Question

If the median of the observations x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈ is m, then the median of the observations x₃, x₄, x₅, x₆ (where x₁ < x₂ < x₃ < x₄ < x₅ < x₆ < x₇ < x₈) is _________.

Answer 1

If the number of observations i.e. n is even, then

Median = Mean of $\left(\frac{n}{2}\right)\mathrm{th}$ observation and $\left(\frac{n}{2}+1\right)\mathrm{th}$ observation

Now,

The median of the observations x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈ is m. Here, n = 8.

$$\begin{gathered} \therefore \frac{4\mathrm{th}\mathrm{observation}+5\mathrm{th}\mathrm{observation}}{2}=m \\ \Rightarrow \frac{x_4+x_5}{2}=m \\ \Rightarrow x_4+x_5=2m.....\left(1\right) \end{gathered}$$


Consider the observations x₃, x₄, x₅, x₆. Here, n = 4.

∴ Median of the observations x₃, x₄, x₅, x₆ = $\frac{2\mathrm{nd}\mathrm{observation}+3\mathrm{rd}\mathrm{observation}}{2}=\frac{x_4+x_5}{2}=\frac{2m}{2}=m$ [Using (1)]

If the median of the observations x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈ is m, then the median of the observations x₃, x₄, x₅, x₆ (where x₁ < x₂ < x₃ < x₄ < x₅ < x₆ < x₇ < x₈) is ____m____.