Question

The mean of n observation is `overlineX`.  If the first observation is increased by 1, the second by 2, the third by 3, and so on, then the new mean is

पर्याय

• `overlineX`\[ + \left( 2n + 1 \right)\]
• `overlineX`\[+ \frac{n + 1}{2}\]
• `overlineX`\[ + \left( n + 1 \right)\]

• `overlineX`\[ - \frac{n + 1}{2}\]

Answer 1

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

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

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

If the first item is increased by 1, the second by 2, the third by 3 and so on.

Then, the new observations are

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

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

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

`= (nx +(n(n+1))/2)/n`

`=overlineX`\[ + \frac{n + 1}{2}\]