First find the median of $ 46, 64, 87, 41, 58, 77, 35, 90, 55, 92, $ and $ 33 $. If $ 92 $ is replaced with $

Question

First find the median of $ 46, 64, 87, 41, 58, 77, 35, 90, 55, 92, $ and $ 33 $. If $ 92 $ is replaced with $ 99 $ and $ 41 $ is replaced with $ 43 $ in the data, then find the new median. What is the difference between the old median and the new median?

Examoogle · Accepted Answer

The correct answer is Option D: 0. Sort the data to identify the middle term. Since there are $n=11$ values, the median is the 6th value in the ordered sequence.

Step-by-Step Solution & Concepts:
Shortcut Trick: Sort the data to identify the middle term. Since there are $n=11$ values, the median is the 6th value in the ordered sequence.

Formula Used: $$Median = 	ext{middle value in an ordered set}$$

Given: Data set: {46, 64, 87, 41, 58, 77, 35, 90, 55, 92, 33}. Replacements: 92 is replaced by 99, and 41 is replaced by 43.

Steps: Sort the original data: List the original data points in ascending order to find the median: {33, 35, 41, 46, 55, 58, 64, 77, 87, 90, 92}.. Find the old median: With $n=11$ terms, the median is the 6th term. The 6th term is 58.. Modify the data set: Replace 92 with 99 and 41 with 43. New data: {46, 64, 87, 43, 58, 77, 35, 90, 55, 99, 33}.. Find the new median: Sorting the new data: {33, 35, 43, 46, 55, 58, 64, 77, 87, 90, 99}. The 6th term remains 58.. Calculate the difference: Difference = $|Median_{old} - Median_{new}| = |58 - 58| = 0$.

Answer: D is correct because both the old median and the new median are 58, resulting in a difference of 0.

Exam Tip: This concept relates to the 'Robustness of Statistics'; note that the median is less sensitive to extreme outliers than the mean.

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