Question

The mean of n observation is `overlineX` .f the first item is increased by 1, second by 2 and so on, then the new mean is

Options

•  `overlineX+n`

•  `overlineX+n/2`

•  `overlineX+(n+1)/2`

• None of these

Answer 1

Let \[x_1 , x_2 , x_3 , . . . , x_n\]  be the n observations.

Mean

`overlineX`\[= \frac{x_1 + x_2 + . . . + x_n}{n}\]

\[\Rightarrow x_1 + x_2 + x_3 + . . . + x_n = \]`noverlineX`

If the first item is increased by 1, second by 2 and so on.
Then, the new observations are

\[x_1 + 1, x_2 + 2, x_3 + 3, . . . , x_n + n\].

New mean = \[\frac{\left( x_1 + 1 \right) + \left( x_2 + 2 \right) + \left( x_3 + 3 \right) + . . . + \left( x_n + n \right)}{n}\]

\[= \frac{x_1 + x_2 + x_3 + . . . + x_n + \left( 1 + 2 + 3 + . . . + n \right)}{n}\]

\[ = \frac{nX + \frac{n\left( n + 1 \right)}{2}}{n}\]

\[ = X + \frac{n + 1}{2}\]