Problem 41 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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 [2003] (A) is increased by 2 (B) is decreased by 2 (C) is two times the original median (D) remains the same as that of the original set

Step-by-Step Solution

Answer

The median remains the same as that of the original set (D).

Step 1: Understanding the Median

For a set with an odd number of observations, like 9, the median is the middle value when all the numbers are arranged in order. Here, the set is given to have 9 observations with the median being 20.5.

Step 2: Identify the Median Position in the Original Set

Since there are 9 observations, the 5th observation when arranged in increasing order will be the median. Therefore, the median of the original set is the 5th observation, which is 20.5.

Step 3: Explore the Impact of Increasing the Largest Four Observations

Increasing the largest four observations doesn't affect the 5th observation in terms of its position in the ordering of the set. This is because these increases affect only the numbers after the 5th observation, maintaining the original middle position.

Step 4: Conclusion about the New Median

Since the position of the median (the 5th observation) and the actual median value (20.5) are unaffected by adding 2 to only the largest four numbers, the median remains the same.

Key Concepts

Distinct Observations

When analyzing a data set, it is crucial to understand the concept of distinct observations. "Distinct" indicates that each observation or data point in the set is unique and there are no repeats. This uniqueness of each data point ensures that the calculation of the median is straightforward without the complexity of duplicate numbers affecting the central value.

In the given exercise, the data set consists of 9 distinct observations. Since all the observations are different from each other, determining the median becomes a direct task. It simplifies the process since you do not have to account for any repeated values, which could skew the calculation and alter the central tendencies of the set.

Odd Number of Observations

When calculating the median of a data set, the number of observations plays a crucial role. In sets with an odd number of observations, the median is the middle value when the data points are sorted in order. This simplicity arises because there is a clear single middle data point.

For our exercise example, the set comprises 9 observations. Since 9 is an odd number, the median is simply the 5th value when all 9 values are ordered. This position is what keeps the calculation straightforward and allows for more efficient identification of the median. No averaging of middle numbers is needed here, unlike even-numbered data sets.

Impact of Changes on Median

Understanding how changes in the data affect the median is vital. In the context of our problem, increasing the largest four observations by a constant, 2 in this case, does not change the position of the original median. The median depends solely on the middle value of the ordered list when the number of data points is odd.

Since this alteration doesn't impact the ordering of the first five observations and hence does not affect the 5th position, the median remains 20.5. The median remains unaffected as long as changes to the data occur to values outside the middle position. The result remains unchanged despite the transformations applied to one side of the data set.

Median Position

The position of the median is essential for determining the central value in any data set. For odd numbers of observations, the median's position is directly the middle one, calculated using the formula:

• - Median Position = \(\frac{n + 1}{2}\)

where \(n\) is the total number of observations.

In our exercise with 9 observations, the median position is the 5th observation (\(\frac{9 + 1}{2} = 5\)). This property of odd-numbered data sets ensures the median is a single unambiguous value. As a result, when data changes that do not affect this crucial position, the median remains consistent.