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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