Question

the mean of 31 result is 6. if the mean of 1st 16 result is 8 delta that of last 16 result is 62. find the 16th result.​

Answer 1

Let the results are $a_1 , a_2 , a_3 ,.... a_{16} ,... a_{31}$

From the properties of mean we know : -

• Mean = ( sum of observations ) / ( number of observations )

Thus,

= > Mean of 31 results = ( $a_1+ a_2</p><p>+ a_3 + ....+ a_{16}....+ a_{31}$ ) / 31

= > 6 = $a_1+ a_2 + a_3 +.,...a_{16} + a_{31}$ / 31

= > 6 x 31 = $a_1+ a_2 + a_3 +.....a_{16}...+ a_{31}$

= > 186 = $a_1+ a_2 + a_3 +....a_{16}.....+ a_{31}$ ----: ( 1 )

= > Mean of 1st 16 results = ( sum of 1st 16 results ) / 16

= > 8 = ( $a_1 + a_2 + a_3 + .....+a_{16}$ ) / 16

= > 8 x 16 = $a_1+ a_2 + a_3+....+a_{16}$

= > 128 = $a_1+ a_2 + a_3+....+ a_{16}$ ----: ( 2 )

= > Mean of last 16 results = ( sum of last 16 results ) / 16

= > 62 = ( ( $a_{16} .....+ a_{31}$ ) / 16

= > 62 x 16 = $a_{16} + ..... + a_{31}$

= > 992 = $a_{16}+.....+a_{31}$ ----: ( 3 )

Adding ( 2 ) and ( 3 ) and subtracting ( 1 ) from the addition of ( 2 ) and ( 3 ) ,

= > 128 + 992 - 186 = [ ( $a_1+ a_2 + a_3+....+ a_{16}$ ) + ( $a_{16}+.....+a_{31}$) - ( $a_1+ a_2 + a_3 +.,...a_{16} + a_{31}$ )

= > 934 = $a_{16}$

Hence the required 16th result is 934.