What a growth percentile actually means
A growth percentile answers one question: out of 100 children of the same sex and exact age in the reference population, how many would measure less than this child? A weight-for-age at the 30th percentile means 30 of those 100 weigh less and 70 weigh more. It says nothing on its own about health, and nothing about whether the child is growing well — a single point on a chart is a snapshot, and the clinically informative signal is the trajectory across several visits.
The z-score, also called the SD score, carries the same information on a different scale: how many standard deviations above or below the reference median the measurement sits. Percentile and z-score are one-to-one, so nothing is lost going between them, but the z-score behaves far better at the extremes. The gap between the 1st and the 0.1st percentile is invisible on a chart and obvious as a z-score (−2.33 against −3.09). That is why growth monitoring in malnutrition settings is done in z-scores and why WHO states its cut-offs that way.
Percentile is also the wrong scale to average or to difference. The change from the 50th to the 60th percentile is a smaller change in kilograms than the change from the 85th to the 95th, because the underlying distribution is denser in the middle. Differences and trends belong on the z-scale.
How the LMS method works, and why growth charts need it
If children's weights at a given age were normally distributed, a percentile would be a two-parameter calculation: subtract the mean, divide by the standard deviation, look up the normal table. They are not. Weight-for-age is strongly right-skewed at every age — there is a floor below which a living child cannot fall and no comparable ceiling above — so a symmetric model puts the low centiles in the wrong place, which is exactly where clinical decisions get made.
Cole's LMS method solves this with three smooth curves rather than two. At each age the reference distribution is summarised by M, its median; by S, a generalised coefficient of variation; and by L, the power of a Box-Cox transformation that makes the distribution normal. Applying that power, subtracting one and dividing by L·S converts the measurement into a z-score on a genuinely normal scale, at which point the standard normal distribution gives the percentile directly.
Each parameter has a readable meaning. M is the 50th centile you see printed on the chart. S is roughly the relative spread: an S of 0.11 says the distribution is about 11% wide in coefficient-of-variation terms. L encodes the skew: L = 1 means no transformation is needed and the distribution is already normal; L = 0 is the log-normal limit, where the formula collapses to ln(X/M)/S; and negative L, which is typical for weight, means right-skewed — the distance from the median up to +1 SD exceeds the distance down to −1 SD.
The published references are exactly these three curves tabulated by sex, measurement and age. The CDC 2000 growth charts for the United States and the WHO Child Growth Standards for children under five are both distributed as LMS tables, one row per half-month or month of age. That is where the three numbers you type into this calculator come from, and reading them off the right row is the only step that requires care.
Worked example: a 13.5 kg measurement against L = −0.2, M = 12.0, S = 0.11
Suppose the reference row for the child's sex and exact age gives L = −0.2, M = 12.0 kg and S = 0.11, and the child weighs 13.5 kg.
- Form the ratio. X/M = 13.5 / 12.0 = 1.125.
- Raise it to the power L. 1.125−0.2. Take logs: ln 1.125 = 0.1177830, multiply by −0.2 to get −0.0235566, exponentiate to get 0.9767187.
- Subtract one. 0.9767187 − 1 = −0.0232813.
- Divide by L × S. −0.2 × 0.11 = −0.022, so −0.0232813 / −0.022 = z = 1.0582. The two negatives cancel, which is what makes an above-median measurement give a positive z even with a negative L.
- Convert to a percentile. Φ(1.0582) = 0.85503, so the child is at the 85.5th percentile.
- Express it as a percentage of the median. 100 × 13.5 / 12.0 = 112.5%.
Now invert the formula to see the skew directly. X(z) = 12(1 − 0.022z)−5, since 1/L = −5. At z = +1 that is 12 × 0.978−5 = 13.412 kg; at z = −1 it is 12 × 1.022−5 = 10.763 kg. The step up from the median is 13.412 − 12.000 = 1.412 kg and the step down is 12.000 − 10.763 = 1.237 kg. Those are not equal, and no mean-and-SD model can reproduce that asymmetry — which is precisely the reason growth charts are published as LMS triples.
How to interpret a percentile once you have one
Start with the trajectory, not the point. A child tracking steadily along the 15th percentile from six months to three years is growing normally; a child who drops from the 75th to the 15th over the same period has crossed two major centile bands and that is the finding worth a clinician's attention. Growth references exist to detect change, and a single measurement gives you no change to detect.
Use the z-scale for anything quantitative. WHO's classification of nutritional status is stated in z-scores: below −2 SD is the moderate category of the relevant indicator and below −3 SD is the severe category, applied to weight-for-age (underweight), height-for-age (stunting) and weight-for-length or weight-for-height (wasting). Above +2 SD on weight-for-length is the overweight screening threshold and above +3 SD is obesity in children under five. These are screening cut-offs designed for population surveillance and for prompting individual assessment — they are not diagnoses.
Choose the right reference for the age. WHO's standards describe how children should grow under recommended conditions and were built from breastfed infants in six countries; they are the appropriate reference from birth to 24 months, and CDC recommends their use for that range in the United States. The CDC 2000 charts describe how a US reference population did grow, and are used from age two upward. Switching between them mid-childhood creates a discontinuity in z-scores, which is why the switch is made at a defined age rather than opportunistically.
Do not compare percentiles across measurement types as if they were the same scale. A child at the 20th percentile for weight and the 20th for height is proportionate; a child at the 20th for weight and the 80th for height is thin for their length, and it is BMI-for-age or weight-for-length that captures that, not either measurement alone.
Z-score to percentile conversion
| Z-score | Percentile | Common name |
|---|---|---|
| −3.00 | 0.13% | WHO severe cut-off |
| −2.50 | 0.62% | |
| −2.00 | 2.28% | WHO moderate cut-off |
| −1.881 | 3.00% | 3rd centile line |
| −1.645 | 5.00% | 5th centile line |
| −1.282 | 10.00% | 10th centile line |
| −1.00 | 15.87% | |
| −0.674 | 25.00% | lower quartile |
| 0.00 | 50.00% | median (M) |
| +0.674 | 75.00% | upper quartile |
| +1.00 | 84.13% | |
| +1.282 | 90.00% | 90th centile line |
| +1.645 | 95.00% | 95th centile line |
| +1.881 | 97.00% | 97th centile line |
| +2.00 | 97.72% | WHO overweight screen |
| +3.00 | 99.87% | WHO obesity screen |
Read the table in either direction: the calculator uses the left-to-right direction for a measured child and the right-to-left direction to place the centile lines for your chosen band.
Mistakes that produce a wrong percentile
- Reading the wrong row. L, M and S must all come from the same reference, the same sex, the same measurement type and the age nearest the child's exact age in months. A weight-for-age M with a height-for-age L is not a mixture of two references; it is nonsense.
- Using age in years. Growth references are tabulated in months, often in half-month steps in the first year, because a two-month-old and a four-month-old differ enormously. Round the age to the nearest tabulated point, or interpolate between the two neighbouring rows.
- Mixing units. The measurement and M must be in the same unit. WHO tables are metric throughout; a weight in pounds against an M in kilograms gives a wildly high z-score that still looks like a number.
- Using length where height is meant, or the reverse. Recumbent length runs slightly greater than standing height in the same child, and the references are built accordingly — length under two years, height from two years.
- Treating a single percentile as a verdict. Roughly 3% of perfectly healthy children sit below the 3rd centile by definition. The reference cannot distinguish a small healthy child from a child failing to thrive; only the growth trajectory and clinical context can.
- Interpolating percentiles instead of z-scores. If you need a value between two tabulated ages, interpolate L, M and S, or interpolate the z-score — never average two percentiles, because the percentile scale is not linear in the measurement.
This is a calculation tool, not a clinical assessment
This page performs the LMS arithmetic and nothing else. It does not hold the CDC or WHO reference tables, so it cannot check that the L, M and S you entered belong together or match your child's age. Growth concerns — a child crossing centile bands, a z-score below −2, a sudden change in trajectory, or any measurement that does not match how the child looks and behaves — belong with a paediatrician, health visitor or family doctor, who will weigh them against feeding, parental heights, prematurity, medical history and repeated measurements over time.
Where this sits among the other growth tools
Weight-for-age is the most familiar measure and the least specific. It cannot separate a child who is short and proportionate from one who is a normal length and underweight, which is why WHO reports stunting from height-for-age and wasting from weight-for-length as separate indicators built from the same LMS machinery. Run this calculator once per indicator, using the matching reference row each time, and read the three z-scores together.
From age two upwards, body mass index for age takes over as the standard adiposity screen, and the child BMI percentile calculator handles that scale directly. Below two years, weight-for-length does the same job, since BMI is unstable when length is changing this fast.
Height percentiles also feed forward. The Khamis-Roche and mid-parental methods estimate adult stature from current height, age and parental heights; the predicted adult height calculator covers that. And if you are working backwards from a pregnancy, the due date calculator gives the gestational age that determines whether a preterm infant's chronological age needs correcting before you look up a reference row at all — correction is conventionally applied until about two years of age for infants born preterm.
For adults the equivalent question is answered on an entirely different footing: adult references are cut-point based rather than centile based, which is why the BMI calculator reports categories rather than percentiles.