovement jump with arm swing (CMJARM).
66
A mathematical model for estimating jumping preparedness
A
B
C
D
Figure 3. Double leg vertical countermovement jumpno arm swing allowed (CMJNOARM).
A
B
C
Figure 4. Double leg concentric vertical squat jump (SJCONARM ).
A
B
C
Figure 5. Double leg concentric vertical squat jump (SJCONCNOARM).
A
B
C
D
Figure 6. Double leg vertical countermovement standing block jump (CM_{BJ}).
67
M. Dopsaj et al.
A
B
C
Figure 7. Standing long jump (SLJ).
Measuring methods
The measurement was performed by the Abalakís method, i.e. belt jump method in which
the standard PVC measuring belt was fixed on the upper side in the position around the
umbiculus (in front of the stomach) of the examined player, and from the lower side on the
parquet in the front projection of the standing point. The belt was pulled through the measuring
fixator from the lower side which was firmly fixed to the floor so that the belt could freely move
(in the direction of pullout movement) through it (Klavora, 2000). The examined players did a
set series consisting of 6 vertical jumps, each jump was repeated twice, while the break between
the jumps lasted 30 seconds. Each jump was made with maximal intensity with a task to jump
back onto the same place. The position of jumpup and jumpdown was marked on the parquet
(Figure 1). In case an examined player jumped down out of the marked jumpdown zone, that
attempt was not recorded and the measuring was repeated. A better result in each jump type was
recorded as the final score of that particular jump type. After that measuring, the examined
players were tested on another place by the standing long jump test which was realized by a
standardized procedure (Zaciorsky & Kraemer, 2006).
Statistical methods
Raw data were analyzed in the first phase by application of descriptive statistics in order
to calculate the basic descriptive indicators, the mean values (mean) and the standard deviations
(sd) in particular. For the purpose of calculating general difference between the jumping ability
in function of examined subsamples, the ANOVA was used (Hair, Anderson, Tatham, & Black,
1998). Definition of the Index of general jumping preparedness (SCOREpoint) was made by
applying the method of mathematical analogy where the value of position of the factor score of
each examined player was turned into an analogue point score defined from 0 points (as
hypothetical minimum) to 100 points (as hypothetical maximum). In the consequent statistical
process of defining a mathematical model the value of the SCORE point represented a criterion
variable, while the individual score obtained by application of the battery of 7 jumping tests
stood for a system of predictable variables. The final form of the model was defined by
application of the Multivariant regressive analysis. All analyses were done in the statistical
package SPSS 12.0 and the difference criterion was defined on the level p=0.05 (Hair et al.,
1998) .
68
A mathematical model for estimating jumping preparedness
Results
Table 2 presents the KMO results of measuring of the sample adequacy where it is
possible to see that the raw data belong to a homogenous group with the reliability level from
88.30% (p=0.000). In that way it was proved that they can be validly analyzed by the method of
multivariant statistics, and that they can be validly interpreted as well.
Table 2
The results of KMO measurement of the sample adequacy KMO and Bartlett's Test
KaiserMeyerOlkin Measure of Sampling
0.883
Adequacy.
Approx. ChiSquare
442.55
Bartlett's Test of
df
21
Sphericity
Sig.
0.000
Table 3 shows the communalities of the variables, and it is evident that all the used
variables are highly projected on to the common measuring variance, i.e. they belong to the same
measurement space. In such a way it can be claimed that the given set of tests can be treated as a
battery of tests for evaluation of general level of jumping abilities. Of all single test tasks,
CMJARM, with its extraction level of
84.3%, has the greatest projection onto the common
measurement space, while SJCONCNOARM at the extraction level of 70.4%, has the lowest level.
Table 3
Communalities of the used variables for general jumping preparedness evaluation
Communalities
Initial Extraction
CMJARM
1.00
0.843
CMJNOARM
1.00
0.830
SJCONARM
1.00
0.840
SJCONCNOARM
1.00
0.704
CM_{BJ}
1.00
0.838
SJCONCBJ
1.00
0.827
SLJ
1.00
0.763
Extraction Method: Principal Component Analysis.
Table 4 shows the result of separate cumulative variance projected onto the first factor.
The used battery of tests has cumulatively accounted for 80.64% of the total measurement
variance, which means that at the level of 80.64%, it defined the general jumping preparedness
of female volleyball players of senior/varsity age.
69
M. Dopsaj et al.
Table 4
Cumulative level of explained measurement variance
Total Variance Explained
Initial Eigenvalues
Extraction Sums of Squared
Component
Total
% of Cumulative % Total
% of Cumulative %
1
5.65
80.64
80.64
5.65
80.64
80.64
Extraction Method: Principal Component Analysis.
Table 5 shows the results of multiple regression analysis where the SCOREpoint represented the
value of criteria variables, and the results of the separate tests showed the predictive variables.
Table 5
Results of multiple regression analysis
Coefficients^{a}
Unstandardized Standardized
95,0% Confidence
Coefficients
Coefficients
Interval for B
Model
t
Sig.
Lower Upper
B Std. Error
Beta
Bound Bound
(Constant)
86.7620
0.005
18939.46
0.000
86.771
86.753
CMJARM
0.4595
0.000
0.163
2625.68
0.000
0.459
0.460
CMJNOARM
0.5158
0.000
0.161
2871.742
0.000
0.515
0.516
SJCONARM
0.4620
0.000
0.162
2912.801
0.000
0.462
0.462
1
SJCONCNOARM
0.4812
0.000
0.149
3255.311
0.000
0.481
0.482
CM_{BJ}
0.5431
0.000
0.162
2914.241
0.000
0.543
0.543
SJCONCBJ
0.5626
0.000
0.161
2990.454
0.000
0.562
0.563
SLJ
0.1381
0.000
0.155
3551.424
0.000
0.138
0.138
Model Summary^{b}
Model
R
R Square
Adj. R Square
Std. Error of the Estimate
1
1.000^{a}
1.000
1.000
0.003
a. Predictors: (Constant), CMJ_{A}
RM, CMJNOARM, SJCONARM, SJCONCNOARM, CMBJ, SJCONCBJ, SLJ
b. Dependent Variable: SCOREpoint
Table 6 presents the results of ANOVA; there are statistically significant differences in
comparison with the index of general jumping preparedness
(SCOREpoint) between the
analyzed groups, i.e. female players from highlevel international competitions (the Olympics),
highlevel national competitions (Super League), and statelevel (First League) and regional
level (Second League ñ North) competitions; at a level of F ñ 17.38, p = 0.000.
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A mathematical model for estimating jumping preparedness
Table 6
The results of ANOVA index of general jumping preparedness (SCOREpoint) in comparison with
the tested groups of examined players
Tests of BetweenSubjects Effects
Dependent Variable:SCOREpoint
Source Type III Sum of Squares df Mean Square
F
Sig.
Team
13963.216
3
4654.405
17.380
0.000
a. R Squared = 0.487 (Adjusted R Squared = 0.459)
Table 7 and Graph 1 show the basic descriptive data of the SCOREpoint with determined
differences between groups of tested female volleyball players from highlevel international
competitions (the Olympics), highlevel national competitions (Super League), and statelevel
(First League) and regionallevel (Second League ñ North).
Table 7
Results of the descriptive statistics
OS (N=15)
SL (N=12)
PL (N=20)
DLS (N=12)
VOLLEYBALL
mean± sd
mean± sd
mean± sd
mean± sd
SCOREpoint
60.62±10.38^{Ü}
67.12±13.70
$▲
48.83±12.02
35.00±11.55
OS vs DLS^{Ü}; SL vs PL^{$}; SL vs DLS^{▲} p> 0.05^{▲};
Graph 1. The overview of the basic descriptive indicator (mean±sd) SCOREpoint with
determined differences between the groups of athletes from elite international competitions (the
Olympics), elite national competitions
(Super League), and statelevel
(First League) and
regionallevel (Second League ñ North).
71
M. Dopsaj et al.
At the end, the finally defined simple mathematical model for evaluation of general
jumping preparedness (SCOREpoint) has the following form:
= 86.762 + (0.4595 · CMJARM + 0.5158 · CMJNOARM + 0.4620 · SJCONARM +
SCOREpoint
0.4812 · SJCONCNOARM + 0.5431 · CM_{BJ }+ 0.5626 · SJCONCBJ + 0.138 · SLJ)
Discussion
A simple mathematical model was defined on basis of the obtained results which
explained the measurement space i.e. a criterion of general jumping preparedness of senior
female volleyball players at the level of 100% (Adj. R^{2} = 1.000, Table 5). In other words, it
explained this with the negligible prediction error (Table 5, Std. Err. Est. = 0.003 SCORE_{point}).
The results also showed that the defined model has a satisfactory level of discrimination, for it
turned out there is a statistically significant criterion difference in comparison with the tested
group that represented players selected with respect to different competition level ñ highlevel
international competitions (the Olympics), highlevel national competitions (Super League), and
statelevel (First League) and regionallevel (Second League ñ North) competitions, at a level of
F ñ 17.38, p = 0.000 (Table 6).
Results of an interaction of influences of separate variables of various types of jumps
have showed that the greatest influence on general jumping ability
(the largest value of
coefficient of regression influence) has SJconc_{BJ }= 0.5626, followed by CM_{BJ }= 0.5431 and so
on, while the smallest one has SLJ = 0.1381 (Table 5). It is obvious that the largest impact on
total variability of general jumping preparedness is by types and variants of jumps that within a
given motoric task bear also a specific structure of movement, that is, they directly represent a
specific volleyball jumping technique. Examples of these are double leg vertical squat standing
block jump (SJ_{ConcBJ}, Image 7) and double leg vertical countermovement standing block jump
(CM_{BJ}, Image 6).
On the other hand, the smallest impact on any given indicator have the jumps that
dominantly represent the motoric task that defines the general level of preparedness when it
comes to jumps without any additional jump technique. Those are double leg vertical
countermovement jump with arm swing (form of excentricconcentric action of leg stretching
muscles with circular arm motion CMJ_{Arm}, Picture 2), and Standing long jump with arm swing
(form of excentricconcentric exertion of legstretching muscles with circular arm motion and
dominantly horizontal body movement ñ SLJ).
In this way it was also shown that the selected battery of jumps is valid both in respect
with its composition and structure since it is not only of statistically significant validity in
evaluating the general jumping preparedness, but also sensitive enough when considering the
level of separate impact of result attained in jumps with motoric structure that bears
informational complexness of jumping techniques specific to volleyball.
Since the results have shown that the defined model has a satisfactory level of
discrimination it is recommended for practical use, for the purpose of a simple testing method in
the process of evaluating given preparedness with respect to phases of preparation of senior
female volleyball players.
72
A mathematical model for estimating jumping preparedness
References
Black, B. (1995). Conditioning for volleyball. Strength and Conditioning Journal, 17(5), 5355.
Ãopi„, N., Dopsaj, M., Neöi„, G., & Sikimi„, M. (2010). Level of jumping performance in elite
female volleyball players with regard to their positions: Multidimenzional assessment
model. International Journal of Volleyball Research. (In Press, december 2010)
Dopsaj, M. (2010). Karakteristike Ft krive: AnalitiËki i dijagnostiËki znaËaj u sportu. U R.
Stankovi„ (Ur.), XIV Meðunarodni nauËni skup FIS Komunikacije u sportu, fiziËkom
vaspitanju i rekreaciji: Zbornik radova, Niö, 22. oktobar 2010 (str. 4769). Niö: Fakultet
sporta i fiziËkog vaspitanja Univerziteta u Niöu.
Dopsaj, M., Neöi„, G., & Ãopi„, N. (2010). The multicentroid position of the anthropomorphological
profile of female volleyball players at different competitive levels. Facta universitatis

series: Physical Education and Sport, 8(1), 4757.
Ercolessi, D. (1999). La caduta dal salto. Super Volley, 1, 7982.
Fleck, S. J., Case, S., Puhl, J., & Van Handle, P.
(1985). Physical and physiological
characteristics of elite women volleyball players. Canadian Journal of Applied Sport
Sciences, 10, 122126.
Gandeken, S. B. (1999). Offseason strength, power, and plyometric training for Kansas State
volleyball. Strength and Conditioning Journal, 21(6), 4955.
Hair, J., Anderson, R., Tatham, R., & Black, W. (1998). Multivariate data analysis with readings
(Fifth Ed.). New Jersey: PrenticeHall International, Inc.
Klavora, P. (2000). Verticaljump test: A critical review. National Strength & Conditioning
Association, 22(5), 7075.
Marques, M. C., GonzalezBadillo, J. J., & Kluka, D. (2006). Inseason strength training male
professional volleyball athletes. Strength and Conditioning Journal, 28(6), 212.
Marques, M. C., GonzalezBadillo, J. J., Cunha, P., Resende, L., Santos, M., & Domingos, P.
(2004). Changes in strength parameters during twelve competitive weeks in top
volleyball athletes. International Journal of Volleyball Research, 7, 2328.
Neöi„, G. (2008). Struktura takmiËarske aktivnosti odbojkaöica, Godiönjak Fakulteta sporta i
fiziËkog vaspitanja,
14, 89112.
Raji„, B., Dopsaj, M., & PablosAbella, C.
(2008). Basic and specific parameters of the
explosive force of leg extensors in high trained serbian female volleyball players:
Characteristics of the isometric forcetime curve model. Serbian Journal of Sports
Sciences, 2(14), 131139.
Sheppard, J. M., Gabbett, T., Taylor, K. L., Dorman, J., Lebedew, A. J., & Borgeaud, R. (2007).
Development of repeatedeffort test for elite men's volleyball, International Journal of
Sports Physiology and Performance, 2(3), 292304.
Spence, D. W., Disch J. G., Fred, H. L., Coleman, A. E. (1980). Descriptive profiles of highly
skilled women volleyball players. Medicine and Science in Sports and Exercise, 12(4),
299302.
Stojanovi„, T., & Kosti„, R. (2002). The effects of the plyometric sport training model on the
development of the vertical jump of volleyball players. Facta Universitatis: Physical
Education and Sport, 1(9): 1125.
73
M. Dopsaj et al.
Suzovi„, D., & Nedeljkovi„, A. (2009). Kratke pulsne kontrakcije: Odnos izmeðu maksimalne
sile i brzine razvoja sile. FiziËka kultura, 63(1), 1725.
Young, W., MacDonald, C., & Flowers, M. A. (2001). Validity of double and singleleg vertical
jumps as tests of leg extensor muscle function. Journal of Strength and Conditioning
Research, 15(1), 611.
Zatsiorsky, M. V., & Kraemer, J. W.
(2006). Science and practice of strength training.
Champaign, IL: Human Kinetics.
Submitted November 15 , 2010
Accepted December 14, 2010
74