Carrier Farrag Method
- 1. Carrier Farrag Method PresentedBy: Soumik Sadman Anwar 3rd Year (Hons.), Roll: SH-105 Department of Statistics, University of Dhaka
- 3. Introduction A mathematical graduation of census age distribution proposed by Carrier and Farrag (1959)
- 4. Why Use Carrier- Farrag Method Carrier-Farrag method ensures smoother values than that of quadratic interpolation by handling the problem of massive age shifting
- 6. Formulae Elementary Differential Equations Let, 𝑢𝑖 = Population in the quinary age groups 5(x+i) to 5(x+i)+4 𝑈𝑖 = Population in the denary age group such that, 𝑼𝒊 = 𝒖𝒊 + 𝒖𝒊+𝟏
- 7. Calculating Denary age group from Quinary age group Enumerated Population Age in years Quinary age group Denary age group (1) (2) (3) 5-9 4655 9270 10-14 4615 15-19 4374 8508 20-24 4134 25-29 3572 6301 30-34 2729 35-39 2288 4495 40-44 2207 45-49 1886 3210 50-54 1324 55-59 1204 2207 60-64 1003 65-69 602 1164 70-74 562
- 8. Assumptions We assume, 𝑢𝑖 is a second degree function of i. So, 𝑢𝑖+1 = 𝑢𝑖 + ∆𝑢𝑖 If i=0, 𝑢1 = 𝑢 𝑜 + ∆𝑢 𝑜 i=1, 𝑢2 = 𝑢1 + ∆𝑢1 = 𝑢 𝑜 + 2∆𝑢 𝑜 + ∆2 𝑢 𝑜 i=2, 𝑢3 = 𝑢2 + ∆𝑢2 = 𝑢 𝑜 + 3∆𝑢 𝑜 + 3∆2 𝑢 𝑜 i=3, 𝑢4 = 𝑢3 + ∆𝑢3 = 𝑢 𝑜 + 4∆𝑢 𝑜 + 6∆2 𝑢 𝑜 i=4, 𝑢5 = 𝑢4 + ∆𝑢4 = 𝑢 𝑜 + 5∆𝑢 𝑜 + 10∆2 𝑢 𝑜 Here, we ignore the higher terms of ∆
- 9. Calculation We know, 𝑈𝑖 = 𝑢𝑖 + 𝑢𝑖+1 Putting, i=0, 𝑈0 = 𝑢0 + 𝑢1 i=2, 𝑈2 = 𝑢2 + 𝑢3 = 2𝑢 𝑜 + 5∆𝑢 𝑜 + 4∆2 𝑢 𝑜 1 2 𝑈2 = 𝑢 𝑜 + 5 2 ∆𝑢 𝑜 + 2∆2 𝑢 𝑜 … … …(𝑖) i=4, 𝑈4 = 𝑢4 + 𝑢5 = 2𝑢 𝑜 + 9∆𝑢 𝑜 + 16∆2 𝑢 𝑜 So, 𝑈0 - 𝑈4 = −8∆𝑢 𝑜 - 16∆2 𝑢 𝑜 1 16 (𝑈0 - 𝑈4) = − 1 2 ∆𝑢 𝑜 - ∆2 𝑢 𝑜… … … (ii)
- 10. By, (i) + (ii), we get, 1 2 𝑈2 + 1 16 (𝑈0 - 𝑈4) = 𝑢 𝑜 + 2∆𝑢 𝑜 + ∆2 𝑢 𝑜 = 𝑢2 By, (i) + (ii), we get, 1 2 𝑈2 - 1 16 (𝑈0 - 𝑈4) = 𝑢 𝑜 + 3∆𝑢 𝑜 + 3∆2 𝑢 𝑜= 𝑢3 So, the key formulae are, 𝒖 𝟐 = 𝟏 𝟐 𝑼 𝟐 + 𝟏 𝟏𝟔 (𝑼 𝟎 - 𝑼 𝟒) 𝒖 𝟑 = 𝟏 𝟐 𝑼 𝟐 - 𝟏 𝟏𝟔 (𝑼 𝟎 - 𝑼 𝟒)
- 11. Drawbacks 1st and last two quinary age group data cannot be graduated
- 12. Thank You