Question

The arithmetic mean of $k$ numbers $y_1,y_2,...y_k$ is $A$. If $y_k$ is replaced by $x_k$, then the new arithmetic mean will be :

• A. $A-y_k+x_k$
• B. $\frac{kA-y_k+x_k}{k}$
• C. $A-(y_k-x_k)$
• D. $\frac{(k-1)A}{k}-y_k+x_k$

Answer 1

Answer: (B)

$\frac{kA-y_k+x_k}{k}$

Mean is the sum of the data values divided by the total number of values.

Given that arithmetic mean of $k$ numbers $y_1,y_2,...y_k$ is $A$

$⟹A=\frac{y_1+y_2+...+y_k}{k}$

$⟹y_1+y_2+...+y_{k-1}+y_k=Ak$

$⟹y_1+y_2+...+y_{k-1}=Ak-y_k$ ------(1)

Given that $y_k$ is replaced by $x_k$

Therefore new mean is $\frac{y_1+y_2+...+y_{k-1}+x_k}{k}$

$⟹\text{new mean}=\frac{Ak-y_k+x_k}{k}$ (from (1))