Yonago Acta medica 2001;44:151–156

How Are Statistical Parameters of the Velocity Vector of Body Sway
Distributed in Normal Human Subjects?
Kei Nakahara, Katsuyuki Kawamoto, Rie Hamasaki, Aya Nosaka, Hiroko
Funakoshi and Hiromi Takeuchi
Department of Oto-Rhino-Laryngology, Tottori University Faculty of Medicine, Yonago
683-0826 JapanThe velocity vector when the human body sways has been qualitatively evaluated inclinical sessions. We quantitatively measured the velocity vector for 1 min in 89 normal
subjects standing in a stable posture, and examined distributions of quantities of thevelocity vector. The velocity vector was measured with a stabilometer, which visualizes
the vector as magnitudes radially projected from the center to the periphery into 36
directions by 10˚. The 3 quantities we calculated from the 36 scalars of the vector per
subject were the coefficient of correlation (CV), skewness and kurtosis, which wereanalyzed statistically. Values of skewness were normally distributed. Values of CV
and kurtosis were log-normally distributed when adjusted with log transformation.
Then, we calculated standardized values of the normal distributions, from which thelower and upper cutoff values in the 95% and 99% areas were available. The 3 quantities
showed statistically significant correlations with one another, although the levels were
low. Thus, in the present study, use of the 3 parameters enabled us to quantitatively
evaluate the whole image of velocity vector, which would simplify the procedures of
examination and shorten the time required for differential diagnosis.
Key words: body sway; stabilometer; statistical characteristic value; velocity vector

Human body sway has been researched fromearlier (Basler, 1929; Hellebrandt, 1938).
However, detailed studies on body sway have
only lately become popularized after thedevelopment of the stabilometer and its introduction into clinical sessions. In examining thelocus of body sway, measurement of length andarea of the locus is a good method which relatively integrates qualitative and quantitativeevaluations (Okawa et al., 1995; Imaoka et al.,1997). On the other hand, in the present-day
clinic, the velocity vector has only been used asa reference to understanding the sway pattern at
most. It is quite difficult to quantitatively evaluate the whole velocity vector with one index,
when a human body sways. So, practical examinations have depended on the qualitative classification of body sway patterns and the partially quantitative analysis of Romberg’s rate
Abbreviation: CV, coefficient of variation(Yamamoto and Komatsuzaki, 1984). Qualitative evaluation is apparently comprehensible,
but the examiners’ bias might spoil the objectivity of analysis. On the other hand, Romberg’s
rate does not reflect the whole velocity vector, but
only many data related to contralateral directions among the 36 magnitudes of velocity vector.
To quantitatively evaluate the velocity vector in a more simplified manner, in this article
we describe how we measured the momentcharacteristic values of the velocity vector in
healthy subjects, and tried to treat them statistically through analyses of their quantities.

Subjects and Methods
Subjects were 89 healthy adult volunteers (47males and 42 females) in their 20’s or 30’s suf-151

K. Nakahara et al.

Vector analysis
Sort from Max
1 (18)
381.342 ( 7)
367.623 (24)
358.97
4 ( 6)
354.715 (27)
345.256 (10)
311.547 (12)
309.678 (25)
299.939 (26)
296.97

0
1
2
3
4
5
6
7
8
9
10
11

=
=
=
=
=
=
=
=
=
=
=
=

216.2
211.3
178.8
189.4
289.4
192.6
354.7
367.6
224.9
292.2
311.5
224.3

7
10

6

12

18

0

Sway locus
25
24

26
27

Velocity vector

12 = 309.7
13 = 149.7
14 = 295.3
15 = 234.9
16 = 221.6
17 = 247.9
18 = 381.3
19 = 229.1
20 = 197.8
21 = 196
22 = 229
23 = 173.6
24 = 359
25 = 299.9
26 = 297
27 = 345.3
28 = 279.4
29 = 218.4
30 = 217.9
31 = 252.4
32 = 254.9
33 = 198.3
34 = 201
35 = 226.1

Fig. 1. Schema of vector analysis consisting of images of the sway locus (left) and the velocity vector (right).
Vectors are displayed as magnitudes radially projected into 36 directions, which represent scalars. The digits
from 0 to 35 on both sides are the means of scalar data obtained from the 89 healthy subjects. Upper left
shows the maximum 9 scalar levels. The digit written in parentheses indicates the number of magnitude
directions.

fering from no disease inappropriate to theobject of this study, and with normal limbs andtrunk. The mean age was 25.8 years with arange of 20 to 38 years. The velocity vector wasmeasured with an SKG-TN-I stabilometer(Nagashima Medical Instrument, Co., Ltd.,Tokyo, Japan) run on its analysis program
under the guidelines determined by the Japan
Society for Equilibrium Research: the subject
stands on the stabilometer with both legs closed
in a well-lighted environment, gazing at an indicator, while the sway of the center of gravity ofthe body is recorded for 1 min (Steering Committee of the Japan Society for Equilibrium
Research, 1996). All subjects well understoodthe object of the study.
The whole image of velocity vector was
visualized as 36 magnitudes radially projectedfrom the center to the periphery, by 10˚ increments between 0˚ and 359˚ (Fig. 1). From thethus-obtained 36 scalars per patient, we calculated the coefficient of variation (CV), skew-

ness and kurtosis in the 89 subjects, and examined types of distributions of the 3 statistics (Table
1): the CV, skewness and kurtosis respectively
show the dispersion, symmetry and centrality ofthe distribution. In addition, histograms of the
3 statistics were examined to find out whether
they were normally distributed or not.
The chi-square goodness-of-fit test, PearsonHartley’s test and Geary’s test were used to examine normality. Pearson’s correlation coefficient was used in testing the correlation. Statistical significance was determined when the probability was less than 0.05.

Table 1. Definitions of statistics used
CV = SD/Mean
Skewness = 36–1•Σ (Datum – Mean)3/SD3
Kurtosis = 36–1•Σ (Datum – Mean)4/SD4CV, coefficient of variation.

152

30

mean
SD
maximum
minimum

A

25
20

0.43
0.11
0.81
0.18

25

Number of subjects

Number of subjects

Velocity vector of human body sway

15
10
5
0
0.2

0.4

0.6

0.8

20
15
10
5
0

Value of CV

15
10
5
0
0

0.4

0.8

20

10
5

-0.4

2.93
0.71
4.84
1.63

15
10
5
0
2.0

3.0

4.0

0.4

0.8

1.2

30
25

5.0

C

20
15
10
5
0

1.0

0

Value of skewness
Number of subjects

Number of subjects

20

0.8

15

0

1.2

mean
SD
maximum
minimum

C

25

0.6

B

Value of skewness
30

0.4

25

Number of subjects

Number of subjects

mean 0.48
SD 0.34
maximum 1.18
minimum —0.45

B

-0.4

0.2

Value of CV

25
20

A

2.0

3.0

4.0

5.0

Value of kurtosis

Value of kurtosis
Fig. 2. Histograms of the CV (A), skewness (B) andkurtosis (C). Distribution is asymmetrical in A andC, and symmetrical in B. CV, coefficient of variation.

Fig. 3. Distribution curves of the CV (A), skewness(B) and kurtosis (C). The CV values are distributed
log-normally (mean –8.73 × l0–1, variance 7.01 × l0–
2), the skewness values normally (4.80 × l0–1, 1.14 ×l0–1) and the kurtosis values log-normally (1.05, 5.85
× l0–2). CV, coefficient of variation.

Results

respectively. The kurtosis levels were distributed asymmetrically between 1.63 and 4.84
with a mean, an SD and a mode of 2.93, 0.71
and near 3, respectively (Fig. 2).

Observed data of statistics
The CV data were distributed asymmetricallybetween values 0.18 and 0.81 with a mean of
0.43, an SD of 0.11 and a mode of about 0.3.
The values of skewness were observed in symmetry between –0.45 and 1.18, with a mean, an
SD and a mode of 0.48, 0.34 and about 0.5,

Application of the normal distribution
In the determination of normality of distributions of the CV, skewness and kurtosis (Fig. 3),
we used 2 quantities, that is, the sample mean
153

K. Nakahara et al.

Table 2. Formula for calculating the cutoff values of the CV, skewness and kurtosis in thestandardized normal and log-normal curves
Statistic

Formula [lower cutoff value, upper cutoff value]

Skewness†[Χ – z*α • σ, Χ + z*α • σ ]CV and kurtosis‡[G • exp(–z*α • σ ln), G • exp(z*α • σ ln)]CV, coefficient of variation; Χ, sample mean; σ, standard deviation of unbiased samples; G, geometricsample mean; σ ln, log-transformed σ ; z*α, two-sided α point in the normal distribution.
†Normally distributed.
‡Log-normally distributed.

and unbiased sample variance. As a result, theskewness was normally distributed (mean 4.80
× l0 –1, variance 1.14 × l0 –1). Conversion tologarithms achieved normality of distribution in
the CV (–8.73 × l0–1, 7.01 × l0–2) and kurtosis(1.05, 5.85 × l0–2).

Determination of the standardized values
Based on these normal distribution models, we
calculated the standardized values of each statistic in 95% and 99% areas in the distribution
curves (Table 2). The lower and upper cutoff
CV values in the 95% area were [0.25, 0.70], andthose in the 99% area were [0.21, 0.83]. As for
skewness, the lower and upper cutoff values
were [–0.18, 1.14] in the 95% area and [–0.39,
1.35] in the 99% area, respectively. The kurtosis values were [1.77, 4.57] in the 95% areaand [1.53, 5.31] in the 99% area, respectively(Table 3).

Correlation of the statistics
Scatter diagrams were made of each pair ofvalues of the CV, skewness and kurtosis, andtheir relationships were examined (Fig. 4). The
CV and kurtosis data were converted to logarithms, and then Pearson’s correlation coefficients were calculated. Correlation between the
CV and skewness or kurtosis was significant,which was positive between the CV and skewness, and negative between kurtosis and theCV. However, their coefficient correlation
values were small and the levels of their significance were low.

Discussion
We tried to simplify the 36 moment-characteristic values which quantify the velocity vector
to only 3 statistics per patient. Of the statistics,
the CV represents the extent of scalar distribution, not affected by the absolute value of thevector. The CV is theoretically null if all sca-

Table 3. Cutoff values of the CV, skewness and kurtosis in 95%
and 99% areas in the standardized normal and log-normal curves
Cutoff value [lower, upper]
95% Area
99% Areain the normal curvein the normal curveCV‡[ 0.25, 0.70]
Skewness†[–0.18, 1.14]
Kurtosis‡[ 1.77, 4.57]
CV, coefficient of variation.
†Normally distributed.
‡Log-normally distributed.

154[ 0.21, 0.83][–0.39, 1.35][ 1.53, 5.31]

Velocity vector of human body swaylars of the velocity are equal in every direction:that is, scalar difference by direction becomeslarger as the CV increases. So, we could treat

1.2

A

Skewness

0.8

0.4

0.0

r = 0.33

-0.4
0.2

0.4

0.6

0.8 1.0

CV
5.0

B
Kurtosis

4.0

3.0

2.0

r = 0.52
-0.4

0

0.4

0.8

1.2

Skewness
1.0
0.8

C

CV

0.6

0.4

0.2

r = —0.21
2.0

3.0

4.0

the CV as an index of directional preponderance of the velocity scalar. However, it is notpossible to evaluate which direction is predominant among the analyzed 36 directions of thevelocity vector.
The kurtosis shows the center of magnitude
distribution of the velocity vector, and is theoretically infinite if all scalars of the velocity are
equal in every direction: namely, velocity scalars come to approximate one another in magnitude as kurtosis grows larger. For these,
kurtosis could be an index of the absolute valuespecificity of the velocity scalar.
As we observed, values of skewness werenormally distributed, and those of CV andkurtosis were log-normally distributed. In thefields of medicine and biology, the standardized
normal or log-normal curves are widely used as
distribution models of measurements in asample population. Besides, the 3 statistics
were significantly correlated with one another,although the levels of significance were comparatively low. Therefore, we can utilize eachstatistic independently as an index.
The whole image of velocity vector can bequantitatively evaluated with these 3 statistics.
Moreover, the statistics are hardly affected by
the magnitude of the vector, because statistics
are independent of measurement units. On theother hand, physiological implications of thestatistics remain unclear; further, the statistics
provide no information on the direction of thevector, which is a defect. The absence of units
also disturbs the intuitive understanding of thestatistics.
In future studies, by collecting basic data
extensively, we would like to clarify the physiological significance of these statistics, and torealize the quantitative evaluation of the direction of velocity vector through correction of thedefect in using statistics.

5.0

Kurtosis

References

Fig. 4. Scatter diagrams of the CV and skewness(A), skewness and kurtosis (B), and kurtosis and CV(C). Pairs in A and C are significantly correlated
lowly: the correlation is positive in A and negative in
C. CV, coefficient of variation.

155

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156

Received May 7, 2001; accepted September 18, 2001
Corresponding author: Dr. Kei Nakahara