Problem 18 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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 (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.

Step-by-Step Solution

Answer

The median remains the same as the original set.

Step 1: Understanding the Median

The median of a set is the middle value when the numbers are arranged in order. For nine observations, the median will be the 5th number in the sorted list. Thus, the original median is 20.5, which is the 5th observation.

Step 2: Analyzing the Change

Since each of the largest 4 observations is increased by 2, this adjustment impacts only the last four numbers, but not the median directly as it is the 5th observation. The numbers before and including the 5th observation remain the same.

Step 3: Finding the New Median

In the modified set, the 5th observation is still unchanged because the median is the middle value, and the changes occurred only in the observations larger than the median. Thus, the middle position has not been affected.

Step 4: Conclusion

Since the median remains unchanged when the largest four numbers are increased, the correct choice is (a) remains the same as that of the original set.

Key Concepts

Observations

Understanding observations is key in statistical analysis. Observations are individual pieces of data collected in a study or experiment. In the context of this exercise, there are 9 distinct numbers making up the observation set. Each observation is a value that is distinct, meaning no two numbers are the same. Observations can be quantitative, like numbers and figures, or qualitative, like categories or characteristics.

• Each observation in a dataset contributes to understanding trends and patterns.
• Observations must be accurately recorded to ensure reliability of results.
• In statistical problems, knowing the details of each observation helps derive meaningful insights such as the median.

Organizing observations effectively allows for more accurate calculation and understanding of statistical measures like median, mean, and mode.

Ordered List

An ordered list is crucial for identifying the median in any dataset. To determine the median, the observations must be arranged in ascending or descending order. In the case of this exercise, arranging the 9 observations from smallest to largest is essential because the median is directly related to the order.

• Ordering the data helps identify the middle value in a structured way.
• It assists in visualizing the spread and distribution of the data points.
• Without ordering, it becomes challenging to define the median accurately.

Think of an ordered list as a way to organize your data so you can easily find central tendencies such as the median, which indicates the middle of your data when arranged properly.

Middle Value

The middle value in a dataset is known as the median. It is the point that separates the higher half from the lower half when the dataset is ordered. In a list of 9 items, like in our exercise, the 5th item is the median as it is in the middle of the ordered data.
Understanding the median is important because:

• It indicates the central point of a dataset, providing a measure of central tendency.
• It is not influenced by extremely high or low values (outliers), making it a robust measure.
• For odd numbers of observations, the middle value is easy to identify, while for even numbers, it is the average of the two middle values.

In the problem given, when changes occur in the values above the median, the median remains the same because the middle value (5th observation) does not change.

Statistical Analysis

Statistical analysis involves examining datasets to extract useful information. Several key tools and measures are used, including the median, to understand data distribution and infer patterns. In this exercise, statistical analysis helps us understand how changes to a dataset affect its median.
By increasing the largest 4 observations by 2, we analyze if and how this impacts the dataset's structure. Through statistical analysis, it is clear:

• The median acts as a stable measure of central tendency even with alterations in the dataset.
• Other statistical measures might change, but the median remains unchanged due to its nature.
• Effective analysis can help predict outcomes with adjustments or manipulation within data sets.

Statistical analysis is not just about computation but understanding how various elements within data relate to each other and influence shared outputs like the median.