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Ex 13.2, 2 Find the mean and variance for the first n natural numbers First n natural numbers = 1, 2, 3, 4, …… n Mean = (1 + 2 + 3 + 4 + ……. + n)/n = (n(n + 1)/2)/𝑛 = (𝐧 + 𝟏)/𝟐 Variance (σ2) = (∑▒(𝑥_𝑖 − 𝑥 ̅ )^2 )/n As the data is very large hence, "(xi – " 𝑥 ̅")" 2 calculation is difficult. Hence, we can use the formula : ∴ Variance (σ)2 = (𝑛(n + 1)(2n + 1))/6𝑛− (n + 1)2/4 = ((n + 1)(2n + 1))/6− (n + 1)2/4 = ((𝑛 + 1))/2 ((2n + 1)/3 " − " (𝑛 + 1)/2) = ((𝑛 + 1))/2 ((4n + 2 − 3𝑛 − 3)/6) = (𝑛 + 1)/2 × (𝑛 − 1)/6 = (𝐧^𝟐 − 𝟏)/𝟏𝟐

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.