The numbers 10, 12, 14, 16, 17 and x are in ascending order. If the mean and median of these observations are same, the value of x is :

1. 16

2. 14

3. 54

4. 21

Numbers in ascending order :

10, 12, 14, 16, 17 and x.

Mean = $\dfrac{\text{Sum of observations}}{\text{No. of observations}}$

Substituting values we get :

$\text{Mean } = \dfrac{10 + 12 + 14 + 16 + 17 + x}{6} \\[1em] = \dfrac{69 + x}{6}.$

No. of terms (n) = 6, which is even.

Median = $\dfrac{\Big(\dfrac{n}{2}\Big) \text{ th term} + \Big(\dfrac{n}{2} + 1\Big) \text{ th term}}{2}$

Substituting values we get :

$\text{Median } = \dfrac{\Big(\dfrac{6}{2}\Big) \text{ th term} + \Big(\dfrac{6}{2} + 1\Big) \text{ th term}}{2} \\[1em] = \dfrac{\text{3rd term + 4th term}}{2} \\[1em] = \dfrac{14 + 16}{2} \\[1em] = \dfrac{30}{2} \\[1em] = 15.$

Given,

Mean = Median

$\therefore \dfrac{69 + x}{6} = 15 \\[1em] \Rightarrow 69 + x = 15 \times 6 \\[1em] \Rightarrow 69 + x = 90 \\[1em] \Rightarrow x = 90 - 69 = 21.$

Hence, Option 4 is the correct option.