𝑽𝑰𝑻𝑨𝑳 𝑺𝑻𝑨𝑻𝑰𝑺𝑻𝑰𝑪𝑺

𝟏. 𝑴𝒆𝒂𝒔𝒖𝒓𝒆𝒎𝒆𝒏𝒕 𝒐𝒇 𝒑𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏 (𝑨𝒏𝒂𝒍𝒚𝒕𝒊𝒄𝒂𝒍 𝒎𝒆𝒕𝒉𝒐𝒅 𝒐𝒇 𝒗𝒊𝒕𝒂𝒍 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄𝒔)

𝑃𝑡 = 𝑃0 + (𝐵 − 𝐷 ) + (𝐼 − 𝐸 )
𝑊ℎ𝑒𝑟𝑒 𝑃𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑡𝑖𝑚𝑒 ,
𝑃0 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑐𝑒𝑛𝑠𝑢𝑠 𝑦𝑒𝑎𝑟 𝑡𝑎𝑘𝑒𝑛 𝑎𝑡 𝑡𝑖𝑚𝑒 0
𝐵 𝑎𝑛𝑑 𝐷 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝐵𝑖𝑟𝑡ℎ𝑠 𝑎𝑛𝑑 𝐷𝑒𝑎𝑡ℎ𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
𝐼 𝑎𝑛𝑑 𝐸 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐼𝑚𝑚𝑖𝑔𝑟𝑎𝑛𝑡𝑠 𝑎𝑛𝑑 𝐸𝑚𝑖𝑔𝑟𝑎𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑

𝑴𝑶𝑹𝑻𝑨𝑳𝑰𝑻𝒀 𝑹𝑨𝑻𝑬 (𝑫𝑬𝑨𝑻𝑯 𝑹𝑨𝑻𝑬)

𝟏. 𝑪𝒓𝒖𝒅𝒆 𝑫𝒆𝒂𝒕𝒉 𝑹𝒂𝒕𝒆 (𝑪𝑫𝑹)
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑎𝑡ℎ𝑠 𝑜𝑐𝑐𝑢𝑟𝑟𝑖𝑛𝑔 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟
𝐶𝐷𝑅 = 𝑥1000
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟

𝟐. 𝑨𝒈𝒆 𝑺𝒑𝒆𝒄𝒊𝒇𝒊𝒄 𝑫𝒆𝒂𝒕𝒉 𝑹𝒂𝒕𝒆 (𝑨𝑺𝑫𝑹)

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑎𝑡ℎ𝑠 𝑜𝑐𝑐𝑢𝑟𝑟𝑖𝑛𝑔 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑒𝑑 𝑎𝑔𝑒 𝑔𝑟𝑜𝑢𝑝
𝐴𝑆𝐷𝑅 = 𝑥1000
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑒𝑑 𝑎𝑔𝑒 𝑔𝑟𝑜𝑢𝑝

𝟑. 𝑺𝒕𝒂𝒏𝒅𝒂𝒓𝒅𝒛𝒆𝒅 𝑫𝒆𝒂𝒕𝒉 𝑹𝒂𝒕𝒆 (𝑺𝒕. 𝑫. 𝑹)

∑𝑃𝐴
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑖𝑧𝑒𝑑 𝑑𝑒𝑎𝑡ℎ 𝑟𝑎𝑡𝑒 𝑓𝑜𝑟 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝐴 =
∑𝑃

∑𝑃𝐵
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑖𝑧𝑒𝑑 𝑑𝑒𝑎𝑡ℎ 𝑟𝑎𝑡𝑒 𝑓𝑜𝑟 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝐴 =
∑𝑃
𝑤ℎ𝑒𝑟𝑒 𝑃 𝑖𝑠 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛, 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑎𝑔𝑒 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑑𝑒𝑡ℎ 𝑟𝑎𝑡𝑒𝑠 𝑓𝑜𝑟
𝑡ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝐴 𝑎𝑛𝑑 𝐵

𝟒. 𝑰𝒏𝒇𝒂𝒏𝒕 𝑴𝒐𝒓𝒕𝒂𝒍𝒊𝒕𝒚 𝑹𝒂𝒕𝒆 (𝑰𝑴𝑹)

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑡ℎ𝑠 𝑎𝑚𝑜𝑛𝑔 𝑖𝑛𝑓𝑎𝑛𝑡𝑠 𝑖𝑛 𝑎 𝑦𝑒𝑎𝑟
𝐼𝑀𝑅 = 𝑥1000
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑖𝑣𝑒 𝑏𝑖𝑟𝑡ℎ𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟

𝟓. 𝑵𝒆𝒐 − 𝒏𝒂𝒕𝒂𝒍 𝑴𝒐𝒓𝒕𝒂𝒍𝒊𝒕𝒚 𝑹𝒂𝒕𝒆(𝑵𝑴𝑹)

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑒𝑜 − 𝑛𝑎𝑡𝑎𝑙 𝑑𝑒𝑎𝑡ℎ𝑠 𝑜𝑐𝑐𝑢𝑟𝑖𝑛𝑔 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟
𝑁𝑀𝑅 = 𝑥1000
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑖𝑣𝑒 𝑏𝑖𝑟𝑡ℎ𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟

𝟔. 𝑴𝒂𝒕𝒆𝒓𝒏𝒂𝒍 𝑴𝒐𝒓𝒕𝒂𝒍𝒊𝒕𝒚 𝑹𝒂𝒕𝒆(𝑴𝑴𝑹)

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑎𝑡ℎ𝑠 𝑜𝑓 𝑚𝑜𝑡ℎ𝑒𝑟𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑐ℎ𝑖𝑙𝑑 𝑏𝑖𝑟𝑡ℎ𝑠 𝑜𝑐𝑐𝑢𝑟𝑖𝑛𝑔 𝑖ℎ 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟
𝑀𝑀𝑅 = 𝑥1000
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑖𝑣𝑒 𝑏𝑖𝑟𝑡ℎ𝑠 𝑜𝑐𝑐𝑢𝑟𝑖𝑛𝑔 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟
 𝑭𝑬𝑹𝑻𝑰𝑳𝑰𝑻𝒀 𝑹𝑨𝑻𝑬(𝑩𝑰𝑹𝑻𝑯 𝑹𝑨𝑻𝑬)
𝟏. 𝑪𝒓𝒖𝒅𝒆 𝑩𝒊𝒓𝒕𝒉 𝑹𝒂𝒕𝒆 (𝑪𝑩𝑹)
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑖𝑟𝑡ℎ𝑠 𝑜𝑐𝑐𝑢𝑟𝑟𝑖𝑛𝑔 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟
𝐶𝐷𝑅 = 𝑥1000
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟

𝟐. 𝑨𝒈𝒆 𝑺𝒑𝒆𝒄𝒊𝒇𝒊𝒄 𝑭𝒆𝒓𝒕𝒊𝒍𝒊𝒕𝒚 𝑹𝒂𝒕𝒆 (𝑨𝑺𝑭𝑹)

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑖𝑣𝑒 𝑏𝑖𝑟𝑡ℎ𝑠 𝑜𝑐𝑐𝑢𝑟𝑟𝑖𝑛𝑔 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑑𝑒 𝑎𝑔𝑒 𝑔𝑟𝑜𝑢𝑝 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟
𝐴𝑆𝐹𝑅 = 𝑥1000
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑤𝑜𝑚𝑒𝑛 𝑜𝑓 𝑡ℎ𝑎𝑡 𝑎𝑔𝑒 𝑔𝑟𝑜𝑢𝑝 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟

𝟑. 𝑮𝒆𝒏𝒆𝒓𝒂𝒍 𝑭𝒆𝒓𝒕𝒊𝒍𝒊𝒕𝒚 𝑹𝒂𝒕𝒆(𝑮𝑭𝑹)

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑖𝑟𝑡ℎ𝑠 𝑜𝑐𝑐𝑢𝑟𝑖𝑛𝑔 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟

𝐺𝐹𝑅 = 𝑥1000
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑤𝑜𝑚𝑒𝑛 𝑜𝑓 𝑐ℎ𝑖𝑙𝑑 𝑏𝑒𝑎𝑟𝑖𝑛𝑔 𝑎𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟

𝟑. 𝑻𝒐𝒕𝒂𝒍 𝑭𝒆𝒓𝒕𝒊𝒍𝒊𝒕𝒚 𝑹𝒂𝒕𝒆(𝑮𝑭𝑹)

𝑇𝐹𝑅 = 5 ∑ 𝐴𝑆𝐹𝑅
𝑹𝑬𝑷𝑹𝑶𝑫𝑼𝑪𝑻𝑰𝑶𝑵 𝑹𝑨𝑻𝑬
𝟏. 𝑮𝒓𝒐𝒔𝒔 𝑹𝒆𝒑𝒓𝒐𝒅𝒖𝒄𝒕𝒊𝒐𝒏 𝑹𝒂𝒕𝒆(𝑮𝑹𝑹)
𝐺𝑅𝑅 = 5 ∑ 𝑊𝑆𝐹𝑅

𝟐. 𝑵𝒆𝒕 𝑹𝒆𝒑𝒓𝒐𝒅𝒖𝒄𝒕𝒊𝒐𝒏 𝑹𝒂𝒕𝒆(𝑵𝑹𝑹)

𝐺𝑅𝑅 = 5 ∑(𝑊𝑆𝐹𝑅 𝑥 𝑆) 𝑤ℎ𝑒𝑟𝑒 𝑠 = 𝑆𝑢𝑟𝑣𝑖𝑣𝑎𝑙 𝑟𝑎𝑡𝑖𝑜

𝟑. 𝑾𝒐𝒎𝒆𝒏 𝑺𝒑𝒆𝒄𝒊𝒇𝒊𝒄 𝑭𝒆𝒓𝒕𝒊𝒍𝒕𝒊𝒚 𝑹𝒂𝒕𝒆(𝑾𝑺𝑭𝑹)

𝐹𝑒𝑚𝑎𝑙𝑒 𝐵𝑖𝑟𝑡ℎ𝑠
𝑊𝑆𝐹𝑅 = 𝑥1000
𝐹𝑒𝑚𝑎𝑙𝑒 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛

𝑪𝑶𝑴𝑷𝑶𝑵𝑬𝑵𝑻𝑺 𝑶𝑭 𝑨 𝑳𝑰𝑭𝑬 𝑻𝑨𝑩𝑳𝑬

Age No of No of Mortality Survival No of years No of years

survivors deaths Ratio ratio lived between lived Expectation of life
x and (x+1) after age x
𝑑𝑥 𝑙 𝑥 + 𝑙 (𝑥+1) 𝑇𝑥
x 𝑙𝑥 𝑑𝑥
𝑞𝑥 =
𝑝𝑥 = 1 − 𝑞𝑥
𝐿𝑥 =
𝑇𝑥 = 𝑙 𝑥 + 𝑙 (𝑋+1) + ⋯
𝑒𝑥 0 =
= 𝑙 𝑥 − 𝑙 𝑥+1 𝑙𝑥 2 𝑙𝑥
 𝑰𝑵𝑫𝑬𝑿 𝑵𝑼𝑴𝑩𝑬𝑹
𝑷𝒓𝒊𝒄𝒆 𝑹𝒆𝒍𝒂𝒕𝒊𝒗𝒆; 𝑰𝒕 𝒊𝒔 𝒕𝒉𝒆 𝒑𝒓𝒊𝒄𝒆 𝒊𝒏 𝒕𝒉𝒆 𝒄𝒖𝒓𝒓𝒆𝒏𝒕 𝒚𝒆𝒂𝒓 𝒆𝒙𝒑𝒓𝒆𝒔𝒔𝒆𝒅 𝒂𝒔 𝒂 𝒑𝒆𝒓𝒄𝒆𝒏𝒕𝒂𝒈𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒑𝒓𝒊𝒄𝒆 𝒊𝒏 𝒕𝒉𝒆
𝒃𝒂𝒔𝒆 𝒚𝒆𝒂𝒓
𝑷𝟏
𝑷= 𝒙𝟏𝟎𝟎
𝑷𝟎

𝟏. 𝑷𝑹𝑰𝑪𝑬 𝑰𝑵𝑫𝑬𝑿 𝑵𝑼𝑴𝑩𝑬𝑹

𝑼𝒏 − 𝑾𝒆𝒊𝒈𝒉𝒕𝒆𝒅 𝑷𝒓𝒊𝒄𝒆 𝑰𝒏𝒅𝒆𝒙 𝑵𝒖𝒎𝒃𝒆𝒓

𝑾𝒆𝒊𝒈𝒉𝒕𝒆𝒅 𝑷𝒓𝒊𝒄𝒆 𝑰𝒏𝒅𝒆𝒙 𝑵𝒖𝒎𝒃𝒆𝒓
𝒂) 𝑺𝒊𝒎𝒑𝒍𝒆 𝒂𝒈𝒓𝒓𝒆𝒈𝒂𝒕𝒊𝒗𝒆 𝑷𝑰𝑵:
∑ 𝑷𝟏 𝑾𝒆𝒊𝒈𝒉𝒕𝒆𝒅 𝑨𝒈𝒈𝒓𝒆𝒈𝒂𝒕𝒊𝒗𝒆 𝑷𝑰𝑵:
𝑷𝟎𝟏 = 𝒙𝟏𝟎𝟎
∑ 𝑷𝟎
𝒃) 𝑺𝒊𝒎𝒑𝒍𝒆 𝑨𝒗𝒆𝒓𝒂𝒈𝒆 𝒐𝒇 𝒑𝒓𝒊𝒄𝒆 𝒓𝒆𝒍𝒂𝒕𝒊𝒗𝒆: ∑𝒑𝟏 𝒒𝟎
𝟏. 𝑳𝒂𝒔𝒑𝒆𝒚𝒓𝒆′ 𝒔 𝑷𝑰𝑵(𝑷𝟎𝟏𝑳𝒂 ) = 𝒙𝟏𝟎𝟎
∑𝒑𝟎 𝒒𝟎
𝟏. 𝑨𝒓𝒊𝒕𝒉𝒎𝒆𝒕𝒊𝒄 𝑴𝒆𝒂𝒏
∑𝑷
𝑷𝟎𝟏 =
𝒏 𝟐. 𝑷𝒂𝒂𝒔𝒄𝒉𝒆′ 𝒔 𝑷𝑰𝑵(𝑷𝟎𝟏𝑷𝒂 ) = ∑
∑ 𝒑𝟏 𝒒 𝟏
𝒙𝟏𝟎𝟎
𝒑𝟎 𝒒 𝟏
𝟐. 𝑮𝒆𝒐𝒎𝒆𝒕𝒓𝒊𝒄 𝑴𝒆𝒂𝒏
∑ 𝒍𝒐𝒈 𝑷
𝑷𝟎𝟏 = 𝑨𝒏𝒕𝒊𝒍𝒐𝒈 [ ]
𝒏 𝟑. 𝑴𝒂𝒓𝒔𝒉𝒂𝒍𝒍 − 𝑬𝒅𝒈𝒆𝒘𝒐𝒓𝒕𝒉′ 𝒔 𝑷𝑰𝑵(𝑷𝟎𝟏 𝑴𝒆 )
∑𝒑 𝒒 + ∑𝒑𝟏 𝒒𝟏
=( 𝟏 𝟎 ) 𝒙𝟏𝟎𝟎
∑𝒑𝟎 𝒒𝟎 + ∑𝒑𝟎 𝒒𝟏
𝑾𝒆𝒊𝒈𝒉𝒕𝒆𝒅 𝑨𝒗𝒆𝒓𝒂𝒈𝒆𝒔 𝒐𝒇 𝑷𝒓𝒊𝒄𝒆 𝑹𝒆𝒍𝒂𝒕𝒊𝒗𝒆𝒔

𝟏. 𝑨𝒓𝒊𝒕𝒉𝒎𝒆𝒕𝒊𝒄 𝑴𝒆𝒂𝒏
∑ 𝑷𝑾 𝟒. 𝑫𝒐𝒓𝒃𝒊𝒔𝒉 − 𝑩𝒐𝒘𝒍𝒆𝒚′ 𝒔 𝑷𝑰𝑵(𝑷𝟎𝟏𝑫𝒃 )
𝑷𝟎𝟏 =
∑𝑾 𝟏 ∑𝒑𝟏 𝒒𝟎 ∑𝒑𝟏 𝒒𝟏
= ( + )𝒙𝟏𝟎𝟎
𝟐 ∑𝒑𝟎 𝒒𝟎 ∑𝒑𝟎 𝒒𝟏
𝟐. 𝑮𝒆𝒐𝒎𝒆𝒕𝒊𝒄 𝑴𝒆𝒂𝒏
∑ 𝑾𝒍𝒐𝒈𝑷
𝑷𝟎𝟏 = 𝑨𝒏𝒕𝒊𝒍𝒐𝒈 [ ]
∑𝑾 𝟓. 𝑭𝒊𝒔𝒉𝒆𝒓′𝒔 𝑷𝑰𝑵(𝑷𝟎𝟏 𝑭 )
∑𝒑𝟏 𝒒𝟎 ∑𝒑𝟏 𝒒𝟏
𝑷𝟏 =√ 𝒙 𝒙𝟏𝟎𝟎
𝑾𝒉𝒆𝒓𝒆 𝑷 = 𝒙𝟏𝟎𝟎 𝒂𝒏𝒅 ∑𝒑𝟎 𝒒𝟎 ∑𝒑𝟎 𝒒𝟏
𝑷𝟎
𝑾 = 𝒘𝒆𝒊𝒈𝒉𝒕 𝒐𝒓 𝑩𝒖𝒅𝒛𝒆𝒕 𝒐𝒓 𝑬𝒙𝒑𝒆𝒏𝒔𝒆𝒔

𝑾𝒉𝒆𝒓𝒆 𝑷𝟎𝟏 = 𝑷𝒓𝒊𝒄𝒆 𝒊𝒏𝒅𝒆𝒙 𝒂𝒏𝒅 𝑵𝒐𝒕𝒆: 𝑺𝒉𝒐𝒓𝒕 𝒄𝒖𝒕 𝒇𝒐𝒓𝒎𝒖𝒍𝒂

𝑾 = 𝒘𝒆𝒊𝒈𝒉𝒕𝒔
𝒑𝟏 = 𝑷𝒓𝒊𝒄𝒆 𝒊𝒏 𝒕𝒉𝒆 𝒄𝒖𝒓𝒓𝒆𝒏𝒕 𝒑𝒆𝒓𝒊𝒐𝒅 𝟏. 𝑫𝒐𝒓𝒃𝒊𝒔𝒉 𝑩𝒐𝒘𝒍𝒆𝒚′𝑰𝒏𝒅𝒆𝒙 𝒏𝒖𝒎𝒃𝒆𝒓
𝒑𝟎 = 𝑷𝒓𝒊𝒄𝒆 𝒊𝒏 𝒕𝒉𝒆 𝒃𝒂𝒔𝒆 𝒑𝒆𝒓𝒊𝒐𝒅 𝟏
= [𝑷𝟎𝟏𝒍𝒂 + 𝑷𝟎𝟏 𝒑𝒂 ]
𝒒𝟏 = 𝑸𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝒊𝒏 𝒕𝒉𝒆 𝒄𝒖𝒓𝒓𝒆𝒏𝒕 𝒑𝒆𝒓𝒊𝒐𝒅 𝟐
𝒒𝟎 = 𝑸𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝒊𝒏 𝒕𝒉𝒆 𝒃𝒂𝒔𝒆 𝒑𝒆𝒓𝒊𝒐𝒅 𝟐. 𝑭𝒊𝒔𝒉𝒆𝒓′ 𝒔 𝑰𝒏𝒅𝒆𝒙 𝒏𝒖𝒎𝒃𝒆𝒓

= √𝑷𝟎𝟏 𝒍𝒂 𝒙 𝑷𝟎𝟏𝒑𝒂
 𝟐. 𝑸𝑼𝑨𝑵𝑻𝑰𝑻𝒀 𝑰𝑵𝑫𝑬𝑿 𝑵𝑼𝑴𝑩𝑬𝑹

𝑼𝒏 − 𝑾𝒆𝒊𝒈𝒉𝒕𝒆𝒅 𝑸𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝑰𝒏𝒅𝒆𝒙 𝑵𝒖𝒎𝒃𝒆𝒓

𝑾𝒆𝒊𝒈𝒉𝒕𝒆𝒅 𝑸𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝑰𝒏𝒅𝒆𝒙 𝑵𝒖𝒎𝒃𝒆𝒓

𝒂) 𝑺𝒊𝒎𝒑𝒍𝒆 𝒂𝒈𝒓𝒓𝒆𝒈𝒂𝒕𝒊𝒗𝒆 𝑸𝑰𝑵: 𝑾𝒆𝒊𝒈𝒉𝒕𝒆𝒅 𝑨𝒈𝒈𝒓𝒆𝒈𝒂𝒕𝒊𝒗𝒆 𝑸𝑰𝑵:

∑ 𝒒𝟏
𝑸𝟎𝟏 = 𝒙𝟏𝟎𝟎 ∑𝒒𝟏 𝒑𝟎
∑ 𝒒𝟎
𝟏. 𝑳𝒂𝒔𝒑𝒆𝒚𝒓𝒆′ 𝒔 𝑸𝑰𝑵(𝑸𝟎𝟏 𝑳𝒂 ) = 𝒙𝟏𝟎𝟎
𝒃) 𝑺𝒊𝒎𝒑𝒍𝒆 𝑨𝒗𝒆𝒓𝒂𝒈𝒆 𝒐𝒇 𝒑𝒓𝒊𝒄𝒆 𝒓𝒆𝒍𝒂𝒕𝒊𝒗𝒆: ∑𝒒𝟎 𝒑𝟎

𝟏. 𝑨𝒓𝒊𝒕𝒉𝒎𝒆𝒕𝒊𝒄 𝑴𝒆𝒂𝒏
∑𝑸 ∑𝒒 𝟏 𝒑𝟏
𝑸𝟎𝟏 = 𝟐. 𝑷𝒂𝒂𝒔𝒄𝒉𝒆′ 𝒔 𝑸𝑰𝑵(𝑸𝟎𝟏𝑷𝒂 ) = ∑𝒒 𝟎 𝒑𝟏
𝒙𝟏𝟎𝟎
𝒏

𝟐. 𝑮𝒆𝒐𝒎𝒆𝒕𝒓𝒊𝒄 𝑴𝒆𝒂𝒏
∑ 𝒍𝒐𝒈 𝑸 𝟑. 𝑴𝒂𝒓𝒔𝒉𝒂𝒍𝒍 − 𝑬𝒅𝒈𝒆𝒘𝒐𝒓𝒕𝒉′ 𝒔 𝑸𝑰𝑵(𝑸𝟎𝟏𝑴𝑬 )
𝑸𝟎𝟏 = 𝑨𝒏𝒕𝒊𝒍𝒐𝒈 [ ]
𝒏 ∑𝒒𝟏 𝒑𝟎 + ∑𝒒𝟏 𝒑𝟏
=( ) 𝒙𝟏𝟎𝟎
∑𝒒𝟎 𝒑𝟎 + ∑𝒒𝟎 𝒑𝟏
𝑾𝒆𝒊𝒈𝒉𝒕𝒆𝒅 𝑨𝒗𝒆𝒓𝒂𝒈𝒆𝒔 𝒐𝒇 𝑷𝒓𝒊𝒄𝒆 𝑹𝒆𝒍𝒂𝒕𝒊𝒗𝒆𝒔

𝟏. 𝑨𝒓𝒊𝒕𝒉𝒎𝒆𝒕𝒊𝒄 𝑴𝒆𝒂𝒏 𝟒. 𝑫𝒐𝒓𝒃𝒊𝒔𝒉 − 𝑩𝒐𝒘𝒍𝒆𝒚′ 𝒔 𝑸𝑰𝑵(𝑸𝟎𝟏 𝑫𝑩 )

∑ 𝑸𝑾 𝟏 ∑𝒒𝟏 𝒑𝟎 ∑𝒒𝟏 𝒑𝟏
𝑸𝟎𝟏 = = ( + )𝒙𝟏𝟎𝟎
∑𝑾 𝟐 ∑𝒒𝟎 𝒑𝟎 ∑𝒒𝟎 𝒑𝟏

𝟐. 𝑮𝒆𝒐𝒎𝒆𝒕𝒊𝒄 𝑴𝒆𝒂𝒏
∑ 𝑾𝒍𝒐𝒈 𝑸
𝑸𝟎𝟏 = 𝑨𝒏𝒕𝒊𝒍𝒐𝒈 [ ] 𝟓. 𝑭𝒊𝒔𝒉𝒆𝒓′𝒔 𝑸𝑰𝑵(𝑸𝟎𝟏𝑭 )
∑𝑾
∑𝒒𝟏 𝒑𝟎 ∑𝒒𝟏 𝒑𝟏
=√ 𝒙 𝒙𝟏𝟎𝟎
𝑸𝟏 ∑𝒒𝟎 𝒑𝟎 ∑𝒒𝟎 𝒑𝟏
𝑾𝒉𝒆𝒓𝒆 𝑸 = 𝒙𝟏𝟎𝟎 𝒂𝒏𝒅
𝑸𝟎
𝑾 = 𝒘𝒆𝒊𝒈𝒉𝒕 𝒐𝒓 𝑩𝒖𝒅𝒛𝒆𝒕 𝒐𝒓 𝑬𝒙𝒑𝒆𝒏𝒔𝒆𝒔
𝑵𝒐𝒕𝒆: 𝑺𝒉𝒐𝒓𝒕 𝒄𝒖𝒕 𝒇𝒐𝒓𝒎𝒖𝒍𝒂
𝑾𝒉𝒆𝒓𝒆 𝑸𝟎𝟏 = 𝑷𝒓𝒊𝒄𝒆 𝒊𝒏𝒅𝒆𝒙 𝒂𝒏𝒅
𝟏. 𝑫𝒐𝒓𝒃𝒊𝒔𝒉 𝑩𝒐𝒘𝒍𝒆𝒚′ 𝑰𝒏𝒅𝒆𝒙 𝒏𝒖𝒎𝒃𝒆𝒓 = 𝑸𝟎𝟏 𝑫𝑩
𝑾 = 𝒘𝒆𝒊𝒈𝒉𝒕𝒔
𝒑𝟏 = 𝑷𝒓𝒊𝒄𝒆 𝒊𝒏 𝒕𝒉𝒆 𝒄𝒖𝒓𝒓𝒆𝒏𝒕 𝒑𝒆𝒓𝒊𝒐𝒅 𝟏
𝒑𝟎 = 𝑷𝒓𝒊𝒄𝒆 𝒊𝒏 𝒕𝒉𝒆 𝒃𝒂𝒔𝒆 𝒑𝒆𝒓𝒊𝒐𝒅 = [𝑸𝟎𝟏𝒍𝒂 + 𝑸𝟎𝟏 𝒑𝒂 ]
𝟐
𝒒𝟏 = 𝑸𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝒊𝒏 𝒕𝒉𝒆 𝒄𝒖𝒓𝒓𝒆𝒏𝒕 𝒑𝒆𝒓𝒊𝒐𝒅
𝟐. 𝑭𝒊𝒔𝒉𝒆𝒓′ 𝒔 𝑰𝒏𝒅𝒆𝒙 𝒏𝒖𝒎𝒃𝒆𝒓 = 𝑸𝟎𝟏 𝑭
𝒒𝟎 = 𝑸𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝒊𝒏 𝒕𝒉𝒆 𝒃𝒂𝒔𝒆 𝒑𝒆𝒓𝒊𝒐𝒅

= √𝑸𝟎𝟏 𝒍𝒂𝒙 𝑸𝟎𝟏𝒑𝒂

𝑻𝒆𝒔𝒕 𝒕𝒐 𝒔𝒆𝒍𝒆𝒄𝒕 𝒂𝒑𝒑𝒓𝒐𝒄𝒊𝒂𝒕𝒆 𝑭𝒓𝒐𝒓𝒎𝒖𝒍𝒂
∑𝒑𝟏 𝒒
𝟏. 𝑻𝒊𝒎𝒆 𝑹𝒆𝒗𝒆𝒓𝒔𝒂𝒍 𝑻𝒆𝒔𝒕 (𝑻𝑹𝑻) 𝑲𝒆𝒍𝒍𝒚𝒆′ 𝒔 𝑰𝑵(𝑷𝟎𝟏𝑲 ) = 𝒙𝟏𝟎𝟎
∑𝒑𝟎 𝒒
𝑻𝑹𝑻 = 𝑷𝟎𝟏 𝒙𝑷𝟎𝟏 = 𝟏
𝟐. 𝑭𝒂𝒄𝒕𝒐𝒓 𝑹𝒆𝒗𝒆𝒓𝒔𝒂𝒍 𝑻𝒆𝒔𝒕 (𝑭𝑹𝑻)
∑𝒑𝟏 𝒒𝟏
𝑭𝑹𝑻 = 𝑷𝟎𝟏 𝒙𝑸𝟎𝟏 = ∑𝒑𝟏 𝒒𝟏
∑𝒑𝟎 𝒒𝟎 𝑽𝒂𝒍𝒖𝒆 𝑰𝑵(𝑷𝟎𝟏𝑽𝒂 ) = 𝒙𝟏𝟎𝟎
∑𝒑𝟎 𝒒𝟎
𝑪𝒊𝒓𝒄𝒖𝒍𝒂𝒓 𝑻𝒆𝒔𝒕 = 𝑷𝟎𝟏 𝒙𝑷𝟏𝟐 𝒙𝑷𝟐𝟎 = 𝟏
𝑻𝒆𝒔𝒕 𝑪𝒉𝒂𝒓𝒕

𝑴𝒆𝒕𝒉𝒐𝒅𝒔 𝑻𝑹𝑻 𝑭𝑹𝑻

𝑷𝟎𝟏 𝑳𝒂 𝑵𝒐 𝑵𝒐
𝑷𝟎𝟏 𝑷𝒂 𝑵𝒐 𝑵𝒐
𝑷𝟎𝟏 𝑴𝑬 𝒀𝒆𝒔 𝑵𝒐
𝑷𝟎𝟏 𝑫𝑩 𝑵𝒐 𝑵𝒐
𝑷𝟎𝟏 𝑭 𝒀𝒆𝒔 𝒀𝒆𝒔
 𝑻𝑰𝑴𝑬 𝑺𝑬𝑹𝑰𝑬𝑺
𝑴𝑬𝑻𝑯𝑶𝑫𝑺 𝑶𝑭 𝑴𝑬𝑨𝑺𝑼𝑹𝑰𝑵𝑮 𝑻𝑹𝑬𝑵𝑫
𝟏. 𝑭𝒊𝒕𝒕𝒊𝒏𝒈 𝒔𝒕𝒓𝒊𝒈𝒉𝒕 𝒍𝒊𝒏𝒆
𝟐. 𝑭𝒊𝒕𝒕𝒊𝒏𝒈 𝑷𝒂𝒓𝒂𝒃𝒐𝒍𝒖𝒊𝒄 𝒕𝒓𝒆𝒏𝒅 𝒐𝒓 𝒒𝒖𝒂𝒓𝒅𝒓𝒂𝒕𝒊𝒄 𝒕𝒓𝒆𝒏𝒅 𝒐𝒓 𝒔𝒆𝒄𝒐𝒏𝒅
𝒅𝒆𝒈𝒓𝒆𝒆 𝒑𝒂𝒓𝒂𝒃𝒐𝒍𝒂
𝟑. 𝑭𝒊𝒕𝒕𝒊𝒏𝒈 𝑬𝒙𝒑𝒐𝒏𝒆𝒏𝒕𝒊𝒂𝒍 𝑻𝒓𝒆𝒏𝒅

𝑵𝒐𝒓𝒎𝒂𝒍 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔
𝟏. 𝑭𝒊𝒕𝒕𝒊𝒏𝒈 𝒔𝒕𝒓𝒊𝒈𝒉𝒕 𝒍𝒊𝒏𝒆
𝒚 =𝒂+𝒃𝒙
∑𝒚 = 𝒏𝒂 + 𝒃∑𝒙
∑𝑿𝒀 = 𝒂∑𝒙 + 𝒃∑𝒙𝟐
∑𝒚 ∑𝒙𝒚
𝒘𝒉𝒆𝒓𝒆 𝒂 = 𝒏
,𝒃 = ∑𝒙𝟐
𝟐. 𝑭𝒊𝒕𝒕𝒊𝒏𝒈 𝑷𝒂𝒓𝒂𝒃𝒐𝒍𝒖𝒊𝒄 𝒕𝒓𝒆𝒏𝒅 𝒐𝒓 𝒒𝒖𝒂𝒓𝒅𝒓𝒂𝒕𝒊𝒄 𝒕𝒓𝒆𝒏𝒅 𝒐𝒓 𝒔
𝒆𝒄𝒐𝒏𝒅 𝒅𝒆𝒈𝒓𝒆𝒆 𝒑𝒂𝒓𝒂𝒃𝒐𝒍𝒂
𝒚 = 𝒂 + 𝒃 𝒙 + 𝒄 𝒙𝟐
∑𝒚 = 𝒏𝒂 + 𝒃∑𝒙 + 𝒄∑ 𝒙𝟐 − (𝟏)
∑𝒙𝒚 = 𝒂∑𝒙 + 𝒃∑𝒙𝟐 + 𝒄∑𝒙𝟑 − (𝟐)
∑𝒙𝟐 𝒀 = 𝒂∑𝒙𝟐 + 𝒃∑𝒙𝟑 + 𝒄∑𝒙𝟒 − (𝟑)

𝒊𝒇 ∑𝒙 = 𝟎, ∑𝒙𝟑 = 𝟎
𝒇𝒓𝒐𝒎 𝟏 𝒂𝒏𝒅 𝟐
∑𝒚 = 𝒏𝒂 + 𝒄 ∑𝒙𝟐
∑𝒙𝒚
𝒃=
∑𝒙𝟐
∑𝒙𝟐 𝒀 = 𝒂∑𝒙 + 𝒄∑𝒙𝟒
𝟐

𝟑. 𝑭𝒊𝒕𝒕𝒊𝒏𝒈 𝑬𝒙𝒑𝒐𝒏𝒆𝒏𝒕𝒊𝒂𝒍 𝑻𝒓𝒆𝒏𝒅

𝒚 = 𝒂 𝒃𝒙
∑𝒍𝒐𝒈𝒚 = 𝒏𝒍𝒐𝒈𝒂 + 𝒍𝒐𝒈𝒃∑𝒙
∑𝒙𝒍𝒐𝒈𝒚 = 𝒍𝒐𝒈𝒂∑𝒙 + 𝒍𝒐𝒈𝒃∑𝒙𝟐
∑𝒍𝒐𝒈𝒚 ∑𝒙𝒍𝒐𝒈𝒚
𝒘𝒉𝒆𝒓𝒆 𝒍𝒐𝒈𝒂 = 𝒏
, 𝒍𝒐𝒈𝒃 = ∑𝒙𝟐
∑𝒍𝒐𝒈𝒚 ∑𝒙𝒍𝒐𝒈𝒚
𝒂 = 𝑨𝒏𝒕𝒊𝒍𝒐𝒈 ( 𝒏
) , 𝒃 = 𝑨𝒏𝒕𝒊𝒍𝒐𝒈 ( ∑𝒙𝟐
)
 INTERPOLATION AND EXTRAPOLATION

BINOMIAL EXPANSION METHOD (PASCAL TRIANGAL)

NETWONS ADVANCING DIFFERENCE METHOD

The formula is given by
𝟏 𝑿(𝑿 − 𝟏)∆𝟎 𝟐 𝑿(𝑿 − 𝟏)(𝑿 − 𝟐)∆𝟎 𝟑
𝒚𝒙 = 𝒚𝒙 + 𝑿∆𝟎 + +
𝟐 𝟔
𝟒
𝑿(𝑿 − 𝟏)(𝑿 − 𝟐)(𝑿 − 𝟑)∆𝟎
+ +⋯
𝟐𝟒
𝑮𝒊𝒗𝒆𝒏 𝒗𝒂𝒍𝒖𝒆 − 𝒙𝟎
𝒘𝒉𝒆𝒓𝒆 𝑿 =
𝒅𝒊𝒇𝒇𝒆𝒓𝒂𝒏𝒄𝒆 𝒐𝒇 𝒙 𝒄𝒐𝒍𝒖𝒎𝒏