| The mean of n observations is \[{{\bar{x}}_{1}}.\]If the first observation is increased by 1, second by 2 and so on, then their mean is \[{{\bar{x}}_{2}}.\]The value of \[{{\bar{x}}_{2}}-{{\bar{x}}_{1}}\]is [Oriental Insurance Company (AAO) 2012] |
A) \[n\]
B) \[\frac{n}{2}+1\]
C) \[\frac{n\,\,(n+1)}{2}\]
D) \[\frac{(n+1)}{2}\]
E) None of these
Correct Answer: D
Solution :
| Here, new mean,\[{{\bar{x}}_{2}}\] |
| \[=\frac{{{x}_{1}}+{{x}_{2}}+...+{{x}_{n}}}{n}+\frac{1+2+3+...+n}{n}\] |
| \[={{\bar{x}}_{1}}+\frac{n\,\,(n+1)}{2n}={{\bar{x}}_{1}}=\frac{n+1}{2}\] |
| \[\therefore \]\[{{\bar{x}}_{2}}+\,\,{{\bar{x}}_{1}}={{\bar{x}}_{1}}+\frac{n+1}{2}-{{\bar{x}}_{1}}=\frac{n+1}{2}\] |
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