The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12 . If μ and σ^2 denote the mean and variance of the correct observations respectively, then 15(μ+μ^2+σ^2 ) is equal to

Ans. (2521)
Sol. Let the incorrect mean be μ’ and standard deviation be σ’
We have
μ’=(Σxi)/15=12⇒Σxi=180
As per given information correct Σxi=180-10+12
⇒μ( correct mean )=182/15
Also
σ’=√((Σxi^2)/15-144)=3⇒Σxi ^2=2295
Correct Σxi ^2=2295-100+144=2339
σ^2 ( correct variance )=2339/15-(182×182)/(15×15)
Required value
=15(μ+μ^2+σ^2 )
=15(182/15+(182×182)/(15×15)+2339/15-(182×182)/(15×15))
=15(182/15+2339/15)
=2521