Question

Median of the data 5, 56, 48, 13, 27, 91, 16, 9, 78 is \(x\). Later the observations 56, 78 are found to be incorrect and they are replaced by 33 and 42 respectively. If the median of the new data is \(y\), then \(x^{3}-y^{3}=\)

• A. 5572
• B. 8127
• C. 3600
• D. 0

Hint:

For an odd number of observations, the median is simply the middle term after arranging the data in ascending order.

Answer 1

The Correct Option is D

Step 1: Find the original median \(x\). Arrange the observations in ascending order: \[ 5,\;9,\;13,\;16,\;27,\;48,\;56,\;78,\;91 \] Since there are 9 observations, \[ x=\text{5th observation}=27 \]

Step 2: Replace the incorrect observations. New data: \[ 5,\;33,\;48,\;13,\;27,\;91,\;16,\;9,\;42 \] Arranging: \[ 5,\;9,\;13,\;16,\;27,\;33,\;42,\;48,\;91 \] Thus, \[ y=\text{5th observation}=27 \]

Step 3: Compute \(x^3-y^3\). \[ x^3-y^3 = 27^3-27^3 = 0 \] 0