Question

If mean of n onservation is x. If the first observation is increased by 2, the second by 4, the third by 6 and so on,find the mean of new observations ​

Answer 1

Answer:

New mean is x + n + 1.

Step-by-step explanation:

From the properties of mean :

• Mean = sum of observations / no. of observations

= > Mean of n observations = ( first observation + second observation + third observation + ..... up to nth observation ) / n

= > x = ( first observation + second observation + third observation + ..... up to nth observation ) / n

= > xn = first observation + second observation + third observation + ..... up to nth observation ... ( 1 )

According to the question : First observation is increased by 2, the second by 4, the third by 6 and so on.

= > New mean = { ( or. first observation + 2 ) + ( or. second observation + 4 ) + ( or. third observation + 3 ) + ..... ( nth observation + 2n ) } / n

= > New mean = { ( or. first observation + or. second observation + third observation + ....upto nth observation ) + ( 2 + 4 + 6 + ....upto 2n ) } / n

= > New mean = { xn + 2( 1 + 2 + 3 + .... upto n ) } / n { from ( 1 ) }

Here, 1 + 2 + 3 + .... upto n forms an AS. Sum of AS = ( n / 2 )( 1 + n ) = ( n )( n + 1 ) / 2.

Thus,

= > New mean = [ xn + 2{ ( n )( n + 1 ) / 2 } ] / n

= > New mean = [ xn + n( n + 1 ) ] / n

= > New mean = x + n + 1

Hence, new mean is x + n + 1.