Problem 7 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The mean of the data set comprising of 16 observations is 16\. If one of the observation valued 16 is deleted and three new observations valued 3,4 and 5 are added to the data, then the mean of the resultant data, is: (a) \(15.8\) (b) \(14.0\) (c) \(16.8\) (d) \(16.0\)

Step-by-Step Solution

Answer

The mean of the resultant data is 14.0.

Step 1: Calculate Original Sum of Observations

Given that the mean of the data set is 16 for 16 observations, calculate the sum of all observations using the formula for mean: \( \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} \). Hence, \( 16 = \frac{S}{16} \), where \( S \) is the sum of all observations. Multiplying both sides by 16 gives \( S = 256 \).

Step 2: Adjust Sum After Modification

After removing an observation valued 16, the new sum becomes \( 256 - 16 = 240 \). Then, add the new observations 3, 4, and 5: \( 240 + 3 + 4 + 5 = 252 \).

Step 3: Determine New Number of Observations

Originally, there were 16 observations. After removing one and then adding three new ones, the total number of observations changes to \( 16 - 1 + 3 = 18 \).

Step 4: Calculate New Mean

Use the formula for mean again with the updated sum and number of observations. The new mean \( \text{New Mean} = \frac{\text{New Sum}}{\text{New Number of Observations}} = \frac{252}{18} = 14 \).

Step 5: Confirm the Calculated Mean Matches Answer Choices

The calculated mean of 14 matches option (b) 14.0 from the given choices.

Key Concepts

Data Set Manipulation

Data set manipulation is all about making changes to a group of numbers we call a data set. These changes involve adding, removing, or altering the data points, known as observations.
In this exercise, we started with a data set of 16 observations. This group averaged out to a mean of 16. But then, something changes.
One observation valued at 16 is taken away. Meanwhile, three new observations with values 3, 4, and 5 are added in. This process showcases the manipulative aspect—changing the size and composition of the data set.
Why is data set manipulation crucial? Because real-world data is dynamic. Understanding how changes affect overall statistics, especially the mean, helps in analyzing such data flexibly and accurately.

Mean Formula

The mean is a simple yet powerful statistic and understanding the mean formula helps us capture the center of any data set. The formula is:

• Mean = \( \frac{\text{Sum of Observations}}{\text{Number of Observations}} \)

Here, we took a data set with 16 observations, each contributing to a mean of 16. By using the formula, you can imagine summing all observations to find a total sum. For this exercise, 16 as the mean and 16 as the number of observations gave us a sum of 256.
This step sets the stage for re-calculating the mean after evolving the data set. It emphasizes how important knowing the total sum and number of items is, allowing us to keep up with dynamic data changes.

Observation Adjustments

Making adjustments to observations in a data set affects the overall mean. Here's how:
First, when we remove an observation, we subtract its value from the total sum. In this case, removing a value of 16 decreased the sum to 240. Then, introducing new observations with values 3, 4, and 5 increased the summed value to 252.
Simultaneously, altering the count of observations, initially 16, later becoming 18, is crucial as it shifts the denominator in the mean formula. Once we adjust both the sum and number of observations, recalculating the mean becomes straightforward.

• New Mean = \( \frac{252}{18} = 14 \)

Understanding this process allows you to see just how sensitive mean can be to changes within a data set and highlights the importance of each single data point.