Question

The average of \(n\) numbers \(y_{1}, y_{2} \ldots \ldots . . y_{n}\) is M. If \(y_{n}\) is replaced by \(\mathrm{y}^{\prime}\) then the new average is (a) \(\left[\left(M+y_{n}-y^{\prime}\right) / n\right]\) (b) \(\left[\left\\{(n-1) M+y^{\prime}\right\\} / n\right]\) (c) \(\left[\left(n M-y_{n}+y^{\prime}\right) / n\right]\) (d) \(M-y_{n}-y^{\prime}\)

Short answer

The new average when \(y_n\) is replaced by \(y'\) will be (c) \(\left[\left(n M-y_{n}+y^{\prime}\right) / n\right]\).

Step-by-step solution

Find the sum of n numbers

To find the sum of the n numbers, we can multiply the average (M) by n: Sum = M * n

Replace yn by y'

Now, we are replacing the number yn with y'. To find the new sum, we can first subtract yn from the sum we calculated in step 1, and then add y': New Sum = Sum - yn + y'

Substitute the expression for Sum

From step 1, we can substitute Sum = M * n in step 2: New Sum = (M * n) - yn + y'

Calculate the new average

Now, we can find the new average by dividing the new sum by n: New Average = New Sum / n

Simplify and find the correct option

Simplify the expression derived in step 4 and compare it with the given options: New Average = [(M * n) - yn + y'] / n = [(n * M - yn + y')] / n Comparing this simplified expression with the given options, we find that the correct answer is (c). So, the new average when yn is replaced by y' will be: (c) \(\left[\left(n M-y_{n}+y^{\prime}\right) / n\right]\)

Key concepts

Replacement of Data

The concept of replacement of data involves changing one element in a set of numbers with another value. In our exercise, this means replacing the number \( y_n \) with a new value \( y' \).

This step is crucial because it directly affects the computations that follow, specifically the sum of numbers and the resulting average. It illustrates the basic principle that altering even a single data point can change the entire dataset's dynamics.

In practical terms, think of it as changing a "member" of the dataset family and observing how it impacts the overall "harmony" or average of the group.

Sum of Numbers

The sum of a series of numbers is found by adding up all the individual elements.

In mathematical terms, if you have \( n \) numbers each represented as \( y_1, y_2, ... , y_n \), then the sum \( S \) can be calculated as:

• \( S = y_1 + y_2 + ... + y_n \)

In our exercise, we initially express this sum using the average \( M \) as \( S = M \times n \). This helps simplify further computations.

Once \( y_n \) is replaced by \( y' \), the new sum changes to \( S_{new} = (M \times n) - y_n + y' \), indicating the adjustment for the replaced value in the total sum.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions to solve equations or find unknown variables.

In this exercise, after replacing \( y_n \), we manipulate the original equation to find the new sum. By replacing \( y_n \) in \( S = M \times n \), we derive the new sum:

• \( S_{new} = (M \times n) - y_n + y' \)

Such manipulation is essential for transforming equations into more manageable forms, making the calculation of the new average straightforward.

Skills in algebraic manipulation enable us to work through complex problems by breaking them down into simpler steps.

New Average Computation

The computation of a new average involves dividing the new sum of numbers by the total number of elements \( n \). This gives us the updated mean after replacing a particular number.

In the exercise, after finding the new sum \( (M \times n) - y_n + y' \), the new average \( A_{new} \) is calculated as:

• \( A_{new} = \frac{(M \times n) - y_n + y'}{n} \)

This computation provides a fresh perspective on the dataset's central tendency.

It’s a reminder that any single change, such as replacing one number, impacts the collective average, illustrating how averages provide a comprehensive overview of data.