The variance of the numbers 8, 21, 34, 47, ..., 320 is __________. [2025]

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Solution

Q.
The variance of the numbers 8, 21, 34, 47, ..., 320 is __________. [2025]

Ans.
(8788)

Since given numbers are in A.P.

$∴ 8 + (n - 1) 13 = 320 \Rightarrow 13 n = 325 \Rightarrow n = 25$

Number of terms = 25

Now, Mean $= \frac{\sum_{}^{} x_{i}^{}}{n} = \frac{8 + 21 + . . . + 320}{25} = \frac{\frac{25}{2} (8 + 320)}{25} = 164$

and variance $(σ^{2}) = \frac{\sum_{}^{} x_{i}^{2}}{n} - {(mean)}^{2}$

$= \frac{8^{2} + 21^{2} + . . . + 320^{2}}{25} - {(164)}^{2} = \frac{\sum_{n = 1}^{25} {(13 n - 5)}^{2}}{25} - 26896$

$= \frac{\sum_{n = 1}^{25} (169 n^{2} + 25 - 130 n)}{25} - 26896$

$= \frac{\frac{169 \times 25 \times 26 \times 51}{6} + 625 - \frac{130 \times 25 \times 26}{2}}{25} - 26896$

= 35684 –26896 = 8788.

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