Question

Mean of a certain number of observation is `overlineX`.  If each observation is divided by m(m ≠ 0) and increased by n, then the mean of new observation is

Options

• `overlineX/m +n`

• `overlineX/n+m`

   

• `overlineX +n/m`

• `overlineX +m/n`

Answer 1

Let

\[y_1 , y_2 , y_3 , . . . , y_k\]

 be k observations.
Mean of the observations = `overlineX`

\[\Rightarrow \frac{y_1 + y_2 + y_3 + . . . + y_k}{k} = x\]

\[ \Rightarrow y_1 + y_2 + y_3 + . . . + y_k = kx . . . . . \left( 1 \right)\]

If each observation is divided by m and increased by n, then the new observations are

\[\frac{y_1}{m} + n, \frac{y_2}{m} + n, \frac{y_3}{m} + n, . . . , \frac{y_k}{m} + n\]

∴ Mean of new observations

\[= \frac{\left( \frac{y_1}{m} + n \right) + \left( \frac{y_2}{m} + n \right) + . . . + \left( \frac{y_k}{m} + n \right)}{k}\]

\[ = \frac{\left( \frac{y_1}{m} + \frac{y_2}{m} + . . . + \frac{y_k}{m} \right) + \left( n + n + . . . + n \right)}{k}\]

\[ = \frac{y_1 + y_2 + . . . + y_k}{mk} + \frac{nk}{k}\]

`(koverlineX)/mk + (nk)/k`

\[ = \fr

`overlineX/m +n`