Weight for Height during Growth, Useful Formulas
to Insert into the Pediatrician’s Smart Phone
Pierre M Braillon*
Department of Pediatric Imaging, University Claude Bernard, France
Submission: February 15, 2017; Published: April 03, 2017
*Corresponding author: Pierre M Braillon, PhD, MD, Department of Pediatric Imaging, Hospital “Femme-Mère-Enfant” and University Claude
Bernard – Lyon, France.
How to cite this article: Pierre M B. Weight for Height during Growth, Useful Formulas to Insert into the Pediatrician’s Smart Phone. Acad J Ped Neonatol.
2017; 4(2): 555689. DOI: 10.19080/AJPN.2017.04.555689
The objective of this work was to define simple formulas to calculate the ideal weight (W) for height (H) of children and adolescents aged 2
to 18 years. Using data from 12 countries, two formulas of the form W=A * exp (B*H) have been established (W in kg, H in m). The values for A
and B were 2.120±0.112 kg and 1.993±0.05 m-1, respectively, in girls, and 2.379±0.130 kg and 1.895±0.060 m-1 in boys. The relevance of these
formulas has been controlled by comparing the calculated weight to the measured weight, for children living in 20 other countries, and with
various ranges of age. The formulas can be easily introduced into a computer or a smartphone. They appear useful for pediatricians to follow
the growth of children, especially those with disease that affects their body weight, the rule being that a gain in H of 5 cm would lead to a weight
gain of 10% and a gain of 35 cm a doubling of the body weight.
In pediatrics, height (H) and weight (W) are the two most
commonly used anthropometric indicators of growth. Usually, the
measured H and W values in a given subject are then compared
to the mean values for age read on two separate specific growth
charts, for H and W, in order to define the so-called “Z scores”. They
are also used to calculate the body mass index (BMI). However,
in many situations, none of these parameters are sufficiently
informative, and their interpretation can be misleading. This
is especially the cases in abnormal growth velocity, or when
comparing the growth of children at the time of puberty as H can
be significantly different for the same age. This is also the case for
children from different ethnic origins or those for whom the age is
not precisely known.
Therefore the relationship between W and H is of real interest
to the pediatrician. However, most of the relationships between H
and W were established for adults. They are usually linear and are
not applicable in pediatrics. Broca  was the first to define such
a relationship. Years later Devine , Robinson et al. , Miller et
al. , and Hamwi  published alterations to the Broca’s formula.
However, these corrections were still not adapted to studies in
children. Finally, few practical data W(H) to be used in pediatrics
can be found. They are usually given as growth charts established
for a limited range of ages, rather than the entire growth period
The aims of this work were
(i) to calculate the best correlations between H and W
during growth by using data published in countries from
diverse geographical regions, and given as numerical data
from ages 2 to 18 years,
(ii) to estimate the possible differences between the mean
W values calculated on this range of ages in these different
(iii) to define general formulae for W against H which can be
easily inserted in a computer, and
(iv) to test the validity of the general W(H) formula in
populations for whom W and H data are given on a much
limited range of ages than 2-18 years or from specific limited
The results obtained here should be considered only as
practical tools for daily practice in pediatrics. No attempt was
made to consider this work as a statistical study.
Anthropometric data from different countries giving H and W
in numerical values (and not as graphical charts) for children and
adolescents aged 2 to 18 years were used. Data from 12 countries
were selected, namely: China , France , Germany , Italy , Korea , Netherlands , Norway , Spain , Sweden
, Turkey , United Kingdom , and USA .
The mean data values given in these studies for W (kg) were
plotted against the mean corresponding values for H (m) in each
male and female population, and they were fitted with the best
correlation. A Kruskal-Wallis one way analysis of variance on
ranks was then performed (SigmaStat 3.5; Systat Software, Inc.
GmbH) to compare the different groups. Finally, general formulae,
W (H), for W against H, that might be used in all these male and
female populations were defined.
In order to test the pertinence of these formulae, data from
20 other countries or limited populations, were used. These data
were given in studies from Argentina , Australia , Brazil
, Cyprus , Denmark , Finland , Paris-France ,
Jena-Germany , Hungary , India , Iran [28, 29], Japan
[30-32], Malaysia , Mexico , Pakistan , Poland ,
Serbia , Seychelles , South Africa , and Taiwan .
They covered a limited range of ages during the growth period of
time. Also were included data from 2 populations with specific
anthropomorphic characteristics or way of life and nutrition,
namely Pygmies , and Aboriginal communities .
When plotted against H, W values showed similar and regular
growth shape for each population studied. They were best fitted
by exponential curves (in all cases r²> 0.99) with equations
computed as W= A * exp (B*H).
Values for the coefficients A and B are given in Table 1. In
their range of data, A and B values were inversely and similarly
correlated together (B=- 0.437 * A + 2.935; r²=0.89 in males, and
B=– 0.438 * A + 2.920; r²=0.81 in females). The mean values for
the product A.B, a main parameter in the change of W with H gain
(e.g. DW/DH), were 4.502±0.122 in males, and 4.218±0.128 in
females, showing low differences between the studied populations
(coefficients of variation: 2.7% and 3.0%, respectively). A
Kruskall-Wallis is one way analysis of variance on ranks indicated
that no statistically significant difference was found in the 12
groups of W values calculated with their own equation for H from
80 cm to 170 cm in females, and from 80 cm to 180 cm in males
(p=0.999). Therefore two general equations were defined by using
A and B mean values (±1 standard deviation): A=2.120±0.112 kg,
B=1.993±0,054 m-1 in girls and A=2.379±0.130 kg, B=1.895±0.060
m-1 in boys (1), H and W being expressed in meters and kilograms,
respectively (H given with 3 digits).
(W and A in kg, H in m and B in m-1).
CV (%): Coefficient of Variation in percent.
The relevance of these equations was tested by comparing W
data given in 20 more studies to the calculated mean weight for
the same corresponding height (Wcalc). This is illustrated by the
results given in table 2. Values close to 1 were found for Wcalc/W
in most cases or they allow to estimate the anthropomorphic
differences between populations or sub-populations when they
differ significantly from 1 (see, for example, South Africa, Table 2).
No precise numerical data for W and H were found for Pygmies
or Aboriginal communities. However, from the estimation that
can be made on the graphs related to the growth trajectories in
Baka Pygmies , it appeared that the mean measured height
and weight at age 10 are close to 1.18m and 22.0kg, in both
girls and boys, while W was calculated as 22.3Kg in both sexes.
At age 18 (H ~1.47 in girls, and 1.49 in boys) the mean weights
were calculated as 39.7Kg in girls, and 40.0kg in boys, values
approximately 6% lower than the measured mean W values.
In Aboriginal populations, from the graphs presented by the
authors , H and W values can be appreciated at ages 5, 10
and 15 years as: 1.08m, 17,5kg - 1.38m, 32,6kg - 1.62m, 52,4kg
in females, and 1,05m, 16.8kg - 1,40m, 33.9kg– 1.68m, 56.4kg in
males. The corresponding calculated values for W were: 18.2kg,
33.2kg, 53.5kg, respectively, in females, and 17.4kg, 33.8kg, 57.4kg
in males, the differences between W and Wcalc ranging from 0%
From the equation W=A *exp (B*H), it can be seen that a
weight gain of approximately 10% is expected for a gain of height
DH=0.05m, while a twofold change of W occurs for DH=0.35 m (for
example, from 1.05 m at age 4-5 years to 1.40 m at the beginning
of the puberty spurt).
The main goal of this work was to define useful relationships
between the weight and height of children aged from 2 years
to 18 years . A similar idea was proposed by Ehrenberg to
estimate the logarithm weight of children aged from 5 to 13
years. Severe criticism was made by Cole (43) regarding this kind
of preoccupation. Clearly, the period of time from birth to age
2 is a period with a particular dynamic of growth (that may be
represented by the first stage of a logistic curve) and should be
considered separately. It was suggested to use expressions of the
form W/Hp, or W=k * Hp. Unfortunately, p value is not constant
along the entire growth period of time [30,44,45]. The choice of
exponential functions greatly simplifies the computing process.
We used data published in several countries as growth
reference, and obtained for most of them by using the Cole &
Green  LMS method. To calculate mean values from such data,
which are still means, is of course a limitation not well accepted
in statistics. However, as previously said, the aim of this work was
first to define simple tools that can be useful to those in charge of
children during their growth, and no attempt to make a precise
statistical study was made.
Two equations of the form W=A* exp (B*H) were obtained.
These equations, easy to enter in any computer or smartphone,
allow pediatricians to estimate the mean ideal weight for height
of a boy or girl from age 2 to age 18 with an accuracy sufficient
in clinical practice. They could be particularly useful to follow the
body weight changes in children with nutrition diseases, especially
anorexia or obesity. It is to be note that it is not necessary to
precisely know the patient’s age, nor his/her Tanner stages ,
and that the changes in growth velocity at puberty are implicitly
taken into account in this method of weight assessment. Moreover,
the equations obtained here appeared correctly adapted for the
analysis of growth whatever the ethnic origin of the subjects
was, even in specific pediatric populations, when growth can
be considered as possibly related to a particular way of life and
To conclude, the ideal weight W for height H could be calculated
with a good accuracy in children and adolescents aged from two
years to eighteen years, with the formulae W=2.120*exp(1.993*H)
in females and W=2.329*exp(1.895*H) in males. These formulae,
which are effective in any population can be easily inserted
in a computer or a smartphone. They could be of help to the
pediatrician to follow the growth of children suffering from
diseases that affect their body weight, the regular rule being that
an increase in the weight of 10%, and its doubling, are expected
for every five and thirty-five centimeters of gain in H, respectively.