Demography and vital statistics

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In short

Demography is the statistical study of human populations - their size, composition, distribution and the three processes that change them: fertility, mortality and migration. Vital statistics are the counts of vital events (births, deaths, marriages, foetal deaths) and the rates and ratios built from them, which turn raw registration data into comparable measures of a population's health and growth. In India the data come from the decennial Census of India (Census Act, 1948), the Civil Registration System under the Registration of Births and Deaths Act, 1969 (substantially amended in 2023), the Sample Registration System (SRS) run by the Office of the Registrar General of India, and large-scale surveys such as the NFHS. The topic is standard in B.Sc. Statistics applied papers, in ISS and UGC-NET, and in Indian Statistical Service, ORGI, NSO, IIPS, insurance and public-health analytics work. Life-table and standardisation questions carry heavy marks because they are also the foundation of actuarial pricing and health programme evaluation.

Explained

Sources of demographic data in India

Census of India - complete enumeration every ten years; conducted under the Census Act, 1948 and Census Rules, 1990 by the Registrar General and Census Commissioner. Census 2011 counted 1,210.85 million persons, decadal growth 17.70%, density 382 per km², sex ratio 943 females per 1000 males, literacy 74.04%.
The 2021 Census was postponed; the Government has announced the next census for 2027 (reference date 1 March 2027, and 1 October 2026 for snow-bound areas), to be digital and with caste enumeration. Treat dates as announced, not final.
Civil Registration System (CRS) - continuous, legally compulsory registration of births and deaths. The RBD (Amendment) Act, 2023 (in force 1 October 2023) put registration on a national digital database, made the digital birth certificate a single document for admission, driving licence, passport etc., and required Aadhaar details of parents/informant.
SRS - a dual-record sample survey of a large nationally representative panel of rural and urban units, giving annual CBR, CDR, IMR, TFR and abridged life tables; the primary source for India's vital rates because CRS completeness varies by State.
MCCD - Medical Certification of Cause of Death, coded by ICD (ICD-10 in Indian practice; the world is moving to ICD-11, India transitioning), covers mainly institutional deaths, so cause-of-death data are not representative.
Surveys - NFHS (NFHS-5, 2019-21 gave TFR 2.0), DLHS, SRS special bulletins on maternal mortality; NSS for migration and employment.
Errors: coverage error (omission/duplication), content error (age misreporting, misstatement of events), non-registration and delayed registration.

Fertility measures

Crude Birth Rate (CBR) - live births per 1000 mid-year population. Simple and available, but affected by age-sex composition.
General Fertility Rate (GFR) - births per 1000 women aged 15-49 (sometimes 15-44); restricts the denominator to women at risk.
Age-Specific Fertility Rate (ASFR) - births to women of age group i per 1000 women of that group; the building block of all refined measures.
Total Fertility Rate (TFR) - average children a woman would bear if she passed through all reproductive ages at current ASFRs. A synthetic cohort measure: independent of age structure, hence used for comparisons. Replacement level ≈ 2.1; India's SRS TFR is about 2.0 (latest published reports), i.e. below replacement.
Gross Reproduction Rate (GRR) - daughters per woman, i.e. TFR restricted to female births; ignores mortality.
Net Reproduction Rate (NRR) - daughters per woman after allowing for the chance that the mother survives to each childbearing age. NRR = 1 means exact replacement, > 1 growth, < 1 eventual decline.
Child-woman ratio - children 0-4 per 1000 women 15-49; used where birth registration is poor.
Period vs cohort: period rates (TFR) describe a calendar year; cohort fertility follows real women and is only complete after age 49. Tempo effects (postponement of births) depress period TFR.

Mortality measures and standardisation

CDR - deaths per 1000 population; a rich district with many elderly can show a higher CDR than a poor one with a young population, so CDR must not be compared directly.
ASDR (m_x) - deaths per 1000 in an age (or age-sex) group; the correct basis for comparison.
Infant Mortality Rate (IMR) - infant (under 1) deaths per 1000 live births - note the denominator is births, not population, so IMR is a ratio, not a true rate. Neonatal (under 28 days), post-neonatal, under-five mortality (U5MR) are its parts. SRS IMR is about 27-28 per 1000 in recent reports, with wide State variation (Kerala low single digits, some northern States above 40).
Maternal Mortality Ratio (MMR) - maternal deaths per 100,000 live births; India's SRS special bulletins report 97 for 2018-20, 93 for 2019-21 and about 88 for 2021-23 (the latest), against the SDG target of below 70 by 2030.
Still Birth Rate, Perinatal Mortality Rate (late foetal deaths + early neonatal deaths per 1000 total births), Case Fatality Rate (deaths among cases of a disease).
Direct standardisation - apply each population's ASDRs to one *standard population* age structure; needs ASDRs for every population compared.
Indirect standardisation - apply a *standard set of ASDRs* to each population's age distribution to get expected deaths; SMR = observed/expected deaths. Used when ASDRs are unavailable but total deaths and age distribution are.
Advantage of standardisation: removes compositional differences. Limitation: the answer depends on the standard chosen, and the standardised rate is an index, not a real rate.

Life table

A life table traces a hypothetical cohort of l₀ (usually 100,000) births through death at current age-specific mortality. Columns: x, n q_x, l_x, d_x, n L_x, T_x, e_x.
Assumptions: a closed cohort (no migration), current mortality unchanged, deaths uniformly distributed within an age interval (except infancy, where deaths cluster early).
Types: *complete* (single year of age) vs *abridged* (5-year groups, as in SRS); *current/period* (cross-sectional rates) vs *cohort/generation* (actual cohort followed); *select* life tables in insurance (mortality depends on duration since selection).
Uses: expectation of life, survival probabilities, premium and annuity calculation, NRR computation, estimating population projections by the component method, person-years for epidemiology.
India's SRS abridged life tables give life expectancy at birth of about 70 years (female higher than male by roughly 3-4 years) for the most recent published period, up from about 32 years in 1947.

Population growth, projection and stable population

Growth models: arithmetic (constant absolute increment), geometric (constant rate per period), exponential (continuous compounding), logistic (Verhulst-Pearl) which allows for a saturation level K and gives an S-shaped curve - the only one of the four that can bend over.
Component (cohort-component) method - the professional method: carry each age-sex group forward with survival ratios, add births from projected ASFRs, adjust for migration. Used in the MoHFW/NCP *Population Projections for India and States 2011-2036*.
Vital index = births/deaths × 100; natural growth rate = CBR − CDR; overall growth adds net migration.
Stable population (Lotka): a population with unchanging ASFRs and ASDRs and no migration converges to a fixed age distribution and a constant growth rate r, the intrinsic rate of natural increase, independent of the starting age structure. A stationary population is the special case r = 0 (the life table's L_x column is its age distribution).
Mean length of generation T ≈ ln(NRR)/r; momentum - a young population keeps growing for decades even after NRR falls to 1, which is why India's population will rise past 2050 although TFR is already 2.0.
Demographic transition: Stage I high birth/high death, Stage II death rate falls (fastest growth), Stage III birth rate falls, Stage IV both low, Stage V decline/ageing. India is in late Stage III with large State differences - Kerala and Tamil Nadu near Stage IV, Bihar and Uttar Pradesh in Stage II-III. The resulting bulge of working-age people is the demographic dividend (roughly to the 2040s).

Population composition and quality of age data

Sex ratio - in India, females per 1000 males (the UN convention is the reverse); Census 2011: 943 overall, 919 child (0-6). Sex ratio at birth from SRS is around 907-910, below the natural 943-952 females per 1000 males, indicating sex selection - addressed by the PCPNDT Act, 1994.
Age pyramid - broad-based (young, high fertility), bell-shaped (ageing), urn-shaped (declining).
Dependency ratio = (population 0-14 + 60 and over) / population 15-59 × 100; child and old-age dependency separately. India's old-age dependency is rising sharply.
Density, urbanisation rate (31.1% urban in 2011), literacy rate, work participation rate.
Age heaping - preference for ages ending in 0 and 5 (and in India for even ages). Measured by Whipple's index (ages 23-62, digits 0 and 5; 100 = no heaping), Myers' blended index (all ten terminal digits, 0 = no preference), and the UN Age-Sex Accuracy Index (age-ratio + sex-ratio scores).
Graduation/smoothing - correcting heaped data by moving averages, Sprague/Karup-King multipliers, or fitting a mathematical curve to l_x (Gompertz, Makeham).
Chandrasekaran-Deming dual-record formula estimates true events and registration completeness by matching registration against an independent survey.

Migration

Definitions in Indian censuses: migrant by *place of birth* and by *place of last residence*, with duration of residence; streams are rural-rural, rural-urban, urban-rural, urban-urban.
Measures: in-migration rate, out-migration rate, net migration rate (per 1000 of the receiving population), gross migration, migration ratio.
Indirect estimation: residual method - net migration = (P_t − P_0) − (births − deaths) over the period; vital statistics or survival-ratio methods where direct data are absent.
Marriage is the largest reason for female migration in India; work and employment for male migration.

Key points

Vital statistics - counts of births, deaths, marriages and the rates derived from them; "vital" means life events, not importance.
Rate vs ratio vs proportion - a rate has a population at risk and a time reference in the denominator; IMR and MMR use births, so they are ratios.
CBR and CDR - crude because they ignore age-sex composition; never compare two populations on CDR alone.
TFR - children per woman at current ASFRs; composition-free, replacement ≈ 2.1, India about 2.0.
GRR vs NRR - GRR ignores female mortality, NRR allows for it, so NRR ≤ GRR always.
Direct standardisation needs each group's ASDRs; indirect needs only a standard set of rates plus age distribution.
SMR - observed deaths ÷ expected deaths; above 1 means worse than the standard.
Life table - q_x, l_x, d_x, L_x, T_x, e_x for a synthetic cohort of 100,000 births under current mortality.
e₀ - expectation of life at birth, the single best summary of mortality; India about 70 years currently.
IMR - the most sensitive indicator of health and living conditions; Kerala lowest among Indian States.
Logistic curve - the only standard growth model with an upper asymptote K; fitted by the three-point or Rhodes method.
Stable population - fixed ASFRs and ASDRs give a constant age distribution and intrinsic growth rate r; stationary is r = 0.
Population momentum - growth continues after NRR reaches 1 because of a young age structure.
Whipple's index > 125 - age data are "rough"; Myers' index measures preference for all ten digits.
Chandrasekaran-Deming formula - estimates true number of events and CRS completeness from two incomplete lists.
CRS vs SRS - CRS is legal and complete-coverage in principle (RBD Act, 1969, amended 2023); SRS is a sample but gives reliable rates.
Dependency ratio - (0-14 plus 60+) per 100 persons aged 15-59; falling child dependency gives the demographic dividend.
Sex ratio at birth below natural levels signals sex selection; PCPNDT Act, 1994 applies.

Formulas and examples

CBR = (B / P) × 1000 - B = live births in the year, P = mid-year population. Unit: per 1000 population per year.
CDR = (D / P) × 1000 - D = deaths in the year.
Natural growth rate = CBR − CDR (per 1000); Vital index = (B / D) × 100.
ASDR m_x = (D_x / P_x) × 1000; ASFR f_i = (B_i / W_i) × 1000, W_i = women in age group i.
GFR = B / W(15-49) × 1000.
TFR = n × Σ f_i / 1000, with n = width of the age group (5 for standard groups); unit: children per woman.
GRR = TFR × (share of female births) = n Σ f_i^female / 1000.
NRR = n Σ (f_i^female / 1000) × (L_i / l₀ · 1/n), i.e. female ASFRs weighted by the survival of women to each childbearing age; dimensionless.
IMR = (deaths under age 1 / live births) × 1000; NMR = deaths under 28 days per 1000 live births.
MMR = (maternal deaths / live births) × 100,000; U5MR = deaths under 5 per 1000 live births.
Direct STDR = Σ (P_i^s × m_i) / Σ P_i^s - P_i^s = standard population in group i, m_i = study population's ASDR.
Indirect: expected deaths E = Σ (P_i × m_i^s); SMR = D_obs / E; indirect STDR = SMR × CDR_standard.
Life table: q_x = d_x / l_x; p_x = 1 − q_x; d_x = l_x − l_{x+1}; L_x = (l_x + l_{x+1})/2 (single year, uniform deaths); T_x = Σ_{y≥x} L_y; e_x = T_x / l_x (years); central death rate m_x = d_x / L_x; relation m_x = 2q_x / (2 − q_x).
Growth: arithmetic P_t = P₀(1 + rt); geometric P_t = P₀(1 + r)^t; exponential P_t = P₀ e^{rt}; doubling time = ln 2 / r.
Logistic: P_t = K / (1 + e^{−(a + bt)}), K = carrying capacity.
Dependency ratio = [P(0-14) + P(60+)] / P(15-59) × 100; Sex ratio (India) = females per 1000 males.
Whipple's index = 5 × Σ(persons aged 25, 30, …, 60) / Σ(persons aged 23 to 62) × 100.
Chandrasekaran-Deming: N̂ = C + U₁ + U₂ + (U₁U₂ / C) - C = events in both lists, U₁ = only in registration, U₂ = only in survey.
Lotka's equation: ∫ e^{−rx} p(x) m(x) dx = 1 defines the intrinsic rate r; mean generation length T ≈ ln(NRR) / r.

Example 1 - crude rates and growth. A district has mid-year population 25,00,000, with 48,000 live births, 18,000 deaths and 1,320 infant deaths in the year; women aged 15-49 number 6,00,000.

CBR = 48,000/25,00,000 × 1000 = 19.2 per 1000
CDR = 18,000/25,00,000 × 1000 = 7.2 per 1000
Natural growth rate = 19.2 − 7.2 = 12.0 per 1000 = 1.20% per year
GFR = 48,000/6,00,000 × 1000 = 80 per 1000 women
IMR = 1,320/48,000 × 1000 = 27.5 per 1000 live births
Vital index = 48,000/18,000 × 100 = 266.7

Example 2 - why standardisation matters (direct method).

Age group	Standard pop	A pop	A deaths	A ASDR	B pop	B deaths	B ASDR
0-14	30,000	40,000	320	8.0	25,000	150	6.0
15-59	60,000	55,000	275	5.0	55,000	330	6.0
60+	10,000	5,000	250	50.0	20,000	800	40.0
Total	1,00,000	1,00,000	845	-	1,00,000	1,280	-

CDR(A) = 845/1,00,000 × 1000 = 8.45; CDR(B) = 1,280/1,00,000 × 1000 = 12.80 - B looks far worse.
STDR(A) = (30,000×8 + 60,000×5 + 10,000×50)/1,00,000 = (2,40,000 + 3,00,000 + 5,00,000)/1,00,000 = 10.40
STDR(B) = (30,000×6 + 60,000×6 + 10,000×40)/1,00,000 = (1,80,000 + 3,60,000 + 4,00,000)/1,00,000 = 9.40
After removing age composition, A has the higher mortality: B's high CDR came only from its large elderly share. This reversal is the classic exam point.
Indirect check for B using A's ASDRs as standard: E = 25,000×0.008 + 55,000×0.005 + 20,000×0.050 = 200 + 275 + 1,000 = 1,475; SMR = 1,280/1,475 = 0.868, again showing B below the standard.

Example 3 - TFR, GRR, NRR. ASFRs per 1000 women: 15-19: 20, 20-24: 110, 25-29: 130, 30-34: 70, 35-39: 30, 40-44: 10, 45-49: 2. Sex ratio at birth = 950 females per 1000 males. Probability that a woman survives to the mean age of childbearing = 0.96.

Σ f_i = 372 per 1000
TFR = 5 × 372/1000 = 1.86 children per woman (below replacement 2.1)
Female share of births = 950/1950 = 0.4872 → GRR = 1.86 × 0.4872 = 0.906 daughters per woman
NRR ≈ 0.906 × 0.96 = 0.870 - each generation of mothers is replaced by 0.87 daughters, so the population will eventually decline, though momentum keeps it growing for decades.

Example 4 - life table at old ages. Given l₈₀ = 1000, l₈₁ = 700, l₈₂ = 400, l₈₃ = 0.

d₈₀ = 300, d₈₁ = 300, d₈₂ = 400
q₈₀ = 300/1000 = 0.30; q₈₁ = 300/700 = 0.4286; q₈₂ = 400/400 = 1.00
L₈₀ = (1000+700)/2 = 850; L₈₁ = (700+400)/2 = 550; L₈₂ = (400+0)/2 = 200
T₈₂ = 200; T₈₁ = 550 + 200 = 750; T₈₀ = 850 + 750 = 1600
e₈₀ = 1600/1000 = 1.60 years; e₈₁ = 750/700 = 1.07 years; e₈₂ = 200/400 = 0.50 year
Central death rate m₈₀ = 300/850 = 0.3529; check m = 2q/(2−q) = 0.60/1.70 = 0.3529 ✓

Example 5 - projection and doubling time. India's population was 1,028.74 million in 2001 and 1,210.85 million in 2011.

Decadal growth = 182.11/1028.74 = 17.70%
Geometric annual rate: r = (1.1770)^{1/10} − 1; ln 1.1770 = 0.16296, /10 = 0.016296, e^{0.016296} = 1.01643 → r ≈ 1.64% per year
Doubling time = 0.6931/0.016296 ≈ 42.5 years (at that rate)
Naive geometric projection for 2021 = 1210.85 × 1.1770 = 1,425 million, well above the official component-method projection of roughly 1,360-1,390 million - because fertility kept falling. Constant-rate models overstate growth in a transitioning population.

Example 6 - completeness of registration. Of births in a block, 1,800 appear in both the CRS register and an independent survey, 200 only in the register, 400 only in the survey.

N̂ = 1800 + 200 + 400 + (200 × 400)/1800 = 2400 + 44.4 = 2,444 births
Registration completeness = (1800 + 200)/2444 = 81.8%; survey completeness = 2200/2444 = 90.0%

Measure	Denominator	Affected by age structure?	Typical use
CBR / CDR	Total population	Yes	Quick annual reporting (SRS)
GFR	Women 15-49	Partly	Intermediate measure
TFR	Synthetic cohort	No	Policy, inter-State comparison
NRR	Synthetic cohort + survival	No	Replacement/growth potential
IMR, MMR	Live births	No	Health programme evaluation
STDR / SMR	Standard population	Removed	Mortality comparison
e₀	Life table cohort	No	Overall mortality summary

Where it is used

Government statistics - ORGI (Census, CRS, SRS), MoSPI/NSO, State Directorates of Census Operations and Chief Registrars of Births and Deaths; the *SRS Statistical Report* and *Vital Statistics of India based on CRS* are produced from exactly these methods.
Health policy and programmes - NHM targets on IMR, MMR, U5MR and institutional delivery; district health rankings and NITI Aayog's Health Index; HMIS and disease-burden studies at ICMR and PHFI.
Insurance and pensions - IRDAI-regulated life insurers price term plans and annuities from mortality tables (the Indian Assured Lives Mortality tables published by the Institute of Actuaries of India); EPFO and pension schemes use survival probabilities.
Planning and infrastructure - population projections drive school and hospital siting, drinking-water and electricity demand, Finance Commission and delimitation debates, and urban master plans.
Market and business analytics - FMCG and telecom segmentation by age pyramid and household size; real-estate and BFSI demand models built on dependency ratios and urbanisation.
Research and teaching - IIPS Mumbai (NFHS), ISI, UNFPA and WHO country offices; job roles include demographer, statistical officer, health data analyst, actuarial analyst, M&E specialist.

Interview questions

Q: Why is the crude death rate a poor basis for comparing two States? A: It depends on age composition - a State with many elderly shows a high CDR even with better age-specific mortality; use standardised rates or ASDRs.
Q: Difference between direct and indirect standardisation? A: Direct applies the study populations' ASDRs to a standard age distribution; indirect applies standard ASDRs to each study population's age distribution and reports an SMR. Indirect is used when ASDRs are unavailable.
Q: Why is TFR preferred over CBR? A: TFR is built from age-specific rates for a synthetic cohort, so it is free of age-sex composition and comparable across populations and years.
Q: What does NRR = 1 mean, and can population still grow? A: Each woman is exactly replaced by one surviving daughter, so growth eventually stops - but population momentum from a young age structure keeps it growing for several decades.
Q: Why is IMR called a ratio rather than a rate? A: Its denominator is live births in the year, not the population at risk of dying in infancy, and some of the deaths belong to the previous year's birth cohort.
Q: What assumptions underlie a current life table? A: A closed cohort with no migration, current age-specific mortality held constant, and deaths distributed uniformly within each age interval except infancy.
Q: What is the difference between SRS and CRS? A: CRS is the legal, continuous, universal registration of births and deaths under the RBD Act, 1969; SRS is a dual-record sample survey that yields reliable national and State vital rates despite CRS under-registration.
Q: State India's current TFR and life expectancy at birth. A: About 2.0 children per woman and about 70 years respectively in the latest SRS reports - just below replacement fertility, with wide State variation; figures are revised annually.
Q: Which growth model would you fit to a population approaching saturation? A: The logistic curve, because it has an upper asymptote K, unlike arithmetic, geometric or exponential models.
Q: What is a stable population and how does it differ from a stationary one? A: A stable population has constant ASFRs and ASDRs, hence a fixed age distribution and a constant intrinsic growth rate r; a stationary population is the special case r = 0 with constant size.
Q: How would you check the quality of reported ages in a census? A: Compute Whipple's index for digits 0 and 5, Myers' blended index across all terminal digits, and the UN age-sex accuracy index; smooth heaped data by graduation.
Q: How do you estimate how complete birth registration is in a district? A: Match the register against an independent household survey and apply the Chandrasekaran-Deming dual-record formula to estimate true events, then divide registered events by that estimate.
Q: What is the demographic dividend and what ends it? A: The growth advantage of a large working-age share and low dependency during the transition; it ends as the same cohorts age and old-age dependency rises.

Common mistakes

Using total population instead of live births in the denominator of IMR, MMR or stillbirth rate, or forgetting that MMR is per 100,000.
Omitting the age-interval width n when converting ASFRs to TFR, or forgetting to divide ASFRs by 1000.
Reporting GRR greater than TFR, or NRR greater than GRR - both are impossible.
Comparing crude rates between populations and concluding one has worse health, without standardising.
Mixing the two standardisation methods: applying standard rates and then calling the result a directly standardised rate.
Using L_x = (l_x + l_{x+1})/2 for the infant age interval, where deaths are heavily concentrated in the first weeks.
Projecting a transitioning population with a constant geometric rate, and quoting it as an official figure instead of using the cohort-component method.
Quoting the Indian sex ratio in the UN direction (males per 1000 females) or confusing overall sex ratio with sex ratio at birth and child sex ratio.