Reading a Newborn's Growth Chart

Birth weights are roughly normal, centered near 3.3 kg with a standard deviation of about 0.55 kg. A doctor converts a baby's weight to a z-score and then a percentile, and can work backwards too: what weight marks the lightest 10%?

How it works

Percentiles and inverse normal calculations

In one sentence

If $X \sim N(\mu, \sigma^2)$ , the percentile of a value $w$ is $100\,\Phi\!\left(\frac{w-\mu}{\sigma}\right)$ , and the value at the $q$ th percentile is $\mu + z_q\sigma$ with $z_q = \Phi^{-1}(q/100)$ .

General formula

\text{percentile} = 100\,{\htmlData{sym=3}{\Phi}}\!\left(\frac{{\htmlData{sym=0}{w}} - {\htmlData{sym=1}{\mu}}}{\htmlData{sym=2}{{\sigma}}}\right), \qquad {\htmlData{sym=4}{w_q}} = {\htmlData{sym=1}{\mu}} + {\htmlData{sym=5}{z_q}}\,{\htmlData{sym=2}{\sigma}}

A percentile answers 'what share is below this?'. The inverse question, 'what value has this share below it?', reads the same area backwards.

What the symbols mean

$w$
Baby's weight
The newborn's weight in kilograms.
$\mu$
Mean birth weight
About 3.3 kg in this model.
$\sigma$
Standard deviation
About 0.55 kg: the typical distance from the mean.
$\Phi$
Standard normal CDF
The area under the standard bell curve to the left of a z-score.
$w_q$
Weight at percentile q
The weight with q% of babies lighter.
$z_q$
z-score for percentile q
The z-score with area q/100 to its left, from the inverse normal.

Model birth weight as normal with mean $\mu \approx 3.3$ kg and standard deviation $\sigma \approx 0.55$ kg.
To place a baby, standardize: $z = (w - \mu)/\sigma$ . The percentile is the area to the left, $100\,\Phi(z)$ .
To find the weight at percentile $q$ , run it backwards: find the $z_q$ with area $q/100$ to its left (the inverse normal), then $w_q = \mu + z_q\sigma$ .
Real birth weights have a heavier left tail than the bell curve, from premature births, so clinics use growth charts built from measured data rather than a pure normal model.

Learn the math

Using the Normal DistributionStats · Lesson 22, Continuous Distributions and the Normal Curve