Mortality Rate

Mortality Rate in Population Statistics and Health Economic Models

A mortality rate sets a count of deaths against the size of the group that produced them, either its population at the middle of the period or the person-years its members spent at risk. Health economists use mortality rates in burden of disease work, in years of life lost and in the death transitions of lifetime decision models, so the choice of denominator and age structure carries through to life-years and QALYs. Official statistics report crude, age-specific and age-standardised versions, and each answers a different question about the same deaths. A rate also differs from the probability of dying within a model cycle, and converting one into the other is a routine source of error.

Numerator, denominator and time at risk

The CDC course Principles of Epidemiology in Public Health Practice gives the general form as deaths during a period divided by the size of the population among which they occurred, multiplied by a power of ten. When the rate comes from death certificates, the denominator most commonly used is the population at the middle of the period.

$$m = \frac{D}{P} \times K$$

where $m$ is the mortality rate, $D$ is the number of deaths during the period, $P$ is the mid-period population among which they occurred, and $K$ is the reporting multiplier, such as 1,000 or 100,000.

In a cohort or trial, people enter and leave follow-up at different times, so the denominator is observation time. Breslow and Day, in the IARC monograph on cohort studies, describe summing each subject's observation time within each age and calendar period category to obtain the person-years that form the denominators of the rates. The CDC course notes that person-time treats 10 persons followed for one year as equal to one person followed for 10 years, an assumption that is often not valid when risk rises with age.

$$\hat{\lambda} = \frac{D}{T}$$

where $\hat{\lambda}$ is the estimated mortality rate per person-year, $D$ is the number of deaths observed, and $T$ is the total person-years at risk summed over all subjects.

The calendar year of a death also needs stating. The Office for National Statistics (ONS) user guide to mortality statistics counts a death only once it is registered, and of the 568,613 deaths registered in England and Wales in 2024, 524,936 occurred in 2024.

Crude, specific and proportionate measures

The crude mortality rate counts deaths from all causes in the whole population. A cause-specific rate keeps the whole mid-period population as its denominator and restricts only the numerator, so it differs from case fatality, which the CDC course describes as a proportion of persons with a condition who die from it.

Age-specific rates restrict both numerator and denominator to one age group. The ONS guide defines them as deaths in the age group per 1,000 population in the same age group, using mid-year population estimates except at ages under 1 year, where live births are used.

Some measures with "mortality" in their name are not rates in this sense. The CDC course notes that the infant mortality rate divides infant deaths by live births in the same period, so some deaths in the numerator belong to children born in the previous year, and that proportionate mortality uses all deaths as its denominator rather than the population in which the deaths occurred. An all-cause rate taken from national life tables is the subject of the all-cause mortality page, which explains how such tables enter models as background risk.

Direct age standardisation

Death rates rise steeply with age, so an older population can have a higher crude rate while its members die at the same or lower rates age band by age band. The World Health Organization (WHO) indicator metadata notes that two populations with the same age-specific rates will have different overall rates if their age distributions differ, and defines the age-standardised rate as a weighted average of the age-specific rates per 100,000 persons, with weights equal to the proportions in each age group of the WHO standard population.

$$ASR = \frac{\sum_{k} w_{k} m_{k}}{\sum_{k} w_{k}}$$

where $ASR$ is the age-standardised mortality rate, $m_{k}$ is the observed mortality rate in age group $k$, $w_{k}$ is the number of people in age group $k$ of the standard population, and the sums run over all age groups.

The CDC course shows the effect with two states in 2002. Alaska's crude mortality rate was 472.2 per 100,000 against 1,005.7 in Florida, because Alaska's population was considerably younger, yet the age-adjusted rates were 794.1 in Alaska and 787.8 in Florida.

The ONS applies the direct method with the European Standard Population (ESP), a hypothetical population that is the same for males and females. Eurostat revised the 1976 ESP in 2013, and the 2013 structure gives greater weight to older ages, so ONS states that rates based on the two versions are not comparable. Breslow and Day add that the ranking of two populations can depend on the standard when their age patterns differ, and recommend published sets of weights so that series can be compared. Indirect standardisation is covered on the standardised mortality ratio page.

Precision and small numbers of deaths

For national rates, Breslow and Day judge sampling error to be of minor importance next to errors in coding the cause of death and in estimating the population denominators. For smaller groups they treat the variance of an age-specific death count as estimated by the count itself, with person-years treated as fixed, so the standard error of an age-specific rate is the square root of the deaths divided by the person-years. With independent age groups, the variance of a directly standardised rate is then the sum of each squared weight times deaths, divided by the squared person-years.

$$SE(ASR) = \sqrt{\sum_{k} \frac{v_{k}^{2} d_{k}}{n_{k}^{2}}}$$

where $SE(ASR)$ is the standard error of the standardised rate, $v_{k}$ is the proportion of the standard population in age group $k$, $d_{k}$ is the number of deaths in that group, and $n_{k}$ is its population or person-years.

Breslow and Day point out that the fixed weights ignore how precisely each age-specific rate is estimated, so a small age group with a large weight can dominate the error. The ONS guide does not calculate age-standardised rates where there are fewer than 10 deaths in a cell or crude rates where there are fewer than 3, and it marks rates based on fewer than 20 deaths as potentially unreliable.

Rates and probabilities in decision models

A state-transition model needs the probability of dying during each cycle, and a mortality rate is not that probability. The ONS guide to national life tables distinguishes the central rate of mortality, written $m_x$, which is deaths at age $x$ over the average population at that age, from $q_x$, the probability that a person aged exactly $x$ dies before reaching $x + 1$. For ages above 1 year, ONS calculates the probability from the rate.

$$q_{x} = \frac{2 m_{x}}{2 + m_{x}}$$

where $q_{x}$ is the probability of death between exact ages $x$ and $x + 1$, and $m_{x}$ is the central rate of mortality at age $x$ last birthday.

Sonnenberg and Beck, in their guide to Markov models in medical decision making, give the corresponding relation for a constant rate. They note that the probability is always less than the rate for the same period, because fewer cohort members remain at risk as others die, and that a yearly probability cannot simply be divided by 12 to obtain a monthly one.

$$p = 1 - e^{-r t}$$

where $p$ is the probability of death within the interval, $r$ is the constant mortality rate per unit of time, and $t$ is the length of the interval in the same time unit.

Conversions between cycle lengths are set out on the rate-to-probability conversion page, and the continuous-time view on the hazard rate page. Sonnenberg and Beck also describe a state's mortality as a baseline rate depending on age, sex and race plus excess mortality from comorbid disease, the structure developed on the background mortality page. Fleurence and Hollenbeak report that confusion over rates and probabilities in economic models persists despite earlier guidance.

Worked example with two districts and two standard populations

All figures in this example are illustrative. Two districts each have 100,000 residents in three broad age bands. District A has 60,000 people aged under 45 with 30 deaths, 30,000 aged 45 to 74 with 150 deaths and 10,000 aged 75 and over with 600 deaths. District B has 40,000, 36,000 and 24,000 people in the same bands, with 24, 234 and 1,200 deaths.

1. Crude rates. District A has 30 + 150 + 600 = 780 deaths, or 780 / 100,000 × 1,000 = 7.80 per 1,000. District B has 24 + 234 + 1,200 = 1,458 deaths, or 14.58 per 1,000, nearly twice as high.

2. Age-specific rates. District A's rates are 30 / 60,000 × 1,000 = 0.5, 150 / 30,000 × 1,000 = 5.0 and 600 / 10,000 × 1,000 = 60.0 per 1,000. District B's are 24 / 40,000 × 1,000 = 0.6, 234 / 36,000 × 1,000 = 6.5 and 1,200 / 24,000 × 1,000 = 50.0. District B has higher rates in the two younger bands and a lower rate at 75 and over.

3. Standardise to the 2013 ESP. The 2013 ESP weights in the ONS table group into these bands as 54,000, 37,000 and 9,000 of 100,000. District A's standardised rate is (54,000 × 0.5 + 37,000 × 5.0 + 9,000 × 60.0) / 100,000 = (27,000 + 185,000 + 540,000) / 100,000 = 7.52 per 1,000. District B's is (54,000 × 0.6 + 37,000 × 6.5 + 9,000 × 50.0) / 100,000 = (32,400 + 240,500 + 450,000) / 100,000 = 7.23 per 1,000.

4. Standardise to the 1976 ESP. The 1976 weights group to 64,000, 32,000 and 4,000. District A's rate becomes (32,000 + 160,000 + 240,000) / 100,000 = 4.32 per 1,000 and District B's (38,400 + 208,000 + 200,000) / 100,000 = 4.46 per 1,000. The younger standard reverses the ranking, as Breslow and Day warn, and both values sit far below the 2013 versions.

5. Precision. With weights 0.54, 0.37 and 0.09, District A's standard error is the square root of 0.54² × 30 / 60,000² + 0.37² × 150 / 30,000² + 0.09² × 600 / 10,000², or 0.27 per 1,000, and District B's is 0.21. The difference of 0.29 per 1,000 has a standard error of the square root of 0.27² + 0.21², about 0.35, so the data cannot separate the districts.

6. A model input for people aged 75 and over. A cohort model of District A residents aged 75 and over needs the band rate of 60.0 per 1,000, or $m = 0.06$. The ONS formula gives 2 × 0.06 / 2.06 = 0.0583 as the annual probability of death, and the constant-rate formula gives $p = 1 - e^{-0.06} = 0.0582$. A monthly cycle needs $p = 1 - e^{-0.06/12} = 0.00499$, while dividing the annual probability by 12 gives 0.00485. Using the district's crude rate of 7.8 per 1,000 instead would give an annual probability of 0.0078, about 13% of the correct figure, and would overstate survival in every arm of the model.

Standardised rates serve comparisons between districts, but only the age-specific rates belong in a model, because the model cohort has its own age and sex mix.

Mortality rates in burden studies and policy evaluation

The mortality side of burden of illness studies usually weights deaths by age at death. The ONS guide calculates potential years of life lost as the sum of age-group mortality rates weighted by remaining life expectancy, and standardises the result to the 2013 ESP so that areas and years can be compared.

In policy evaluation the main risk is reading a change in crude mortality as a change in health. An ageing population raises the crude rate with no change in age-specific risk, while changes in coding rules, registration practice or population estimates can shift published rates. The ONS guide records that age-standardised rates for 2012 to 2021 were revised after mid-year population estimates were recalculated from Census 2021.

Common misreadings of mortality rates

Most mistakes come from treating different mortality figures as interchangeable. Four recur in health economic work.

1. Comparing standardised rates built on different standards. Rates on the 1976 and 2013 ESP, or on the ESP and the WHO standard, are not on the same scale.
2. Entering a rate as a cycle probability. The difference is small for low rates and short cycles but grows at older ages and over long cycles.
3. Reading a trial's percentage of patients who died as a rate. A one-year cumulative proportion is a risk, and a rate per person-year needs the time at risk.
4. Ignoring small numbers. Age-specific and cause-specific rates for small areas carry wide sampling error, which is why ONS suppresses or flags rates built on few deaths.

Sources

• Breslow NE, Day NE. Statistical Methods in Cancer Research. Volume II: The Design and Analysis of Cohort Studies. IARC Scientific Publications No. 82. Lyon: International Agency for Research on Cancer; 1987. Chapter 2, Rates and rate standardization. https://publications.iarc.who.int/download/SP82_vol2-2.pdf
• Centers for Disease Control and Prevention. Principles of Epidemiology in Public Health Practice. 3rd ed. Lesson 3, Sections 2 and 3. Last reviewed 18 May 2012. https://archive.cdc.gov/www_cdc_gov/csels/dsepd/ss1978/lesson3/section3.html
• Fleurence RL, Hollenbeak CS. Rates and probabilities in economic modelling: transformation, translation and appropriate application. PharmacoEconomics. 2007;25(1):3-6. https://doi.org/10.2165/00019053-200725010-00002
• Office for National Statistics. Guide to calculating national life tables. Last revised 21 January 2019. https://www.ons.gov.uk/peoplepopulationandcommunity/healthandsocialcare/healthandlifeexpectancies/methodologies/guidetocalculatingnationallifetables
• Office for National Statistics. User guide to mortality statistics. Last revised 9 October 2025. https://www.ons.gov.uk/peoplepopulationandcommunity/birthsdeathsandmarriages/deaths/methodologies/userguidetomortalitystatisticsjuly2017
• Sonnenberg FA, Beck JR. Markov models in medical decision making: a practical guide. Medical Decision Making. 1993;13(4):322-338. https://doi.org/10.1177/0272989X9301300409
• World Health Organization. Global Health Observatory indicator metadata registry. Mortality rates, age standardized (per 100 000 population), IMR ID 78. Accessed 31 October 2026. https://www.who.int/data/gho/indicator-metadata-registry/imr-details/78