Problem 24 - A Complete Resource Book in Mathematics for JEE Main

Question

If the mean of a set of observations \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) is \(\bar{x}\), then mean of observations \(x_{i}+3 i \forall i=1,2,3, \ldots n\) equals (A) \(\bar{x}+3(n+1)\) (B) \(\bar{x}+\frac{3(n+1)}{2}\) (C) \(\bar{x}+\frac{n+1}{2 n}\) (D) None of these

Step-by-Step Solution

Answer

The correct answer is (B) \(\bar{x} + \frac{3(n+1)}{2}\).

Step 1: Understanding the Problem

We are given that the mean of a set of observations \(x_1, x_2, x_3, \ldots, x_n\) is \(\bar{x}\). We need to find the mean of the modified observations \(x_i + 3i\) for all \(i = 1, 2, 3, \ldots, n\).

Step 2: Expressing the Mean of Original Observations

The mean \(\bar{x}\) of the original set of observations can be expressed as \(\bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n}\).

Step 3: Calculate the Sum of Modified Observations

For each observation, we add \(3i\), so the sum of modified observations becomes \((x_1 + 3 \, (1)) + (x_2 + 3 \, (2)) + ... + (x_n + 3 \, (n))\). This simplifies to \((x_1 + x_2 + ... + x_n) + 3(1 + 2 + ... + n)\).

Step 4: Summing the Series

The sum of the series \(1 + 2 + ... + n\) can be calculated using the formula for the sum of the first \(n\) natural numbers: \(\frac{n(n+1)}{2}\).

Step 5: Combine and Simplify

The sum of the modified observations is \((x_1 + x_2 + ... + x_n) + 3 \cdot \frac{n(n+1)}{2}\). This simplifies to \(\sum_{i=1}^{n} x_i + \frac{3n(n+1)}{2}\).

Step 6: Find the New Mean

Divide this new sum by \(n\) to get the new mean: \[\text{New Mean} = \frac{\sum_{i=1}^{n} x_i + \frac{3n(n+1)}{2}}{n} = \bar{x} + \frac{3(n+1)}{2}\].

Step 7: Identify the Correct Answer

The calculated new mean, \(\bar{x} + \frac{3(n+1)}{2}\), matches option (B).

Key Concepts

Summation of natural numbers

Summation of natural numbers is a fundamental concept in mathematics. To find the sum of the first \( n \) natural numbers, we use a handy formula:

• \( rac{n(n+1)}{2} \)

This formula is derived from the pairing method. Imagine pairing the first and last numbers, such as 1 and \( n \), while working towards the center
For example, if \( n = 5 \), the sum is: 1 + 2 + 3 + 4 + 5.
Using the formula, you get:

• \( rac{5(5+1)}{2} = 15 \)

This approach works because each pair totals \( n + 1 \). There are \( rac{n}{2} \) such pairs when \( n \) is even, ensuring accuracy and efficiency in calculating sums.

Arithmetic operations in statistics

Arithmetic operations are essential in calculating various statistical measures. The mean, or average, is a primary metric used for summarizing data. To calculate the mean of a set of observations \( x_1, x_2, ..., x_n \), you perform the following steps:

• Sum the observations: \( ext{Sum} = x_1 + x_2 + ... + x_n \)
• Divide by the number of observations: \( ar{x} = \frac{ ext{Sum}}{n} \)

The mean provides a central value for data, helping to understand overall trends. Besides this, sometimes data needs modification or transformation. For example, each observation could be adjusted by a constant value like 3 times the rank number, say \( x_i + 3i \). This operation shifts the data but retains the form, positioning the altered set of values into the same arithmetic framework.

Calculation of modified means

The calculation of modified means involves adjusting the original set of observations by a specified change. Suppose each observation \( x_i \) is modified by adding \( 3i \) — where \( i \) represents the position or index in the data set. To find the new mean, follow these steps:

• Calculate the original mean: \( \bar{x} = \frac{x_1 + x_2 + ... + x_n}{n} \)
• Add \( 3i \) to each observation and find the new sum: \( ext{New Sum} = x_1 + 3(1) + x_2 + 3(2) + ... + x_n + 3(n) \)
• Apply the summation of natural numbers: \( rac{n(n+1)}{2} \)
• Combine the sums: \( ext{New Sum} = \sum_{i=1}^{n} x_i + 3 \cdot \frac{n(n+1)}{2} \)
• Calculate the new mean: \( ext{New Mean} = \frac{ ext{New Sum}}{n} = \bar{x} + \frac{3(n+1)}{2} \)

The modified mean accounts for individual adjustment across the data set, revealing how systematic changes to individual observations impact the statistical measure of the mean.