# THE EVOLUTION OF BODYMASS ADJUSTMENT FORMULAE

2-21-00

Periodically, the issue arises of comparing the strength of athletes of
different bodymass.   The following article summarises some of the more
important events in the evolution of strength-bodymass relationships and
their use in lifting or strength comparison.

Some of us have found the equations involved to be interesting and useful in
comparing treadmill-based performances.  Instead of simply dividing by
bodymass, we have used these lifting-based comparison methods to compare
results obtained for athletes of different bodymass on  treadmill or cycle
ergometers. Years ago, some exercise physiologists applied the two-thirds
power law (mentioned in the first paragraph  below) in an attempt to improve
upon bodymass adjustments made by simple division by bodymass. Since the
ability to execute work is related to lean body or muscle mass, we felt that
using strength-based formulae might be highly relevant.  Has anyone else
examined ergometry and other bodymass-dependent results in this way?

THE EVOLUTION OF BODYMASS ADJUSTMENT FORMULAE

(Ref: Siff M C & Verkhoshansky Y V "Supertraining", 1999)

Strength is related to the cross-sectional area of the muscles and,
consequently, indirectly to bodymass.  Therefore, the heavier the athlete,
the larger the load he can lift.   The athlete's bodymass is proportional to
the cube of its linear dimensions, whereas  a muscle's cross-sectional area
is proportional only to its square. From this basic dimensional analysis, the
mathematical relationship between maximum strength (F) and bodymass (B) may
be expressed as F = a.B ^ 2/3, where a is a constant, which characterises the
athlete's level of strength fitness (Lietzke, 1956).  Lietzke found that the
most accurate fit to data was obtained for an exponent of 0.6748, which was
close to the theoretical value of 0.6667.  This equation expresses with
modest accuracy the relationship between bodymass and results in the Olympic
lifts.

In the practical setting, Hoffman had already appreciated from 1937 the value
of the two-thirds power law in comparing the performances of weightlifters of
different bodymass and he annexed this equation as the 'Hoffman formula'.
More than ten years later, Austin considered the theoretical 2/3 exponent as
insufficiently accurate to describe the records of his day, so he produced
his 'Austin formula' with an exponent of 3/4.  More recently, several
researchers persisted with the two-thirds power law, including Karpovich and
Sinning (1971), who used current weightlifting records to demonstrate that
the exponent still remained fairly close to two-thirds.  Their equation,
however, offered only modest accuracy, with a mean error over all the
bodymass classes of 5.2% in interpolation and major inaccuracies in
extrapolation for the heavier lifters (e.g. the error at 125 kg bodymass was
14.7%).

Numerous attempts have been made since then to derive the closest possible
mathematical relationship between the Olympic lifts and bodymass (e.g. by
O'Carroll, Vorobyev and Sukhanov), but all equations invariably favoured
certain bodymass classes and competitive weightlifters strongly opposed to
comparisons of performance based on relative scores using any of the extant
formulae.

Consequently, in 1971, Siff and McSorley, an engineering student at the
University of Cape Town, South Africa, examined the possibility of fitting
different equations to current weightlifting records for all bodymass
divisions up to 110 kg.  Soon afterwards, McSorley prepared
computer-generated parabolic-fit tables to compare performances by
weightlifters of different bodymass.  In 1972 these tables were adopted by
the South African Weightlifting Union and were used for nearly a decade to
award trophies and select national teams.  In 1976 Sinclair of Canada
concluded similarly that a parabolic system offered the best means of
com-paring the strength of lifters of different bodymasses (Sinclair &
Christensen, 1976).

The McSorley and Sinclair parabolic systems were limited in that both were
most accurate for bodymasses up to 110 kg and, since they were based on world
records of no more than three successive years, the tables became inaccurate
whenever world records were broken.  To avoid these difficulties, it is
preferable to collect a database comprising the mean of the ten best lifts
ever achieved in each of the 11 bodymass classes in weightlifting history for
bodymasses up to about 165 kg (Siff, 1988).  Statistical regression
techniques revealed that various sigmoid (S-shaped) curves, such as the
logistic, hyperbolic tan and Gompertz functions, and a power law provide
highly accurate fits to the data (correlation coefficient R > 0.998).  The
simplest equation for practical application was found to be the following
power law equation:

Total lifted        T = a - b*B ^(-c)

where B = bodymass and a, b and c are numerical constants.

For weightlifting data up to 1988, the values of the constants for adult
lifters are:

a  =  512.245, b =  146230  and c =  1.605

The same power law equation applies accurately to powerlifting records (Siff,
1988).
For powerlifting data up to 1987, the values of the constants are:

Powerlifting Total:                    a   =  1270.4,  b = 172970,  c = 1.3925
Powerlifting Squat:                   a   =  638.01,  b = 9517.7,   c = 0.7911
Powerlifting Bench Press:        a  =   408.15,  b = 11047,     c = 0.9371
Powerlifting Deadlift:               a  =   433.14,  b = 493825,   c = 1.9712

To compare the performances of lifters of different bodymass, simply
substitute each lifter's bodymass in the relevant equations above to
calculate the Total (or lift) expected for a top world class lifter.  Then
divide the each lifter's actual Total by this value and multiply by 100 to
obtain the percentage of the world class lift achieved by each lifter.  This
method is also useful for monitoring the progress of an athlete whose lifts
and bodymass increase over a period of time, because it is pointless to do so
by referring simply to the increase in absolute mass lifted if the athlete's
bodymass has changed significantly.

The following websites discuss the application of several formulae for
comparing weightlifting and powerlifting performances, with some of the
details of the equations from "Supertraining" (Ch 3.3.5). The last quoted
website allows you to automatically calculate several bodymass corrected
Totals.

http://www.geocities.com/Colosseum/8682/formulas.htm
http://www.isu.edu/~andesean/wform.htm
http://www.qwa.org/stats/pwrrankings.asp

Similar formulae for strength comparison of women and juveniles appear in
"Supertraining" (some details of this book, including its Contents, are on
this website:  http://www.geocities.com/Colosseum/8682/siff.htm).

Dr Mel C Siff
Denver, USA
mcsiff@aol.com

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