Assumptions for Reliable Interpolation and Extrapolation

Every five-year plan, welfare scheme, and infrastructure project in the country leans on a single number: the estimated population for a given year. But censuses happen only once a decade, and sometimes even later. To fill that gap, demographers use interpolation and extrapolation with growth rate methods. These mathematical techniques are powerful, but only as reliable as the assumptions sitting underneath them. Get the assumptions wrong, and you get the policy wrong.

Why assumptions matter more than the formula 🔗

Growth rate methods, whether arithmetic, geometric, or exponential, look deceptively simple. You plug in two known population counts, apply a formula, and out comes an estimate. The catch is that every formula carries a hidden contract with reality. It assumes the world behaves in a particular way between and beyond the data points you have. When reality breaks that contract, the estimate breaks with it.

This is why understanding the assumptions is not an academic exercise. The Public Distribution System still operates on 2011 Census figures, and millions of newly eligible beneficiaries have reportedly been left out because the 2021 Census was indefinitely delayed. In this environment, intercensal and post-census estimates are not just classroom exercises; they are the de facto basis for resource allocation.

The essential assumptions for reliable estimates 🔗

For interpolation (estimating values between known points) and extrapolation (estimating values beyond them) to produce trustworthy numbers, several conditions must hold. These assumptions are usually treated as standard in textbooks, but each one deserves a closer look.

Sufficient number of observations 🔗

You cannot draw a line from a single point. Most growth rate methods require at least two reliable observations, and ideally three or more, to detect a meaningful pattern. With only two census counts, a demographer can compute a growth rate, but cannot tell whether that rate is stable, accelerating, or decelerating.

For a newly carved district or a recently urbanised village, two data points often do not exist. In such cases, the standard practice is to borrow growth patterns from demographically similar regions rather than rely on a thin numerical record. The more observations available, the more confidently a trend can be distinguished from noise.

Equal time intervals between observations 🔗

Standard interpolation formulas assume that observations are spaced at equal intervals, typically the ten-year gap between censuses. Equal intervals matter for three practical reasons. They make growth rate formulas straightforward to apply, they allow direct comparison across periods, and they reduce the systematic errors that creep in when intervals vary.

When the interval is broken, as is happening now with a census originally due in 2021 that has been rescheduled to 2026-27, the gap between the previous and next data point will stretch to roughly sixteen years. Applying a standard decadal formula to an irregular gap of this size, without adjustment, will distort the estimate.

Uniform rate of change 🔗

This is perhaps the most consequential assumption. Arithmetic growth methods assume a constant numerical increase in every equal period, while geometric methods assume a constant percentage increase, similar to compound interest. Linear interpolation between two census points carries the same idea in a milder form: that the change between the two points was steady and predictable.

Real populations rarely behave so politely. Fertility transitions, urban migration waves, and mortality shocks can all shift the growth rate within a single decade. The assumption of a uniform rate is a working approximation, useful for short stretches but increasingly fragile over longer horizons.

A stable underlying demographic regime 🔗

Growth rate methods implicitly assume that the forces driving population change, fertility, mortality, and migration, remain broadly stable across the estimation period. Extrapolation, in particular, leans on the idea that past trends will continue into the projection window.

This is a strong assumption. It rules out structural breaks. It assumes no sudden surge in international migration, no abrupt collapse in fertility, no pandemic-scale mortality event. Whenever any of these occur, projections built on the stability assumption need to be revised, sometimes drastically.

Data reliability: the silent assumption 🔗

Even the most elegant formula collapses on poor data. The reliability of any interpolation or extrapolation depends on three quiet but crucial conditions.

Consistency in definitions and coverage 🔗

The boundaries of a district, the definition of “urban,” and the way migrants are counted should remain consistent across the observations being compared. If a town was classified as rural in one census and urban in the next, its apparent population growth will reflect a change in definition rather than actual demographic change. Researchers have noted, for instance, that parts of Delhi’s Hauz Khas neighbourhood were classified as rural in the 2011 Census, even though they are now firmly urban. Without recognising such reclassifications, growth rates can be wildly misleading.

Absence of disruptive external factors 🔗

The assumption of a uniform rate of change implicitly assumes that the population was not buffeted by extraordinary events during the estimation window. A famine, a war, a major epidemic, a large refugee inflow, or a sudden policy-driven migration can all break the trend. The COVID-19 pandemic is the most recent example. Projections made before 2020 could not anticipate the pandemic’s impact on mortality, fertility postponement, and reverse migration to rural areas, and many of those projections turned out to be incorrect.

Accurate base data 🔗

Interpolation and extrapolation amplify any error in the base figures. If the starting census undercounted certain groups, slum dwellers, migrant labourers, or tribal communities, every subsequent estimate built on it inherits that error. Quality assurance in the original enumeration is therefore not a separate concern from estimation; it is the foundation of it.

Implications for population estimates 🔗

The assumptions above are not just methodological footnotes. They shape what we can and cannot say about the country’s population.

Post-census estimates and the widening cone of uncertainty 🔗

Post-census estimates, the figures used for years after the most recent enumeration, are produced by extrapolation. They sit at the riskiest end of the estimation spectrum because every assumption is being stretched into the unknown. The further away from the base year, the wider the cone of uncertainty.

This matters concretely. The National Food Security Act has been distributing benefits based on 2011 population figures for over a decade, even though the actual population has grown substantially and shifted geographically. The longer the post-census gap, the more such schemes risk allocating resources to the wrong places.

Missing data scenarios 🔗

Interpolation is the workhorse for intercensal years and for filling in regions where data collection was disrupted. In the 1951 Census, for example, no enumeration was conducted in Jammu and Kashmir, and the state’s figures were interpolated from the 1941 and 1961 censuses. The assumptions of uniform rate, equal intervals, and a stable demographic regime made that interpolation defensible. In areas where those assumptions are weak, such as conflict-affected regions or rapidly urbanising peripheries, interpolated figures should be treated as approximate rather than definitive.

Choice of method shapes the answer 🔗

Even when all assumptions are met, the choice between arithmetic, geometric, and exponential methods produces different answers. Arithmetic methods tend to underestimate growth in rapidly expanding populations and overestimate it in declining ones. Geometric methods better capture the compounding effect typical of developing economies but can run away into unrealistically large figures over long horizons. Picking the right method is itself an act of assumption: you are assuming the population behaves more like one model than another.

Working responsibly with the assumptions 🔗

Good demographic practice does not pretend the assumptions are always satisfied. Instead, it makes them explicit and works around them. Three habits help.

Triangulate methods. Apply more than one growth rate method and compare the results. If arithmetic and geometric projections converge, confidence rises. If they diverge sharply, the divergence itself is information about how unstable the underlying trend is.

Pair mathematical estimates with component data. When fertility, mortality, and migration data are available from Sample Registration System bulletins or the National Family Health Survey, they can validate or correct the pure mathematical extrapolation. The cohort-component method, which builds projections from these three components separately, generally outperforms simple growth rate extrapolation over long horizons.

Build scenarios, not point estimates. Statistical agencies like the UK’s Office for National Statistics produce projections under low, medium, and high assumption sets rather than a single number. The range communicates the inherent uncertainty and prevents users from treating an estimate as a fact.

The deeper lesson is that interpolation and extrapolation are not neutral arithmetic. Each formula encodes a worldview about how populations change. Recognising the assumptions is what separates a careful estimate from a misleading one, and what protects the policies built on those estimates from being built on sand.

What do you think? If you had to estimate your district’s population for the year 2025 with only the 2001 and 2011 Census figures available, which assumption do you think would be the hardest to defend, and why? And given the delays in the upcoming census, should welfare schemes continue to rely on extrapolations, or should they switch to administrative records like Aadhaar and ration card data?

References

1. http://papp.iussp.org/sessions/papp101_s10/PAPP101_s10_050_020.html
2. https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(23)01477-0/fulltext
3. https://en.wikipedia.org/wiki/2021_Census_of_India
4. https://www.planning.org/pas/reports/report17.htm
5. https://www.ons.gov.uk/peoplepopulationandcommunity/populationandmigration/populationprojections/methodologies/nationalpopulationprojectionsqmi
6. https://www.nature.com/articles/d41586-024-02321-9
7. https://www.policycircle.org/policy/india-census-2021-and-policy/
8. https://en.wikipedia.org/wiki/Census_of_India
9. https://track2training.com/2025/08/07/arithmetic-geometrical-and-decadal-population-growth-methods/