EFW Z-Score and Centile Calculator
EFW Z-Score and Centile Calculator
The z-score and the exact centile for an estimated fetal weight, against the reference mean and standard deviation from YOUR growth standard. The page holds no growth chart — it does the arithmetic the report leaves out.
EFW z-score and centile
Your reference mean and SDEFW 2,100 g; illustrative reference mean 2,400 g and standard deviation 300 g
Formula
centile = 100 × Φ(z), where Φ is the standard normal cumulative distribution
- the reference mean and SD are yours
- from the growth standard your unit uses, at the gestational age of this scan. This page holds no growth chart, deliberately: every published standard is an age-specific distribution given as a table or a skewed model, and a curve fitted to one would misclassify fetuses at the extremes, which is the only place a centile changes anything
- Φ(z)
- computed here from Abramowitz and Stegun’s rational approximation 26.2.17, built out of exp, sqrt and arithmetic because this engine has no normal distribution function. The published bound on the approximation is an absolute error under 7.5 × 10⁻⁸; measured against the true function across six standard deviations either side of the mean, the realised maximum error here is 7.45 × 10⁻⁸. That is about a ten-millionth of a centile, which is far finer than any growth standard
- deriving an SD from centiles
- divide the distance from the mean to a published centile by that centile’s z: 1.2816 for the 10th and 90th, 1.6449 for the 5th and 95th, 1.8808 for the 3rd and 97th. If the lower and upper derivations disagree, the standard is skewed there
- the normality assumption
- the one real limitation. Turning a mean and an SD into a centile assumes the reference distribution is normal at that age. Most published fetal weight standards are mildly skewed and several model it explicitly, so agreement with the standard’s own centiles is close in the middle and loosens in the tails
- refuses beyond six SD
- a z beyond plus or minus 6 returns nothing: the centile would print as 0.0 or 100.0 and imply a precision no growth standard has, and the likeliest explanation is a reference mean from the wrong gestational age
Worked example
EFW 2,100 g; illustrative reference mean 2,400 g and standard deviation 300 g
z = (2100 − 2400) ÷ 300 = −1.00
Φ(−1.00) = 0.158655, so the centile is 15.9
That is the teaching point of this example: a fetus 300 g below the reference mean, which reads as noticeably small, sits at the 16th centile and is NOT small for gestational age on the conventional 10th-centile definition
At z = 0 the page returns exactly 50.0; at z = −1.6449 it returns 5.0 and at z = +1.6449, 95.0 — the checks worth making against any implementation of a normal centile
To reach the 10th centile at this mean and SD the weight would have to fall to 2,015 g, and the 3rd centile is 1,836 g
The caveat: the estimated fetal weight itself has a 95% interval wider than plus or minus 14%, about 1,800 to 2,400 g here — from below the 3rd centile to the 50th. The arithmetic is exact; the input is not
Z-scores and centiles, computed by this page’s own approximation
| Z-score | Centile |
|---|---|
| -2.5758 | 0.50 |
| -1.9600 | 2.50 |
| -1.8808 | 3.00 |
| -1.6449 | 5.00 |
| -1.2816 | 10.00 |
| -1.0000 | 15.87 |
| -0.5000 | 30.85 |
| 0.0000 | 50.00 |
| 0.5000 | 69.15 |
| 1.0000 | 84.13 |
| 1.2816 | 90.00 |
| 1.6449 | 95.00 |
| 1.9600 | 97.50 |
Deriving a standard deviation when your standard publishes only centiles
| Published centile you have | Derivation | Divisor |
|---|---|---|
| 3rd centile | SD = (mean − C3) ÷ 1.8808 | 1.8808 |
| 10th centile | SD = (mean − C10) ÷ 1.2816 | 1.2816 |
| 90th centile | SD = (C90 − mean) ÷ 1.2816 | 1.2816 |
| 97th centile | SD = (C97 − mean) ÷ 1.8808 | 1.8808 |
| 5th or 95th centile | SD = |C − mean| ÷ 1.6449 | 1.6449 |
The arithmetic a growth report leaves out, without pretending to hold a growth chart
A growth scan report prints an estimated fetal weight and usually a centile. What it rarely prints is the z-score, the distance from the reference mean in standard deviations, which is the form the consensus growth-restriction criteria and most of the research literature are written in. This page converts between them, exactly, from the reference mean and standard deviation of whatever standard your unit uses.
It deliberately does not hold a growth chart. Hadlock’s weight-for-age centiles, INTERGROWTH-21st, the WHO charts and customised standards such as GROW are all age-specific distributions published as tables or as parametric models with skewness. A page that fitted a smooth curve to one would return a confident centile that agreed near the median and diverged at the extremes — the only place a centile changes anything. Taking the mean and standard deviation as inputs keeps the standard in the reader’s hands and makes the arithmetic auditable.
Two things will mislead a reader of this number more often than the mathematics will. The input: an estimated fetal weight has a 95% interval wider than 14% of birth weight and overestimation is the commoner direction, so a centile computed from it is a point on a wide distribution and not a measurement. And the choice of standard. Fetal size standards are not interchangeable: Hadlock, INTERGROWTH-21st, the WHO charts and customised standards such as GROW were built on different populations, and a fetus can cross a centile threshold purely by changing standard. In 117,027 UK singleton pregnancies 7.2% were small for gestational age by GROW and 2.4% by INTERGROWTH-21st, two-thirds of the GROW cases being missed (Hugh et al., Ultrasound Obstet Gynecol 2025; the authors work for the institute that developed GROW). Record which standard a centile came from. So record the standard alongside the centile, read the consensus criteria rather than a centile alone, and check the gestational age the reference mean was taken from before concluding anything — a mean from the wrong week makes a normally grown fetus look small, and the redating rules explain why the answer is not to move the dates. This page computes a number, or applies published criteria, and names the body that publishes them. It renders no clinical decision: nothing here says whether this fetus is growth restricted, whether a scan should be repeated, whether labour should be induced or when a pregnancy should be delivered. An obstetric measurement is read alongside the whole clinical picture — history, serial trend, the other biometry — and never instead of it. It supports a clinician’s judgement rather than replacing it.
Frequently asked questions
Where do I get the reference mean and standard deviation?
From the growth standard your unit uses, at the gestational age of the scan. If it publishes centiles instead of a standard deviation, divide the distance from the mean to a published centile by that centile’s z: 1.2816 for the 10th or 90th, 1.6449 for the 5th or 95th, 1.8808 for the 3rd or 97th.
Why does this page not include a growth chart?
Because every published fetal growth standard is an age-specific distribution given as a table or a skewed model, and this engine has no table interpolation. Fitting a curve to one would produce a centile that looked right near the median and was wrong at the extremes, where the decision sits.
How is the centile computed without a normal distribution function?
From Abramowitz and Stegun’s rational approximation 26.2.17, built out of exp, sqrt and arithmetic. Its published error bound is under 7.5 times ten to the minus eight; measured against the true distribution over six standard deviations either side of the mean, the realised maximum error is 7.45 times ten to the minus eight — about a ten-millionth of a centile.
Is a z-score from a mean and SD the same as the standard’s own centile?
Close in the middle, looser in the tails. It assumes the reference distribution is normal at that age, and most published standards are mildly skewed. Past about three standard deviations, prefer the standard’s own centile.
Does a centile below the 10th mean growth restriction?
No. Below the 10th centile is small for gestational age, which is not the same thing. In the Delphi consensus definition the 10th centile is a contributory criterion that must be combined with other findings; only below the 3rd is it solitary.
Related calculators
References
- Zelen M, Severo NC. Probability functions. In: Abramowitz M, Stegun IA, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington, DC: National Bureau of Standards; 1964: eq. 26.2.17.
- Bryc W. A Uniform Approximation to the Right Normal Tail Integral. Preprint 1/2000, University of Cincinnati.
- Gordijn SJ, Beune IM, Thilaganathan B, et al. Consensus definition of fetal growth restriction: a Delphi procedure. Ultrasound Obstet Gynecol. 2016;48(3):333-9.
- Hugh O, Butler E, Ellson H, Mytton J, Gardosi J. Small-for-gestational age according to INTERGROWTH-21st fetal weight standard misses most pregnancies at risk of stillbirth identified by GROW. Ultrasound Obstet Gynecol. 2025.
Not medical advice. For healthcare professionals and education. Reference intervals vary by laboratory and assay — always use your own laboratory's. Never base a dose or a treatment decision on this page alone. Full disclaimer at calcengines.com/disclaimer/