Question

Four important measurements were taken in harsh environmental conditions. Two of the four measurements were lost in the harsh conditions and the remaining two are

5       3

However a scientist recalls the mean and variance of the four measurements were 6 and 20/3, respectively. Find the two missing measurements.

Answer 1

Mean = 6

Variance = 20/3

First observation = 5

Second observation = 3

Let, the third and fourth observations be a and b

number of observations = 4

Mean = sum of observations / number of observations

6 = ( 5 + 3 + a + b )/4

24 = 8 + a + b

a + b = 24 - 8

a + b = 16

Variance =

20/3 = [ ( 5 - 6 )^2 + ( 3 - 6 )^2 + ( a - 6 )^2 + ( b - 6 )^2 ]/3

20/3 = [ ( -1 )^2 + ( -3 )^2 + ( a - 6 )^2 + ( b - 6 )^2 ]/3

( 20/3 )*3= 1 + 9 + ( a - 6 )^2 + ( b - 6 )^2

20 = 10 + ( a - 6 )^2 + ( b - 6 )^2

From mean, a + b = 16

b = 16 - a

20 - 10 = ( a - 6 )^2 + ( 16 - a - 6 )^2

10 =   ( a - 6 )^2 + ( 10 - a )^2

10 =  a^2 + 6^2 - 2*6*a + a^2 + 10^2 - 2*10*a

10 = 2*a^2 + 36 + 100 - 12*a - 20*a

10 = 2*a^2 - 32*a + 136

5 = a^2 - 16*a + 68

a^2 - 16*a + 68 - 5 = 0

a^2 - 16*a + 63 = 0

a^2 - 7*a - 9*a + 63 = 0

a( a - 7 ) - 9( a - 7 ) = 0

( a - 7 )( a - 9 )

Two possible values are 7 and 9

Similarly for b values are 9 or 7

Third and fourth observations are 7 and 9.