Question

Following are the population of a district

Year (x)	1881	1891	1901	1911	1921	1931
Population (y) Thousands	363	391	421	-	467	501

Find the population of the year 1911

Answer 1

Since only five values of fix) are given, the polynomial which fits the data is of degree four.

Hence fifth differences are zeros.

i.e. Δ⁵y₀ = 0

(E – 1)⁵y₀ = 0

(E⁵ – 5E⁴ + 10E³ – 10E² + 5E – 1)y₀ = 0

E⁵y₀ – 5E⁴y₀ + 10E³y₀ – 10E²y₀ + 5Ey₀ – y₀ = 0

y₅ – 5y₄ + 10y₃ – 10y₂ + 5y₁ – y₀ = 0

501 – 5(467) + 10(y₃) – 10(421) + 5(391) – 363 = 0

2456 – 6908 + 10y₃ = 0

– 4452 + 10y₃ = 0

⇒ 10y₃ = 4452

y = `4452/10` = 445.2

The population of the year 1911 is 445.2 thousands