Introduction to statistics

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The variance of first 20-natural numbers is

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$\begin{array}{l} σ^{2} = \frac{1}{20} [1^{2} + 2^{2} + \dots + 20^{2}] - {[\frac{1}{20} (1 + 2 + \dots + 20)]}^{2} \\ = \frac{1}{20} \frac{20 \times 21 \times (2 \times 20 + 1)}{6} - {[\frac{1}{20} \frac{20 \times 21}{2}]}^{2} \\ = \frac{7 \times 41}{2} - \frac{441}{4} = \frac{133}{4} . \end{array}$
In fact, the variance of first $n$ -natural numbers is $\frac{n^{2} - 1}{12}$ .

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Introduction to statistics

Remember concepts with our Masterclasses.

The variance of first 20-natural numbers is

σ2=12012+22+…+202−120(1+2+…+20)2=12020×21×(2×20+1)6−12020×2122=7×412−4414=1334.In fact, the variance of first n-natural numbers is n2−112.

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