The median of a set of 9 distinct...

MathematicsMeasures of Central Tendency (Mean, Median, Mode)JEE Main 2003Easy

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The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set

(A)

remains the same as that of the original set

(B)

is increased by 2

(C)

is decreased by 2

(D)

is two times the original median.

Answer:A

Visualized Solution (English)

Understanding the Median Concept

The Median is the middle value of a data set when arranged in ascending or descending order.
Number of observations ( $n$ ) = $9$ (which is an odd number).
For an odd number of terms, the median is a specific term in the sorted sequence.

Locating the Median Position

Position of Median = $(\frac{n + 1}{2})^{t h}$ term.
Substituting $n = 9$ : Position $= \frac{9 + 1}{2} = 5^{t h}$ term.
Let the sorted observations be $x_{1} < x_{2} < x_{3} < x_{4} < x_{5} < x_{6} < x_{7} < x_{8} < x_{9}$ .
Given: Median $= x_{5} = 20.5$ .

Applying the Change to Observations

The largest 4 observations are $x_{6}, x_{7}, x_{8},$ and $x_{9}$ .
New values become: $x_{6}^{'} = x_{6} + 2$ , $x_{7}^{'} = x_{7} + 2$ , $x_{8}^{'} = x_{8} + 2$ , $x_{9}^{'} = x_{9} + 2$ .
Since we only increased values that were already greater than $x_{5}$ , the relative order of the $5^{t h}$ term remains unchanged.

Final Conclusion

The $5^{t h}$ observation ( $x_{5}$ ) is still $20.5$ .
The number of observations is still $9$ , so the median is still the $5^{t h}$ term.
Conclusion: The median remains the same as that of the original set.
Key Takeaway: Changing values on either side of the median without crossing the median position does not affect the median value.

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