Problem 56 - A Complete Resource Book in Mathematics for JEE Main

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) \(16.0\) (B) \(15.8\) (C) \(14.0\) (D) \(16.8\)

Step-by-Step Solution

Answer

The mean of the resultant data is 14.

Step 1: Understand the Initial Mean

The mean of the data set with 16 observations is given as 16. This implies that the total sum of all observations in the data set is given by the formula for mean: \[ \text{Total Sum} = \text{Mean} \times \text{Number of Observations} = 16 \times 16 = 256 \].

Step 2: Recalculate the Total with Modifications

The data set is modified by removing one observation valued 16 and adding three new observations valued 3, 4, and 5. Calculate the new total sum: \[ \text{New Total Sum} = 256 - 16 + 3 + 4 + 5 = 252 \].

Step 3: Determine the New Number of Observations

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

Step 4: Calculate the New Mean

Using the new total sum and the new number of observations, calculate the new mean: \[ \text{New Mean} = \frac{252}{18} = 14 \].

Key Concepts

Mean of data set

The mean of a data set is a measure of its central tendency. It provides a single value representative of the entire data set. To find the mean, sum up all the values in the data set and then divide by the number of values. This calculation helps us understand what the typical value in a data set might be.

Finding the mean serves as a basic tool in statistics to summarize data. It is widely used in various fields ranging from economics to natural sciences. The formula for mean is:

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

For example, in our exercise, the initial mean was 16, calculated from 16 observations. The sum of these observations was essential to solving the problem.
By recalculating the mean after modifying the data set, you get valuable insights into how changes in observations affect the central tendency of the data.

Number of observations

The number of observations in a data set reflects the count of individual data points available for analysis. It's a fundamental aspect of any data set and plays a crucial role in statistical calculations. When a data point is added or removed, the number of observations changes, impacting measures like the mean.

In the provided exercise, originally, there were 16 observations. After performing some modifications, we removed one observation and added three new ones, resulting in a revised total of 18 observations. Calculating how the mean changes with these adjustments requires understanding the updated number of data points.

• Initial Observations: 16
• Adjusted Observations: 18 (after removing one and adding three)

Recognizing the number of observations is essential for applying statistical formulas correctly, and it ensures the accuracy of results.

Sum of observations

The sum of observations in a data set is simply the total of all its values. It gives us a baseline for calculating the mean and understanding the data's overall magnitude. Changes to the observations, like adding or removing data points, directly affect this sum.

In our exercise, the initial sum of the data set was computed using the mean and number of observations:

• Original Sum = Mean \( \times \) Number of Observations = \( 16 \times 16 = 256 \)

After making modifications like removing one value of 16 and adding numbers 3, 4, and 5, the new sum became:

• New Sum = \( 256 - 16 + 3 + 4 + 5 = 252 \)

By recalculating the sum, you're able to find the new mean, providing insight into how the data set's central position changes with variation.