Figures published c. 2005 by the World Bank. The Millennium Development Goals program concluded in 2015; this site is preserved as a historical record.
The WHO Global Health Observatory says something about the under-five mortality rate that most explainers leave out. In its own metadata, the indicator is “strictly speaking not a rate” but a probability of death derived from a life table. The infant mortality rate formula carries the same caveat.
Three rates get quoted together and confused constantly. The neonatal mortality rate covers the first 28 completed days of life. The infant mortality rate covers birth to exact age one. The under-five mortality rate covers birth to exact age five. All three are expressed per 1,000 live births.
What follows is each formula, the denominator all three share, a worked example that can be reproduced by hand, the sources, and the reasons a calculated figure will not match a published one.
Neonatal, infant and under-five mortality: what each rate measures
The three rates differ in one respect: the age window. Those windows are nested rather than separate, so a death on day three counts in all three measures.
| Rate | Age window | Denominator | Unit | SDG indicator |
|---|---|---|---|---|
| Neonatal mortality rate (NMR) | First 28 completed days | Live births | Per 1,000 live births | 3.2.2 |
| Infant mortality rate (IMR) | Birth to exact age 1 | Live births | Per 1,000 live births | None |
| Under-five mortality rate (U5MR) | Birth to exact age 5 | Live births | Per 1,000 live births | 3.2.1 |
Definitions follow the WHO Global Health Observatory indicator metadata registry. Indicator numbers follow SDG Target 3.2.
The nesting shows in the figures. In the UN Inter-agency Group for Child Mortality Estimation (UN IGME) series for 2024, the global neonatal rate was 17.2 per 1,000 live births, the infant rate 27.7, and the under-five rate 37.4, as published in the World Bank World Development Indicators. Each figure exceeds the last because each window contains the one before it.
Two derived measures fill the gaps. The post-neonatal mortality rate covers 28 days to the first birthday, calculated as the infant rate minus the neonatal rate. The child mortality rate covers ages 1 to 4, the part of child mortality above the infant window.
Why the denominator is live births and not total population
Writing “per 1,000 population” is the most common error in this query set. All three rates divide by live births. Not by mid-year population, not by the child population, not by total pregnancies.
The logic is exposure. The group at risk of dying before age five is the cohort born alive. The crude death rate uses mid-year population, which is where the confusion starts.
The denominator has a precise definition. WHO defines a live birth as “the complete expulsion or extraction from its mother of a product of conception, irrespective of the duration of the pregnancy, which, after such separation, breathes or shows any other evidence of life,” quoted in The DHS Program Guide to DHS Statistics and attributed to WHO, 1950 and 1992.
Because the test is evidence of life after separation, a very early birth recorded as a stillbirth in one country and a live birth in another shifts both the numerator and the denominator.
Neonatal mortality rate formula
The neonatal rate is the one place where the simple ratio and the life-table probability effectively agree.
NMR = (deaths during the first 28 completed days of life / live births in the same period) x 1,000WHO states it in exactly those terms. Neonatal deaths subdivide into early neonatal deaths in the first 7 days and late neonatal deaths after the 7th day but before the 28th completed day.
The DHS Program treats the rate as the component death probability for the age segment 0 months, multiplied by 1,000. Across 28 days the ratio and the probability sit close enough that the count-based statement is standard. That is not true of the under-five rate.
Global figures for 2024, UN IGME series: 17.2 per 1,000 live births, and 2.3 million neonatal deaths, 47 per cent of all under-five deaths. Care at birth dominates this window, which ties it closely to maternal health.
Infant mortality rate formula
Two formulas are in circulation. Both are legitimate, and they answer slightly different questions.
The first is the vital statistics approximation, and it is what a country with near-complete civil registration publishes:
IMR (approximate) = (infant deaths in a year / live births in the same year) x 1,000The second is the definition used by WHO, the World Bank and UN IGME:
IMR = (1 - probability of surviving from birth to exact age 1) x 1,000WHO words it as the probability of a child born in a specific year or period dying before age one, if subject to the age-specific mortality rates of that period.
The DHS Program method: take the component death probabilities for the age segments 0, 1-2, 3-5 and 6-11 months, subtract each from 1 for a survival probability, multiply the four together, subtract the product from 1, and multiply by 1,000.
Post-neonatal mortality falls out as a residual: PNMR = IMR minus NMR. In the UN IGME series, the global infant mortality rate was 27.7 per 1,000 live births in 2024, down from 64.0 in 1990.
Under-five mortality rate formula: a probability, not a simple ratio
WHO writes: “Under-five mortality rate as defined here is strictly speaking not a rate (i.e. the number of deaths divided by the number of population at risk during a certain period of time) but a probability of death derived from a life table and expressed as rate per 1000 live births.”
The World Bank agrees independently. Its definition of series SH.DYN.MORT is the probability per 1,000 that a newborn will die before reaching age five, if subject to the age-specific mortality rates of the specified year.
So the formula is a survival product, not a division:
U5MR = (1 - [p1 x p2 x p3 x ... x pn]) x 1,000Each p is the probability of surviving one age segment. Together they span birth to exact age five. The DHS Program method:
- Split the first 60 months into age segments. DHS uses 0, 1-2, 3-5, 6-11, 12-23, 24-35, 36-47 and 48-59 months of completed age.
- For each segment, calculate the component death probability: deaths in that range divided by children surviving at its start.
- Subtract each death probability from 1 to get a survival probability.
- Multiply all eight survival probabilities together.
- Subtract the product from 1, then multiply by 1,000.
The simple ratio fails for two reasons. The denominator does not line up in time: children dying before age five in one year were born across five birth cohorts, but the ratio divides by a single year of births. DHS is direct about it. The vital statistics approach estimates “a rate of mortality but not a probability,” because “a variation in the number of births with time will change the rate without changes in the underlying probabilities.” Correcting that requires separation factors.
Probabilities also compound rather than add, and the example below shows the size of that gap. The approximation is acceptable where annual births are stable and registration is close to complete. It degrades as births move and as mortality rises.
A worked example of the under-five mortality rate calculation
The component probabilities below are hypothetical. They show the arithmetic and are not any country’s published data.
| Age segment (months) | Death probability (q) | Survival probability (1 – q) |
|---|---|---|
| 0 | 0.0180 | 0.9820 |
| 1-2 | 0.0060 | 0.9940 |
| 3-5 | 0.0045 | 0.9955 |
| 6-11 | 0.0050 | 0.9950 |
| 12-23 | 0.0055 | 0.9945 |
| 24-35 | 0.0025 | 0.9975 |
| 36-47 | 0.0015 | 0.9985 |
| 48-59 | 0.0010 | 0.9990 |
Work through the four rates in order.
- Neonatal. 0.0180 x 1,000 = 18.0 per 1,000 live births.
- Infant. Multiply the first four survival probabilities: 0.9820 x 0.9940 x 0.9955 x 0.9950 = 0.966857. Then (1 – 0.966857) x 1,000 = 33.1.
- Post-neonatal. 33.1 – 18.0 = 15.1.
- Under-five. The product of all eight survival probabilities is 0.956739. Then (1 – 0.956739) x 1,000 = 43.3.
Compare that with the shortcuts. Adding the eight death probabilities gives 44.0, not 43.3. Adding the infant rate of 33.1 to the child mortality rate for ages 1 to 4, which is 10.5 here, gives 43.6, not 43.3.
A child has to survive each earlier segment to be exposed to the next, so the survival probabilities multiply. At these levels the gap is under one death per 1,000 live births. It widens as mortality rises, so the distinction matters most where child mortality is highest.
What these rates do not measure, and why your figure may differ
Three things sit outside the measure. Stillbirths, because all three rates take live births as the denominator. Deaths after exact age five. Cause of death and non-fatal illness, since mortality rates are used partly because data on the incidence and prevalence of disease are frequently unavailable. For what the rate does not show, see the infections that cause most under-five deaths.
Four reasons explain a gap between a hand calculation and a published figure.
- The published figures are modeled. UN IGME comprises UNICEF, WHO, the World Bank and the UN Population Division, with universities and research institutes. It fits a country-specific regression of mortality rates against their reference dates to reconcile sources that disagree.
- Source quality varies. A complete vital registration system covers at least 90 per cent of vital events and is the best source of age-specific mortality data, but complete systems remain uncommon in developing countries. Survey-based rates are period rates, usually for the five or ten years before the survey.
- Age at death heaps at 12 months. Because age at death is recorded in completed months or years, deaths reported at 12 months are classified as child deaths rather than infant deaths, biasing infant mortality down and child mortality up. Under-five rates are little affected, because the boundary falls inside the interval.
- Series get revised. The UN Millennium Development Goals page reports the global under-five mortality rate falling from 90 in 1990 to 43 in 2015. The current UN IGME series gives 93.5 for 1990 and 42.9 for 2015. Same indicator, different vintage.
MDG Target 4.A ran from 1990 to 2015. Its successor, SDG Target 3.2, carries the under-five rate as indicator 3.2.1 and the neonatal rate as indicator 3.2.2, alongside the rest of the development indicators that made the transition.
FAQ
Is the infant mortality rate per 1,000 live births or per 1,000 population?
Per 1,000 live births. So are the neonatal and under-five mortality rates. The crude death rate is the measure that uses mid-year population as its denominator, which is where the mix-up usually starts. Substituting population for live births produces a figure wrong by roughly an order of magnitude.
Why is the under-five mortality rate called a rate if it is a probability?
The name is conventional rather than technically accurate. WHO’s indicator metadata states that the under-five mortality rate is strictly speaking not a rate, but a probability of death derived from a life table and expressed per 1,000 live births. The same caveat appears in the WHO entry for the infant mortality rate.
Can I calculate the under-five mortality rate by dividing under-five deaths by live births?
That gives an approximation, not the indicator as defined. It produces a rate rather than a probability, and it moves when the annual number of births moves even if underlying survival is unchanged. The approximation holds where births are stable and civil registration is close to complete.
What is the difference between neonatal, infant and under-five mortality?
The age window, and the windows nest. Neonatal covers the first 28 completed days of life, infant covers birth to exact age one, and under-five covers birth to exact age five. Every neonatal death is also an infant death and an under-five death, which is why the three rates rise in that order.
Are these still Millennium Development Goal indicators?
No. MDG Target 4.A ran to 2015. Under-five mortality is now SDG indicator 3.2.1 and neonatal mortality is SDG indicator 3.2.2, with 2030 country targets of 25 and 12 per 1,000 live births. The infant mortality rate is still published by WHO, the World Bank and UN IGME, but it is not an SDG indicator.