First slide

Introduction to statistics

Question

If the mean of the squares of first n natural numbers is 105, then the median of first n natural number is

Moderate

A

8
B

9
C

10
D

11

Solution

We have,
$\begin{array}{l} \frac{1^{2} + 2^{2} + \dots + n^{2}}{n} = 105 \\ \Rightarrow \frac{(n + 1) (2 n + 1)}{6} = 105 \\ \Rightarrow 2 n^{2} + 3 n - 629 = 0 \Rightarrow (2 n + 37) (n - 17) = 0 \Rightarrow n = 17 \end{array}$
Thus, given numbers are $1, 2, 3, 4, \dots, 16, 17$
Clearly, 9 is their median.

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