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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 _________.

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Solution

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. $\therefore \frac{4\mathrm{th}\mathrm{observation}+5\mathrm{th}\mathrm{observation}}{2}=m\phantom{\rule{0ex}{0ex}}⇒\frac{{x}_{4}+{x}_{5}}{2}=m\phantom{\rule{0ex}{0ex}}⇒{x}_{4}+{x}_{5}=2m.....\left(1\right)$ 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____.

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