Ex 15.2, 2 - Find mean and variance for first n natural numbers - Standard deviation and variance - Ungrouped data

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  1. Chapter 15 Class 11 Statistics
  2. Serial order wise
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Ex 15.2, 2 Find the mean and variance for the first n natural numbers First n natural numbers = 1, 2, 3, 4, โ€ฆโ€ฆ n Mean = (1 + 2 + 3 + 4 + โ€ฆโ€ฆ. + n)/n = (n(n + 1)/2)/๐‘› = (๐ง + ๐Ÿ)/๐Ÿ Variance (ฯƒ2) = (โˆ‘โ–’(๐‘ฅ_๐‘– โˆ’ ๐‘ฅ ฬ… )^2 )/n As the data is very large hence, "(xi โ€“ " ๐‘ฅ ฬ…")" 2 calculation is difficult. Hence, we can use the formula : โˆด Variance (ฯƒ)2 = (๐‘›(n + 1)(2n + 1))/6๐‘›โˆ’ (n + 1)2/4 = ((n + 1)(2n + 1))/6โˆ’ (n + 1)2/4 = ((๐‘› + 1))/2 ((2n + 1)/3 " โˆ’ " (๐‘› + 1)/2) = ((๐‘› + 1))/2 ((4n + 2 โˆ’ 3๐‘› โˆ’ 3)/6) = (๐‘› + 1)/2 ร— (๐‘› โˆ’ 1)/6 = (๐ง^๐Ÿ โˆ’ ๐Ÿ)/๐Ÿ๐Ÿ

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